Abstract Convex Analysis (Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts)

Abstract Convex Analysis CANADIAN MATHEMATICAL SOCIETY SERIES OF MONOGRAPHS AND ADVANCED TEXTS Monographies et Études...

Author: Ivan Singer

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Abstract Convex Analysis

CANADIAN MATHEMATICAL SOCIETY SERIES OF MONOGRAPHS AND ADVANCED TEXTS Monographies et Études de la Société mathématique du Canada EDITORS/RÉDACTEURS: Jonathan M. Borwein and Peter B. Borwein

A complete list of titles in this series appears at the end of this volume. Tous les titres de cette collection sont énumérés à la fin du volume.

Abstract Convex Analysis IVAN SINGER Romanian Academy Bucharest, Romania

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Weinheim • Brisbane • Singapore • Toronto

This text is printed on acid-free paper. Copyright 0 1997 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY

10158-0012. Library of Congress Cataloging in Publication Data: Singer, Ivan. Abstract convex analysis / Ivan Singer. p. cm.—(Canadian Mathematical Society series of mongraphs and advanced texts) "A Wiley-Interscience publication." Includes bibliographical references (p. - ) and indexes. ISBN 0-471-16015-6 (cloth: alk. paper) 1. Convex functions. 2. Convex sets. 3. Mathematical optimization. I. Title. II. Series.

QA331.5.S53 1997 515'.8–DC20 Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

96-32000

Contents

xi

Foreword

xiii

Preface Introduction: From Convex Analysis to Abstract Convex Analysis 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Abstract Convexity of Sets Inner Approaches 0.1a 0.1b Intersectional and Separational Approaches 0.1c Approaches via Convexity Systems and Hull Operators Abstract Convexity of Functions Abstract Convexity of Elements of Complete Lattices Abstract Quasi-Convexity of Functions Dualities Abstract Conjugations Abstract Subdifferentials Some Applications of Abstract Convex Analysis to Optimization Theory 0.8a Applications to Abstract Lagrangian Duality 0.8b Applications to Abstract Surrogate Duality

Chapter One

Abstract Convexity of Elements of a Complete Lattice

1 1 2 4 6 9 12 14 14 16 19 20 20 28 34

1.1

The Main (Supremal) Approach: M-Convexity of Elements of a Complete Lattice E, Where M c E 1.2 Infimal and Supremal Generators and M-Convexity 1.3 An Equivalent Approach: Convexity Systems 1.4 Another Equivalent Approach: Convexity with Respect to a Hull Operator

Chapter Two Abstract Convexity of Subsets of a

Set

2.1 M-Convexity of Subsets of a Set X, Where M c 2' 2.2 Some Particular Cases 2.2a Convex Subsets of a Linear Space X 2.2b Closed Convex Subsets of a Locally Convex Space X V

34 40 44 47

50 50 56 56 58

Contents

vi

Evenly Convex Subsets of a Locally Convex Space X Closed Affine Subsets of a Locally Convex Space X Evenly Coaffine Subsets of a Locally Convex Space X Spherically Convex Subsets of a Metric Space X Closed Subsets of a Topological Space X Order Ideals and Order Convex Subsets of a Poset X Parametrizations of Families M c 2 x , Where M Is a Set An Equivalent Approach, via Separation by Functions: W-Convexity of Subsets of a Set X, Where W c R x A Particular Case: Closed Convex Sets Revisited Other Concepts of Convexity of Subsets of a Set X, with Respect to a Set of Functions W c RA' (W, )-Convexity of Subsets of a Set X, Where W Is a Set and ça : X x W —> R Is a Coupling Function 2.2c 2.2d 2.2e 2.2f 2.2g 2.2h 2.2i

2.3 2.4 2.5 2.6

Chapter Three Abstract Convexity of Functions

on a Set

3.1 W-Convexity of Functions on a Set X, Where W c R x 3.2 Some Particular Cases 3.2a C (X* ± R), Where X Is a Locally Convex Space C(X*), Where X Is a Locally Convex Space 3.2b 3.2c The Case Where X = {0, 1}" and W c (R n ) * I X 3.2d The Case Where X = {0, 1}" and W , (R")* i x ± R a-Holder Continuous Functions with Constant N, 3.2e Where 0 < a ( 1 and 0 < N < -i- co 3.2f Suprema of a-Wilder Continuous Functions, Where 1 0< a 3.2g The Case Where a > 1 3.3 (W, 0-Convexity of Functions on a Set X, Where W Is a Set and ço : X x W —> R Is a Coupling Function Chapter Four Abstract Quasi-Convexity of Functions

on a Set

4.1 M-Quasi-Convexity of Functions on a Set X, Where M c 2' 4.2 Some Particular Cases 4.2a Quasi-Convex Functions on a Linear Space X 4.2b Lower Semicontinuous Quasi-Convex Functions on a Locally Convex Space X 4.2c Evenly Quasi-Convex Functions on a Locally Convex Space X 4.2d Evenly Quasi-Coaffine Functions on a Locally Convex Space X 4.2e Lower Semicontinuous Functions on a Topological Space X 4.2f Nondecreasing Functions on a Poset X 4.3 An Equivalent Approach: W-Quasi-Convexity of Functions on a Set X, Where W c R x

60 61 62 63 64 64 70 72 83 83 85

92 92 104 104 108 111 114 119 121 125 127

129 129 140 140 142 143 144 145 145 146

Contents

Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X, Where W c R x 4.5 Some Particular Cases 4.5a Lower Semicontinuous Quasi-Convex Functions Revisited 4.5b Evenly Quasi-Convex Functions Revisited Evenly Quasi-Coaffine Functions Revisited 4.5c 4.6 (W, yo)-Quasi-Convexity of Functions on a Set X, Where W Is a Set and ço : X x W —> R Is a Coupling Function 4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on a Set X, with Respect to Convexity Systems B c 2x and Hull Operators u : 2x ---> 2x 4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators on R x

vii

4.4

Chapter Five Dualities Between Complete Lattices 5.1 5.2 5.3 5.4

Dualities and Infimal Generators Duals of Dualities Relations Between Dualities and M-Convex Hulls Partial Order and Lattice Operations for Dualities

Chapter Six

Dualities Between Families of Subsets

Dualities A : 2 x --> 2 w , Where X and W Are Two Sets Some Particular Cases Some Minkowski-Type Dualities 6.2a Some Dualities Obtained from the Minkowski-Type 6.2b Dualities Am, by Parametrizing the Family M 6.3 Representations of Dualities A : 2 x —> 2 w with the Aid of Subsets Q of X x W and Coupling Functions ça : X x W —> R 6.4 Some Particular Cases 6.4a Representations with the Aid of Subsets Q of X x W 6.4b Representations with the Aid of Coupling Functions ça :XxW--->R 6.1 6.2

Chapter Seven Dualities Between Sets of Functions 7.1 7.2

Dualities A : R x --> R w , Where X and W Are Two Sets Representations of Dualities A : Ax —> F, Where X Is a Set and (A, ) c (T?, ) and F Are Complete Lattices 7.3 Dualities A : A x —> B w , Where X Is a Set and (A, ..,), (B, .) c (R, .) Are Complete Lattices 7.4 Some Particular Cases 7.4a The Case Where A = {0, +00} 7.4b The Case Where A = B = [0, 1] 7.5 Strict Dualities A : A x —> B w 7.6 Dualitylike Mappings A : A "' —> B w

151 154 154 158 161 162

165 166 172 172 176 182 186 190 190 200 201 201 208 216 216 217 219 219 224 230 237 237 239 240 241

viii

Contents

Chapter Eight Conjugations 8.1 8.2

Conjugations c : R x —> R w , Where X and W Are Two Sets Representations of Conjugations c : R x —> R w with the Aid of Coupling Functions ço:XxW—> R 8.3 Biconjugates and Abstract Convex Hulls 8.4 Some Particular Cases 8.4a The Case Where X = (0, 1)", W = (R")* I x and (P = (Pnat 8.4b The Case Where X Is a Metric Space, W = X, and yo = (pa, N 8.4c The Case Where X Is a Metric Space, W = X x (R ± \ (0)), and cp = (pa Conjugate of f -.F —h, Where f, h e R x 8.5 The 8.6 Conjugations of Type Lau 8.7 Some Particular Cases Conjugations of Type Lau Associated to a Family M of 8.7a Subsets of a Set X Quasi-Conjugation 8.7b 8.7c Semiconjugation 8.7d Pseudoconjugation 8.7e Some Extensions of the Preceding Conjugations 8.8 Relations Between Conjugations c : R x —> R w and Dualities A : 2 x —> 2 w , Where X and W Are Two Sets 8.9 Some Particular Cases The Conjugation of Type Lau Associated to a 8.9a Minkowski-Type Duality 8.9b Conjugations of Type Lau Associated to Parametrized Minkowski-Type Dualities 8.10 The Conjugate of Type Lau of max{ f, —h), Where f, —h E R x 8.11 Conjugate Functions and Level Sets Chapter Nine v-Dualities and _L-Dualities 9.1 The Binary Operations I and T 9.2 v-Dualities 9.3 I-Dualities 9.4 The Duals of v-Dualities 9.5 The Duals of I-Dualities 9.6 Characterizations of Conjugations of Type Lau with the Aid of v-Dualities and I-Dualities Chapter Ten Abstract Subdifferentials 10.1 Subdifferentials with Respect to a Duality A : R x —> R w , Where X and W Are Two Sets 10.2 Subdifferentials with Respect to a Conjugation c : R x —> R w , Where X and W Are Two Sets

242 242 246 256 261 261 263 266 267 274 284 283 287 290 291 292 296 304 304 304 308 318 335 335 338 342 347 351 355 359 359 364

Contents

10.3

Some Particular Cases 10.3a The Case Where X = (0, 1)??, W = (R n ) * Ix, and ço — (Pnat 10.3b The Case Where X Is a Metric Space, W = X, and (i0 = 40«,N 10.3c The Case Where X Is a Metric Space, W = X x (R ± \ {0}), and cp = (pa, 10.4 The Subdifferential of f + —h at xo, Where f, h E Te and xo E X 10.5 Subdifferentials with Respect to Conjugations of Type Lau 10.6 Some Particular Cases 10.6a L (A)-Subdifferentials, for Minkowski-Type and Parametrized Minkowski-Type Set-Dualities A 10.6b Subdifferentials with Respect to Quasi-Conjugations; Quasi-Subdifferentials 10.6c Subdifferentials with Respect to Semiconjugations; Semisubdifferentials 10.6d Subdifferentials with Respect to Pseudoconjugations; Pseudosubdifferentials 10.7 Subdifferentials with Respect to v-Dualities and I-Dualities

ix

367 367 368 369 370 371 375 375 376 379 383 384

Notes and Remarks

387

References

461

Notation Index

475

Author Index

481

Subject Index

485

Foreword One of the principal methods used in mathematics to represent complex objects involves the application of certain operations to finite or especially infinite sets of simpler objects which can be used to essentially approximate the complex objects. Classical examples of such methods include, among many others, infinite series representations of functions in mathematical analysis and series expansions with respect to a Schauder basis in the study of separable Banach spaces in functional analysis. The author of this monograph is a prominent expert in the study of Schauder bases, but the present book is devoted to a different kind of application of the approach described above, namely to the representation of complex objects through the operation of taking suprema (or infima) of families of elements in an ordered space. In the late 1960s and early 1970s it was realized that it is possible, and relatively straightforward, to obtain many of the principal results of convex duality theory using the representation of a convex function as the pointwise supremum of the set of its affine minorants. Moreover, many of these results do not depend on the linear structure of the class of minorizing functions. The corresponding observation for closed convex sets was made even earlier, that is, many results for these sets easily follow from their "outer" representation as intersections of closed half-spaces. Also, many of these results can be generalized to sets which can be represented as the intersections of other families of sets that are not necessarily half-spaces. Such observations stimulated the development of what has become known as "abstract convex analysis." It is very interesting, from a purely mathematical standpoint, and important from the perspective of applications, to understand exactly which properties of convex sets and functions do not depend on the linear structure of the underlying space. Little more than twenty five years ago it was discovered that several basic results in functional analysis in ordered spaces, and even in classical mathematical analysis (for example, properties of Chebyshev systems), can be studied in a natural way within the framework of abstract convexity. Later many new applications in analysis, algebra and other fields were found. However, the further development of abstract convex analysis, to which the author of this monograph has made valuable and deep contributions (especially concerning conjugations and dualities) was mainly driven by applications to optimization, utilizing various types of abstract duality constructions and subdifferentials. The present book contains an exhaustive survey of the state-of-the-art in contemporary abstract convex analysis. In this book the reader can find a presentation of several approaches to abstract convexity and comparisons between them. Also presented are results concerning abstract convex sets and functions, abstract quasiconvex functions xi

xii

Foreword

and related notions of subdifferentiability. Various types of abstract duality and conjugations are discussed in detail. The book also contains some new contributions by the author which have not previously been published. Many examples are given that help the reader to better understand the main concepts and techniques being developed. The author mentions some applications of abstract convexity to optimization theory which will hopefully be the focus of a future contribution. As a researcher in this field over many years I am convinced that this book will be an extremely useful reference both for new and for established researchers working in areas such as convexity (not only abstract convexity), the theory of ordered spaces, and optimization theory. It will also be a fundamental reference for those who want to learn the essence of abstract convex analysis. The book presents powerful tools that can be exploited in the investigation of many problems which arise in a diverse range of interesting applications. A. M. RUBINOV

Preface

Convex optimization is an important field of application of usual "convex analysis," (that is, of the theory of convex sets and convex functions, conjugate functions, subdifferentials, and the like, in locally convex spaces). Indeed, expositions of convex analysis, together with applications to convex optimization, can be found in numerous books, for example, those of Rockafellar ([235], [236]), Stoer and Witzgall [281], Laurent [156], Holmes [120], Ioffe and Tikhomirov [124], Ekeland and Temam [68], Barbu and Precupanu [21], Bazaraa and Shetty [22], Wets [308], Elster, Reinhardt, Schaible and Donath [75], Zeidler ([316 1, [317]), Aubin [12], Pshenichnyi [231], Tikhomirov [287], Ekeland and Turnbull [69], Aubin and Ekeland [13], Walk [304], and, more recently, in Hiriart-Un-uty and Lemaréchal [118] (although, let us mention that there are also books on convex analysis, not directly concerned with applications to optimization, such as those of Moreau [205], Castaing and Valadier [35], Giles [98], Levin [1581). At a certain stage of development, there has appeared the need of a more general theory of "nonconvex optimization", such as, of minimizing an arbitrary (extended real-valued) function on an arbitrary set without assuming any one of the usual (topological or algebraical) structures. For example, in his book of 1976, Avriel wrote (1171, p. 106): "Little can be said about duality in general nonconvex programming, for this subject is at about the same stage as convex duality was in the early 1950's. A few works on the subject exist, such as . . . , but results are not very satisfactory . . . ." Later, Jefferson and Scott ([134], p. 519) wrote: "An essential concept in the analysis of mathematical programs is the idea of duality. This has furnished new approaches and interpretational insights and has provided the basis for many powerful algorithms. To date, most of this theory has concentrated on convex mathematical programs, whereas the vast spectrum of nonconvex programs has remained relatively untouched . . . ." The problem of constructing a theory of nonconvex optimization, based on suitable extensions of the methods of (usual) convex analysis, has been attacked by several authors, independently at about the same time, such as Dolecki and Kurcyusz ([62], [631), Lindberg [159], and Balder [19]. For example, Dolecki and Kurcyusz [63] wrote: "But only quite recently has it become evident . . . that for much more general extremal problems and Lagrangians the duality relations are expressible in terms very similar to those used in convex analysis." Balder [19] wrote: "By an effective *This term, which means, in fact, "not necessarily convex" optimization, although widely adopted, is somewhat of a misnomer, since convex optimization is also a special case of "nonconvex optimization."

xiv

Preface

extension of the conjugate function concept a general framework for duality-stability relations in nonconvex optimization can be studied . . . ." Some of the concepts and results generalizing those of convex analysis to arbitrary sets and functions, had existed before (and had been used in) the above-mentioned papers, such as the concepts of "convex subsets" of an arbitrary set (Danzer, Griinbaum, and Klee [45]; Fan [82]), of the "conjugate" of a function associated to an arbitrary (not necessarily bilinear) coupling function (Moreau [205] and others), and some were introduced in those papers, such as the concept of a "convex function" on an arbitrary set (Dolecki and Kurcyusz [62], [63]), and of the "subdifferential" of a function on a set, at a point of the set, with respect to an arbitrary coupling function (Balder [19], etc.). These generalized concepts and results were applied in [62], [63], [159], [19], etc., to nonconvex optimization (e.g., to develop a unified theory of "augmented Lagrangians"), using the observation that many classes of sets and functions are in fact (e.g., by a suitable choice of the "coupling function" mentioned above) some sort of "abstract convex" sets or functions, respectively. In a series of papers by several authors, further contributions have been made in these directions. In this monograph we present this new theory of "abstract convex analysis" which consists in extensions of the concepts and results of (usual) convex analysis to arbitrary sets and functions. Also we give many examples of "abstract" convex sets, functions, conjugations, subdifferentials, and the like, and of applications of the general theory to these particular cases. Furthermore we show the applicability of various parts of abstract convex analysis to some topics in nonconvex optimization (let us mention that our subsequent monograph, in preparation, on duality in optimization theory, is based on a systematic application of abstract convex analysis). Most of the concepts and results presented here can be found in articles scattered in journals and proceedings volumes, but we have also included some new results (e.g., on parametrizations of Minkowski-type dualities and of conjugations of type Lau). Up to the present writing, there exists no book on abstract convex analysis. The only book near to this subject is that of Soltan, Introduction to the Axiomatic Theory of Convexity [279], but its intersection with the second part of our present monograph (Chapters 5-10) is empty, and its coverage relative to our first part (Chapters 1-4) is very small (partly, because [279] was written with a different focus, not having applicability to nonconvex optimization as its aim, and partly because most of the results presented here are more recent). The recent book of van de Vel, Theory of Convex Structures [293], is near to (but mostly disjoint from) our Chapter 2. Also, after the present monograph has been completed, we received the preliminary version of the forthcoming book by Pallaschke and Rolewicz on optimization in metric spaces and in Banach spaces [211], that has (in its §§1.1, 1.2, and 1.4) a very small intersection with our Sections 2.3, 3.1, 3.2, 6.1, 6.3, 8.3, and 10.2. In order to limit the size of this monograph, we present here the theory of abstract convex analysis only in the framework of arbitrary sets X, possibly endowed with a given (arbitrary) family .A4 of subsets M c X or a given (arbitrary) set W of functions w: X W. The cases where some usual structures (topological, algebraical, etc.) are assumed are mentioned as particular cases of the general theory, either in the text or (often) in the Notes and Remarks section at the end of the book, where we refer to existing books or papers about these subjects. Let us outline, briefly, chapter by chapter the contents of this book.

Preface

XV

In the Introduction we explain some of the main concepts of abstract convex analysis by showing how they arise, in a natural way, from those of (usual) convex analysis. Let us note that in the other chapters we proceed in the opposite way, namely, we give first the definitions of the concepts of abstract convex analysis and results on them, and after these we give examples of various "particular cases" and some results in these particular cases. To save space (to avoid repetitions), these examples are not given immediately after the definition of each concept but only later, after some related definitions and results, in order to treat them simultaneously (e.g., the particular cases of M-convex sets are given together with those of C(M)-semispaces for the same M, etc.). In the last section of the Introduction we present some examples of applications of abstract convex analysis to optimization theory. First, we show how abstract convex analysis leads to a general theory of perturbational Lagrangian duality, for the primal problem of infimization of an arbitrary function on an arbitrary set, encompassing as particular cases various results about the so-called modified Lagrangians. Next we show how the tools of abstract convex analysis yield some characterizations of general "surrogate dual problems" among general dual problems, regarding the dual objective function as a function of the dual variable and of the primal parameters. Various applications of abstract convex analysis to optimization theory are also mentioned in other chapters and in the Notes and Remarks. In Chapter 1 we study abstract convexity in the framework of arbitrary complete since many concepts and results of Chapters 2 and 3 are deduced, lattices E = (E, in a unified way, by applying those of Chapter 1 to the complete lattice E = ( 2X, D) of all subsets of a set X, ordered by containment (i.e., for G1, G2 E 2, GI G2 of if and only if G1 D G2) and, respectively, to the complete lattice E = (Rx , all functions on a set X, with values in R = [—oc, +Do], endowed with the usual f2(x) for all pointwise order (i.e., for fi, f2 E R X fi f2 if and only if fi (x) X E X). The basic concepts of a convex element x of a complete lattice E with respect to a given subset M of E, and of an infimal generator (actually, the dual concept of a supremal generator) of E are due to Kutateladze and Rubinov (see 11511, [152] and the references therein; see also Dolecki and Kurcyusz [62], [63 1). In Chapter 1 we present the theory of M-convexity of elements of E, using mainly infimal generators of E. We show in Chapter 1 two further equivalent approaches to abstract convexity in a complete lattice E, namely those of convexity systems (called also "dual cyrtologies." by Dolecki and Greco [60]) and of hull operators (or, in the terminology of Birkhoff [27], "closure operators"). One of the main questions in abstract convex analysis is the following: Given an arbitrary set X, which subsets of X should be called "convex"? In Chapter 2 we present several equivalent approaches to this problem, two of which are particularly useful for applications to optimization theory: convexity of a subset of X with respect to a given family M of subsets of X (Danzer, Griinbaum, and Klee [451) and with respect to a given set W of functions on X (Fan [821); both are "outer" approaches, requiring that for each outside point there should exist a member of M, respectively, of W, separating the set from the point, in some sense. We show the equivalence of these two approaches, with the aid of suitable families of extended real-valued functions. Two other (historically older) equivalent approaches, mentioned in Chapter 2, are axiomatical, introducing convex subsets of X as the members of a given ,

xvi

Preface

intersectionally closed family 8 of subsets of X, respectively, as the subsets of X that coincide with their "hull" for a given hull operator (i.e., a "Moore closure operator" [202], [27]) u on 2x , the family of all subsets of X. These are generalizations of the closed subsets of a topological space X obtained by discarding the axiom that the union of any two sets in B should belong to B and, respectively, the axiom that u(G i U G2) = u (G 1 ) U u (G2) for all GI, G2 C X. Finally, another useful equivalent approach, considered in Chapter 2, is that of convexity of a subset of X with respect to a pair (W, yo), where W is another set and yo : X x W --> R is a function (called a "coupling function"). It turns out that many classes of subsets of X (the convex subsets of a linear space X, the closed convex subsets of a locally convex space X, the closed linear subspaces thereof, the closed subsets of a topological space X, the order ideals of a poset X, the order convex subsets of a poset X, etc.) are classes of abstract convex sets in the above senses. The infimal generator of the complete lattice E = (2X, D), where X is a set, used in Chapter 2, is simply the family of all singletons {x }. where x e X. Another main question in abstract convex analysis is the following: Given an arbitrary set X, which (extended real-valued) functions on X should be called "convex"? In Chapter 3 we present the concept of convexity of a function on X with respect to a given set W of functions on X (Dolecki and Kurcyusz [62], [631; see also [151], [1521), which has useful applications to optimization theory (e.g., to optimal value functions and duality-stability relations); this is the supremal approach, requiring that the function should be the (pointwise) supremum of some subset of the given set W. Another useful equivalent approach, considered in Chapter 3, is that of convexity of a function on X, with respect to a pair (W, ço), where W is a set and ço:XxW-- > R is a coupling function. As mentioned above, it turns out that many classes of functions on X are classes of abstract convex functions. The particular case where X is a finite set, i.e., the problem of abstract convexity in discrete structures, is also considered in Chapter 3; however, the problem of finding the "appropriate" convexity concepts in such structures seems to be still open. The main infimal generator of the complete lattice E = (Rx , where X is a set, used in Chapter 3, is the family of all functions on X of the form x{ x } ± d, with x{x } denoting the indicator function of the singleton {x}, where x E X and d E R. Similarly to usual convex analysis, where the first main step in passing from convex functions to more general classes of functions is the study of quasi-convex functions, in Chapter 4 we study abstract quasi-convex functions f on a set X, defined by requiring that all (sub)level sets of f should be convex, for various concepts of abstract convex subsets of X introduced in Chapter 2. We show that these functions are characterized by the fact that they are the suprema of suitable sets of functions of the form —x G + d (with x G denoting the indicator function of G), where G c X and d E R. Also, it turns out that many classes of functions on X are actually classes of abstract quasiconvex functions, and hence, for example, our formulas on the abstract quasi-convex hull of a function encompass, as particular cases, various known formulae for the usual quasi-convex hull, the lower semi-continuous quasi-convex hull, the evenly quasi-convex hull, and other hulls. In Chapter 5, returning again to the general framework of arbitrary complete lattices, we study dualities between two complete lattices E = (E,) and F = (F, that is, mappings t : E ---> F such that A (inf, xi) = supi , / A (x i ) for all

Preface

xvii

{x i } iel c E and all index sets I, including the empty set I, since many results of Chapters 6 and 7 are deduced by applying those of Chapter 5 to the pairs of complete lattices E = (2X, D), F = (2 W , D) and E = (Ax, F (B w , C (R, are two respectively, where X and W are two sets and (A, (B, complete lattices. We study, among several problems, the expression of an arbitrary hull operator u : E ---> E as a second dual, that is, its decomposition in the form u = A' A where A : E --> F is a duality and A' : F E is its dual, our main tools being the infimal generators of E and F. Dualities between families of subsets of two sets X and W, and dualities between sets of functions on two sets X and W, are of importance for duality in general optimization theory, where the primal problem is to minimize a function f over a subset G of a set X and the dual problem is to maximize (or minimize) a function g over a subset P of a set W (with P Ç W and g : W ---> R related to G c X and f : X --> R). Chapter 6 is devoted to dualities (called also "polarities," e.g., in [271) between families of subsets of two sets X and W. We study representations (i.e., expressions of the general form) of dualities A : (2 x , D) ---> (2 w , D) where X and W are two sets, with the aid of subsets Q of X x W and of coupling functions go : X x W ---> T?, Minkowski-type dualities and their parametrizations, equivalence of dualities, decompositions of hull operators u : (2x, D) ---> (2x , D) in the form A'A, where A : (2x , D) --> (2 w , D) is a duality, with dual A', and so on. In Chapter 7 we give some results on dualities between the function spaces (A x , (B, C (R, are two where X and W are two sets and (A, and (B w , complete lattices, the main emphasis being on the representations of dualities A : with the aid of functions e:Xx WxA --> B. We also give ---> (B w , (A x , a generalized Fenchel-Young inequality. for any duality A : (A x , ---> (B w , Most of the applications concern the particular case A = B = R, but we also mention some applications for A = ( 0, +oc) and A = B = [0, 11. Chapter 8 is devoted to the special class of dualities between the function spaces (— Rx , and (— Rw, (where X and W are two sets), called "conjugations," defined by a condition on the conjugate of the sum of a function and a constant function. We show that conjugations coincide with the usual Fenchel-Moreau conjugations associated to arbitrary coupling functions yo : X x W ---> R and that the dual of a conjugation is again a conjugation. Also we present some results on the relations between biconjugates and abstract convex hulls of functions and on the conjugate of the difference of two functions. Furthermore we study a special class of conjugations, called "conjugations of type Lau," that plays among conjugations a similar role to the one played by the Minkowski-type dualities among dualities between subsets of X and W and that contains, as particular cases, the quasi-conjugations of GreenbergPierskalla–Crouzeix ([1051, [41], [43]), semi-conjugations, pseudoconjugations, and other useful conjugations. Next we give some results on relations between conjugations c : --f?x ---> R w and dualities A : 2 x —> 2 w , and on the conjugate of type Lau of the maximum of two functions. Finally, we show that the conjugates and biconjugates of a function f : X ---> R can be expressed with the aid of the (sub)level sets of f and we compute the (sub)level sets of conjugate and biconjugate functions. In Chapter 9 we consider two further special classes of dualities between the function spaces (R x , and (R w,), where X and W are two sets. The dualities in the first class, called "v-dualities," are defined by a condition on the dual of the

Preface

xviii

maximum of a function and a constant function. To determine the dual of a v-duality, we introduce two new binary operations on R, denoted I and T, respectively, and then, extending them (pointwise) to function spaces, we introduce the second class of dualities studied in Chapter 9, called "I-dualities," defined by a condition on the dual of f Id, where f E R X and d is a constant function on X. We show that vdualities and I-dualities are dual to each other and that the conjugations of type Lau are simultaneously conjugations, v-dualities, and I-dualities, and conversely, they are the only dualities having any two (and hence all three) of these properties. Chapter 10 is devoted to abstract subdifferentials. For any duality A : (R x , ,) ----> (R w , ..), where X and W are two sets, the subdifferential f (x 0 ) of a function f E R X at a point xo E X. with respect to A is defined with the aid of the function e' : Wx X x R ---> R representing (by Chapter 7) the dual A' of the duality A, namely by requiring equality in the generalized Fenchel-Young inequality. We show that this concept generalizes the usual subdifferential and preserves some of its main properties. In fact, when A = c(go), the Fenchel-Moreau conjugation associated to a coupling function go : X x W ---> R and when f(x0) E R, a ° f (x o ) becomes the subdifferential of f at x0, with respect to go, in the sense of Balder [19]. Also we compute f (x0) for conjugations of type Lau and, in particular, for quasi-conjugations, pseudoconjugations, and semiconjugations, the latter ones being closely related to the "quasi-subdifferentials" of Greenberg—Pierskalla [105] and Zabotin—KorablevKhabibullin [315], the semisubdifferentials of [260] and the pseudosubdifferentials of [257], respectively. We also give the expressions of the subdifferentials with respect to v-dualities and I-dualities. Finally, in the Notes and Remarks we give some bibliographical references and additional results, some of them with complete proofs, to each of the chapters; the numberings in the Notes and Remarks continue the numberings of the respective chapters. The bibliography refers mainly to the concepts and results of abstract convex analysis, presented here, without tracing completely the history of their roots (the latter can be found in the literature on usual convex analysis, in the books of Moreau [205], Rockafellar [235], Hiriart-Urruty and Lemaréchal [118], etc.). We hope that this book will be useful to a large circle of readers, including, besides researchers in the field, those who want to apply abstract convex analysis (e.g., to nonconvex optimization theory) and those who want to learn it (e.g., graduate students). To this end, we have exemplified the main concepts by various particular cases, and we have given detailed proofs of the results. Also for each result of functional analysis used in the examples, we have given a reference to a treatise or to an article containing a proof. We assume that the reader has some knowledge of (usual) convex analysis, since it constitutes one of the most important particular cases of the general theory. Also the knowledge of some elements of functional analysis, lattice theory, etc. will be useful (but not indispensable, since otherwise the reader may simply consider, instead of locally convex spaces X, only X = R" and may skip Chapters 1 and 5 involving complete lattices, at the price of having to prove with more direct methods some of the results of the other chapters; however, in the latter case, the same type of arguments would be repeated separately for subsets of X and for functions on X). We wish to express our gratitude to several colleagues and friends for many stimulating discussions on usual convex analysis and generalized convex analysis. Thus, we thank P.-J. Laurent, J.-J. Moreau, and R. T. Rockafellar for raising our inter-

a°

aA

Preface

xix

est in usual convex analysis. We are grateful to the late G. Godini for her seminar lectures on the book of Holmes [120], and to S. Dolecki, J.-P. Crouzeix, V. Klee, H. J. Greenberg, the late P. C. Hammer, and others for discussions about their pioneering papers on abstract convex analysis at the time of their publication. We also extend our thanks to the participants of our talks on some parts of this book, at various universities and conferences, for their interest and comments. Our special thanks are due to J.-E. Martinez-Legaz for the pleasure of a long-standing cooperation and for the excitement of discovering jointly some new results on abstract convex analysis. Finally, we wish to express our deep gratitude to J. M. Borwein for his continual support in the final phase of the writing of this book. Our thanks are also due to A. M. Rubinov, one of the founders of abstract convex analysis, for his generous Foreword. We are most grateful to the referees for valuable suggestions and improvements. We also thank warmly G. Arsene for his help in various technical problems, V. Dundreanu for the careful typing of the text in TEX, and our editors at John Wiley & Sons, Inc., for all their assistance with this book. IVAN SINGER

Bucharest, Romania April 1996

Abstract Convex Analysis

Introduction: From Convex Analysis to Abstract Convex Analysis The term "abstract convex analysis" is a combination of "abstract convexity" and "convex analysis." Roughly speaking, it means an extension of the usual convex analysis to the case of abstract convexity. This preliminary chapter explains some of the main concepts of abstract convex analysis. It shows how these concepts arise, in a natural way, from those of (usual) convex analysis. The chapter also gives some examples of applications of abstract convex analysis to optimization theory. At this point we need to emphasize that although the term "convex" occurs frequently in both (usual) convex analysis and abstract convex analysis, there is no danger of confusion because the choice of terminology is different in the latter case, where we have "order convex," "metrically convex," "convex with respect to a family of subsets," "M-convex," and so on.

0.1

Abstract Convexity of Sets

Let us begin by explaining the sense in which we use the terms "generalized convexity," "axiomatic convexity," and "abstract convexity" of sets. Our remarks here will also be valid for the case of functions. A concept of generalized convexity of sets is obtained by selecting an important property of the usual (i.e., classical, traditional, standard, ordinary) convex sets in the linear space R", or in R" endowed with some of its usual structures, and taking it as a definition, in a possibly more general framework. In other words, one selects a certain property that usual convex sets in Rti have, but many other objects in possibly other settings also have, and one uses that property to define a "generalized" sort of convexity." It should be mentioned that many authors use the term "generalized convexity" in a more restrictive sense; namely they allow only the framework of the linear space

2

Introduction: From Convex Analysis to Abstract Convex Analysis

(or, at most, of an arbitrary linear space X). An example is the concept of "quasiconvex" sets (see (0.6) below). However, in the less restrictive sense noted above, we allow more general settings, even that of an arbitrary set X. Then there are some authors (e.g., Danzer, Grtinbaum, and Klee [45], §9) who use the term "generalized convexity" in a much wider sense which includes the generalization of a variant (or an analog) of a property of usual convex sets in R". We will not use this wider sense of "generalized convexity", because if one "generalizes" a concept, then it is natural to expect that the initial concept sould be a particular case of the general concept. For example, "order convexity" is a "generalized convexity" in the wider sense of [45], but it is not in the sense used here, since usual convexity in R" is not a particular case of order convexity in R" for n 2 (see Section 0.1a below). However, "generalized convexity" in the wider sense of [45] (hence, in particular, order convexity) is an abstract sort of convexity. An axiomatic convexity theory is obtained by selecting some important properties of the family of all usual convex sets in the linear space R", or R" endowed with the Euclidean norm, and using them, possibly in a more general framework, as a set of axioms to define the family of all (abstract) "convex" subsets of a set or a subfamily thereof. Alternatively, one can introduce a set of axioms to define the family of all (abstract) "segments" in some settings (e.g., in an arbitrary set X) and then, using these segments, define a concept of "convex sets" (see Section 0.1a). In the sequel we will consider abstract convexity of sets (this term has been used, e.g., in the comprehensive paper of Jamison-Waldner [133]), which includes both generalized convexity (in the sense described above and in the wider, more permissive sense of [45]) and axiomatic convexity.

Inner Approaches

0.1a

Recall that if X = R" , or more generally, if X is a (real) linear space, and x, y E X, then, by the usual definition, the segment (the closed straight line segment) joining x and y (or with "end points" x and y) is the set (x, y) = (Xx + (1 — X)y E X I 0

X

1).

(0.1)

A set G C X is said to be convex, if it contains the whole segment joining each of its two points; that is, if we have the implication x, y

E

G

(x, y) c G.

(0.2)

By defining other concepts of "segment" (x, y) and requiring a condition of type (0.2), one arrives at various concepts of abstract "convexity," in which the underlying set X may be a linear space, or a set endowed with some structure(s), or just an arbitrary set. For example, if X = (X, is a partially ordered set (briefly, a poset) and x, y E X are such that x y, one defines the order segment (x, y) by

(x,y)=1.zeXixzy)

(0.3)

(the condition x y is required to ensure that (x, y) 0 0 and x, y E (x, y)). Thus the order segments are defined only for some pairs of elements x, y E X, namely

0.1 Abstract Convexity of Sets

3

y. A set G C X satisfying (0.2) for all x, y E G with x 2x defined by

u(G) = com G =

n

(G

C

X)

(0.20)

MEM GCM

is a hull operator such that a set G C X is M-convex if and only if it is u-convex. Note that, in general, com G V M. Also, if X is a set and W is any set of functions on X, then the set B = 1C( W) of all W-convex subsets of X is a convexity system, and the mapping u = co w : 2 x --> 2 x defined by

u(G) = co w G = Ix G X I w(x) sup w(G) (w

W)}

X) (0.21) is a hull operator such that a set G C X is W-convex if and only if it is u-convex. Let us mention that in the theory of functions of several complex variables, for a subset G of a domain Q in C" (the space of n complex variables) and a set W of functions that are holomorphic in Q, the "W-convex hull" of G. defined by aW = -

E

E

X I iw(x)i(sup lw(G)I (w

E

(G

W)},

C

(0.22)

is used to define the "W-convexity" of Q by the condition that the W-convex hull of each compact subset of Q should be compact. In particular, when W is the set

0.2 Abstract Convexity of Functions

9

of all functions that are holomorphic in Q, one obtains the notion of "holomorphic convexity" of Q (e.g., see Fuks [93]). The W-convex hull (0.21) with W = P (Q), the set of all functions that are plurisubharmonic in Q, is also used in the theory of complex convexity, to define "pseudoconvexity" of open sets Q c C" (e.g., see Hiirmander [122], [123]). However, in the present book we will not consider complex convexity of sets and functions. The approach to abstract convexity of sets via hull operators is equivalent to the approach via convexity systems. Indeed, it is not hard to show (see Chapters 1 and 2) that for any set X, if u : 2x —> 2x is a hull operator, then the family B = C(u) of all u-convex subsets of X is a convexity system such that a set G c X is u-convex if and only if it is B-convex, and conversely, if B is a convexity system, then the mapping u = coB : 2x —> 2x defined by u(G) = coB G =

n B (E B)

(G c X)

(0.23)

BEB

GCB

is a hull operator such that a set G c X is B-convex if and only if it is u-convex. Thus both approaches encompass the same particular cases. For example, the closed subsets of a topological space X are those that are "convex" with respect to the usual closure operator u(G) = G (the closure of G in X). Similarly to the above remark on the connection between the axiom (0.14) for a convexity system and the axioms (0.14), (0.15) for the family of closed subsets of a topological space, if X is a set and u : 2x —> 2 x is a hull operator satisfying also the axiom u(G 1 U G2) = u(G 1 ) U u(G 2 )

(G 1 ,

G2 C X),

(0.24)

then X is a topological space and u is the usual closure operator u(G) = G. (In fact (0.17)—(0.19) and (0.24) are the classical Kuratowski axioms defining a topological closure operator.) Historically the theory of abstract hull operators (0.17)—(0.19) was first developed as a generalization of the axioms (0.17)—(0.19) and (0.24) of topological closure operators and only later have been found interactions with the theory of convex hull operators in linear spaces (applications of results on abstract hull operators to usual convex hulls and, conversely, generalizations of the theory of convex hull operators in linear spaces to abstract hull operators).

0.2

Abstract Convexity of Functions

We recall that if X = R", or, more generally, if X is a (real) linear space, a function f —oc, -koo] is said to be convex, if

.f (xx + (1 — X)y)

Xf (x)

(1 — X) f (y)

(x, yEX,OX

1), (0.25)

where -i-- is the "upper addition" on R [205], that is, the extension of the usual addition + by the convention +00

(-00) =

(+00 = +00.

(0.26)

10


Since (0.25) is a condition on the values of f at the points of the segment (x, y) (of (0.1)), it is natural that one can use some abstract concepts of segments defined with the aid of a scalar X (see Section 0.1a) to define corresponding segmental concepts of "convexity" of functions, replacing the left hand side of (0.25) by the corresponding points of the abstract segments, for example, requiring that

f (Ç0x,y( )0)

)4' (x)

(1 - x)y

(x, y G X, 0

A ‹, 1),

(0.27)

with (Px y of (0.7). Also, one can define "segmental" concepts of order convex functions and metrically convex functions. However, similarly to the case of sets, in this book we will only study some nonsegmental approaches to abstract convexity of functions, which are more relevant for applications to "nonconvex" optimization theory. Let us now describe them briefly. It is well-known that a function f defined on R" (or, more generally, on any locally convex space X), with values in the extended real line R = [—oc, +oo], is either ±oo or convex, lower semicontinuous, and proper (i.e., f +oo and f (x) > —oc for all x) if and only if it is the supremum (pointwise) of a set of continuous affine functions (e.g., see Moreau [205], p. 28, or Ekeland and Temam [68], ch. 1, prop. 3.1). This fact, together with some observations of Kutateladze and Rubinov [151], [152], have led Dolecki and Kurcyusz ([62], [631) to introduce and study the following abstract sort of "convexity" of functions: Let X be a set and let W be a set of extended real-valued functions on X; that is, W c R X (actually in [62], [63] it has been assumed that the functions in W are finite-valued and that for each w E W and d e R we have w ± d e W, but these restrictions can be discarded). A function f : X —> R is called convex with respect to W (or, briefly, W-convex), if there exists a subset W' of W such that f = sup w weir

(0.28)

(pointwise; i.e., f (x) = sup„) , w , w(x) for each x e X). Note that, by the usual convention sup 0 = —oc (the constant function f (x) = —oc for all x e X), condition (0.28) yields that —oc is convex with respect to W. It is obvious (by taking W' to be a singleton {w}) that every w E W is W-convex. In the case where X is a compact topological space, W is a cone of continuous functions on X, and f is continuous, the above concept of a W-convex function is due to Kutateladze and Rubinov [151], [152]. By the discussion above, the functions f -Foe, f —oc and the lower semicontinuous proper convex functions f : R" —> R are nothing else than the W-convex functions, where W is the set of all affine functions on R". It is natural to ask whether the concept of W-convex functions on a set X is also a generalization of the usual convexity of functions on R", that is, whether there exists a nontrivial set W of functions w : R" R such that each convex function f : R" —> satisfies (0.28) for some subset W' of W (depending on f). For finite-valued convex functions f : R" —> R, the answer is affirmative, since it is well-known that every such function is continuous on R " , and thus, by the above discussion, one can take W to be the set of all affine functions on R". Recently it has been shown (see the Notes and Remarks to Section 3.2) that the answer is affirmative for the convex functions

0.2 Abstract Convexity of Functions

11

f : Rn —> R too, with a suitable set W containing all affine functions; the proof uses the "lexicographical separation" (2.41) of convex sets from outside points. In fact W-convexity is a far-reaching generalization of the usual convexity of functions, encompassing, as particular cases, many different concepts (see Chapter 3). As in the case of abstract convex sets, one can also give an equivalent "separational" approach (in a certain sense) to abstract convexity of functions (see Remark 3.3(a)). There also exist axiomatic approaches to abstract convexity of functions, corresponding to the axiomatic approaches to abstract convexity of sets, mentioned in Section 0.1c. Indeed, let X be a set. Then a set .F of functions on X is called a convexity system if it is closed for sup, that is, if for any index set / we have the implication E F (i G I) = sup fi

ier

(0.29)

E F,

and the functions f G F are called convex with respect to T, or T-convex. In particular, the set F of all (usual) convex functions on Rn, and many other sets of functions, are convexity systems. On the other hand, observe that the (usual) convex —> R are those that coincide with their convex hull fc„, defined functions f : (for every function f : R" —> R) by

L ni

f(x) =- inf /

rn

Xi E

X, Jk. ;

0 (i = 1, . . . , m),

E= 1,

j=1

= x, 1 ., in < -Fool

(x E R a ),

(0.30)

i=1

f we have (1) fe0 and that for any functions f, f : R" —> R with f fc0, (2) fc0 f, and (3) (f))co -= fa, Therefore, if X is any set, one can define an abstract sort of hull operator for functions as a mapping v of R x (the set of all functions f : X —> R) into itself, satisfying for any f, f E — R X the axioms (where instead of v(f ) we use the notation fv )

(0.31)

f

(0.32)

f,

(0.33)

(fu)v = f,

and an abstract sort of "convexity" by calling a function f : X —> R convex with respect to y, or v-convex, if f = f„. Then, by the above discussion, the usual convex functions f : R" —> R are nothing else than the v -convex functions, where y is the usual convex hull operator (0.30). Moreover, if X is a set and W is any set of functions on X, then the set .F = C(W) of all W-convex functions on X is a convexity system, and the mapping y = co(W) defined by ft) = fc0(w) = sup f EW wf

(f E R x )

(0.34)

12


is a hull operator for functions such that a function f : X —> R is W-convex if and only if it is v-convex. However, in general, fco(w) V W. The approach to abstract convexity of functions via hull operators is equivalent to the approach via convexity systems. Indeed, one can show (see Chapters 1 and 3) that, for any set X, if y : R x —> R x is a hull operator, then the set .F of all v-convex functions on X is a convexity system such that a function f : X ---> R is v-convex if and only if it is .T-convex, and conversely, if F is a convexity system of functions on X, then the mapping y = co(F) : R x R x defined by (f E R x )

= fc0(y) = max w (E .F)

(0.35)

f

is a hull operator for functions such that a function f : X —> R is F-convex if and only if it is v-convex. Let us also note that for the case of functions on a set X, the condition "corresponding to" (0.15) is

f2 E F = inf

(0.36)

,(f f2) c F

(which is satisfied, e.g., by the set of all continuous functions on R" but not by the convexity system of all convex functions on R"), and the condition "corresponding to" (0.24) is [inf (f, h)1,= inf (f,„ h„)

(f, h

E

R x ).

(0.37)

As in the case of subsets (see the end of Section 0.1b), one can extend the concept of W-convexity of functions f : X —> R, where W c R x , to that of (W, (p)-convexity, where W is any set and ço : X x W --> R is any coupling function, replacing w(x) by ga(x, w). Again it turns out that this extenion can be reduced to the particular case of V-convexity, where V c R x , by taking V of (0.13).

0.3 Abstract Convexity of Elements

of Complete Lattices

For simplicity of the presentation, in this section and in Section 0.5 we will use a slightly different notation for the elements of a complete lattice E than in the subsequent chapters. One can observe that the approaches of Section 0.2 to abstract convexity of functions are, in a certain sense, "parallel" to the approaches of Sections 0.1b and 0.1c to abstract convexity of sets. For example, the "supremal" approach (0.28) for functions "corresponds to" the intersectional approach (0.9) for sets, "replacing" M' c M and n mEm, by W' c W and sup w , w„ respectively, and similar remarks can be made for the other approaches. The reason for this parallelism lies deeper; namely there exists a general approach to abstract convexity of sets and functions that unifies these analogies. Indeed Kutateladze and Rubinov [151], [152] have introduced the following abstract sort of "convexity" of elements of a complete lattice: Let E = (E,

0.3 Abstract Convexity of Elements of Complete Lattices

13

be a complete lattice, and let M be a subset of E. An element e G E is said to be convex with respect to M (or, briefly, M-convex) if there exists a subset M' of M such that e = sup M'.

(0.38)

Note that, by the usual convention sup 0 = —oo, the least element of E, condition (0.38) yields that —oc is convex with respect to M. Also, it is obvious (by taking M' to be a singleton {m}), that every ni G »1 is M-convex. If X is a set, then, applying the above definition to the complete lattice (E, = (2x , D) of all subsets of X, ordered by containment (i.e., for G I , G2 C X, G2), and observing that supremum in this G2 in E if and only if G1 G1 complete lattice is nothing else than the (usual) intersection of sets, we obtain the definition (0.9) of convexity of subsets G of X with respect to a given family M of subsets of X. On the other hand, applying the above definition (0.38) to the complete of all functions f : X —> R, ordered pointwise (i.e., for lattice (E, = (Rx , f2(x) for all x G X), we , f2 X —> T? , ( f2 in E if and only if fi (x) obtain the definition (0.28) of convexity of functions f : X —> R with respect to a given set W of functions w : X —> R. Thus the concept of M-convex elements of a complete lattice E is a generalization both of the M-convex subsets of a set X and of the W-convex functions on a set X. This generalization leads to a fruitful interaction between the general case of complete lattices E and the particular cases of subsets of a set X and functions on a set X. Indeed, on the one hand, by extending the concepts and results from these particular cases, one can study abstract convexity of elements in arbitrary complete lattices. For example, the "separational" approaches mentioned in Sections 0.1b and 0.2 can be generalized to a "separational" approach to convexity of elements of a complete lattice E, equivalent to (0.38), using "infimal generators" of E (see Chapter 1). Also, since the intersection of a family of subsets of a set X and the (pointwise) supremum of a set of functions on a set X can be extended to the operation sup in a complete lattice E, one can define a convexity system B in E as a subset of E closed for sup, i.e., such that for any index set I we have the implication ei E B (i E I) = sup ei E B,

(0.39)

E

and the elements e E B can be called convex with respect to B (or B-convex). Finally, one can define in a complete lattice E a "hull operator" u : E —> E by three axioms, extending (0.17)—(0.19) and (0.31)—(0.33), and then call the elements x c E satisfying x = u(x), convex with respect to u (or, briefly, u-convex). On the other hand, in the converse direction, one can apply the general theory of complete lattices to the particular case of complete lattices of subsets of X and of functions on X and obtain, in this way, new results in these cases. For example, the application of the results on "infimal generators" in arbitrary complete lattices to some suitable infimal generators in the complete lattices of sets and functions on a set X leads to some new results in these two cases (see Chapters 1-3). Our exposition will also exploit this converse direction. Namely, instead of presenting first the concepts, results, and proofs for abstract convexity of sets and then repeating separately the analogous

14


concepts, results, and proofs for abstract convexity of functions, we will present them in a unified way in Chapters 2 and 3 as particular cases of a first chapter on abstract convexity of elements of a complete lattice.

0.4 Abstract Quasi-Convexity of Functions Many results about (usual) convex functions on R" remain valid for larger sets of functions on R'1 , defined by some weaker properties than convexity, that form the object of the theory of "generalized convexity" of functions (for recent developments, see, e.g., the volume edited by Koml6si, Rapcsdk, and Schaible [147] and the references therein). The first classical generalization of (usual) convex functions is the concept of quasi-convex functions (i.e., such that all sublevel sets are convex subsets of R n ). The theory of quasi-convexity of functions on R" can be extended to the abstract framework of functions on a set X. Indeed, whenever there is defined a family of abstract "convex subsets" of a set X, one can define a corresponding concept of abstract "quasi-convex functions" f : X —> R by requiring that all (sub)level sets Sd(f) =

EXI f (x)

(d E R)

(0.40)

be "convex" subsets of X, in the given sense. We will study these concepts of abstract quasi-convexity of functions on a set X in Chapter 4. In the present book we will not consider other existing concepts of generalized convexity of functions f : R" —> R that involve differentiability, such as pseudoconvexity, invexity, quasi-invexity, and pseudoinvexity, nor will we consider the generalizations of convex functions f : X —> R on a normed linear space X, such as paraconvex functions, since they are not defined in the abstract framework of functions f : X —> R on an arbitrary set X.

0.5

Dualities

It is well-known that the classical concept of polarity from n-dimensional analytic geometry can be extended to subsets of two arbitrary sets X and W. Indeed, in this extension (e.g., see [261), to each subset G of X there corresponds a subset G ° of W (called the polar of G) and to each subset P of W there corresponds a subset PIof X (the "polar" of P), satisfying the following axioms, for any subsets G and G of X and any subsets P and P of W: (1) If G C G, then G ° D G () ; (2) if i; C P, then TH- D P; (3) G C (G 0 ) + and P C (P±) ° . This concept has been further extended to that of a "Galois connection" between two posets E = (E,) and F = as follows (e.g., see 1261): A pair of mappings (A : E —> F, C : F —> E) is said to be a Galois connection if for any e, E E and z, E F we have (1) if e Z., then A (e) A(; (2) if z then Cl(z) e; (3) CA (e) z e and AC(z) sets, (clearly, if (E, = (2X, D), (F, = (2' , D), where X and W are two then (A, e) is a Galois connection if and only if for any sets G C X and P C W, A (G) = G° and O(P) = P± are "polars" of G and P, respectively). Pickert [224]

0.5 Dualities

15

has shown that if E and F are two complete lattices, then already one of the two mappings A and 8, say, A determines the Galois connection; namely (A, 8) is a Galois connection if and only if for any index set / we have A(inf ei) = sup A(e)

({eib E i c E),

8(z) = inf [x c E I A(x) (

(Z E

F).

(0.41) (0.42)

= (2X, D) and (F, = (2w , D), where X and W are two In particular, if (E, sets, then (0.41) and (0.42) become, respectively, A iEz

G i) = n A(Gi) e(P) = Gcx

G

({Gi}j E i C 2x ),

(0.43)

(1) c W).

(0.44)

pCA(G)

= (2X, 2) Indeed, we have already mentioned in Section 0.3 that the sup in (E, and one can observe similarly that the inf is U. If we call is nothing else than the "dual" of the mapping A and denote it by A', then, by the symmetry of conditions (1)—(3) above, e' = A. For two sets X and W, Evers and van Maaren [80] have called "duality between X and W" any mapping A : 2 x —> 2 w satisfying (0.43). Here we will call it a "duality between 2x and 2 141 ." Evers and van Maaren [80] have given many examples of dualities A : 2x —> 2 w occurring in various branches of mathematics (functional analysis, projective geometry, mathematical logic, the theory of ordered fields, etc.) and made an axiomatic study of dualities for two arbitrary sets X and W. Dualities are closely related to abstract convexity. Indeed, since for any duality A : 2 x --> 2 w the mapping A'A : 2 x —> 2 x is a hull operator in the sense of Section 0.1c (this follows easily from conditions (1)—(3) above), it is natural to study A' A-convex subsets of X. In the converse direction, one of the aims is to find, for every hull operator u : 2x —> 2 x , a set W and a duality A : 2 x —> 2 w (not unique) such that u = A' A. Also, Evers and van Maaren [80] have observed that, identifying the functions f : R" R with their epigraphs, the general FenchelMoreau conjugation (see (0.47) below) is a duality A : 2 Rn R —> 2 (Rn)* X R so one can study dual optimization problems and stability in optimization theory via A'A. Nevertheless, in Sections 0.6 and 0.8 we will use some more direct axiomatic approaches. Extending the above concepts to two arbitrary complete lattices (E, and (F,), any mapping A : E —> F satisfying (0.41) is called [272], 1181 1 a duality between E and F, and the mapping A' = C : F —> E defined by (0.42) is called the dual of A. It turns out that A' is likewise a duality and that A" = (A')' = A. The abovementioned relations between dualities A, hull operators A'A, and A'A-convexity extend to this more general framework. One can apply the results on dualities A between two complete lattices (E, () and (F,) to dualities between complete lattices of functions (E, = (Rx ,

n,

16


(F, = (R w , where X and W are two sets, that is, to mappings A : R x —> T?"' satisfying, for any index set I, A

(inf fi )

= sup fi° iE iEi

(0.45)

({,fi}iE/ g Te),

where instead of A (f) we use the notation f A and where inf ia and sup / are understood pointwise on X and W, respectively. For the same reasons as in the case of abstract convexity (see the final part of Section 0.3), in our exposition we will first present (in Chapter 5) the concepts, results, and proofs for dualities between complete lattices; then we will deduce from them, as particular cases, the corresponding facts about dualities for sets and dualities for functions, respectively (in Chapters 6 and 7). In the next section we describe some particular classes of dualities for functions, that will be further studied in Chapters 8 and 9.

0.6 Abstract Conjugations One of the basic tools in the theory of dual optimization problems is the conjugation of functions. Let us recall that if X = R", or, more generally, if X is a locally convex space with conjugate space X*, and f : X —> R the Fenchel conjugate of f is the function f* : X* —> T? defined by ,

f* (w) = sup [w(x) — f (x)) X

(w E X * ).

(0.46)

EX

Moreau [205] has observed that one can extend this concept to functions defined on two arbitrary sets X and W (instead of X and X* above), replacing the bilinear function (x, w) e X x X* —> w(x) G R which occurs in (0.46) by an arbitrary "coupling function" go : X x W —> R. Then the general Fenchel-Moreau conjugate of f : X —> R, associated to ço, is the function fc (0 W —> R defined by f c(° (w) = sup {yo(x, w) ± — f (x)} X

(w

G

W),

(0.47)

EX

where ± is the "lower addition" on R [205], that is, the extension of the usual addition + by the convention: -koo ± (—oc) = —oc ± (-Foc) = —oc.

(0.48)

The generalization (0.47) encompasses, as particular cases, by taking a suitable set W and a suitable coupling function go, various other conjugations different from (0.46), such as those "of type Lau" (in which go takes only the values 0 and —oc). An equivalent axiomatic approach to Fenchel-Moreau conjugates (0.47) was given in [267]. Namely, if X and W are two sets, a mapping c : R x --> R w is called [267]

0.6 Abstract Conjugations

17

a conjugation (or a conjugation operator) if for any index set I we have (inf. fi )=

sup fic

d) c = f + —d

(0.49)

((fib,/ g R x ),

(f

E

Te, d

E

R),

(0.50)

where infia , + and supia , ± are understood pointwise on X and W, respectively, with + and ± being the upper and lower addition on R (see (0.26) and (0.48)), respectively. It turns out [267] that a mapping c : R x ----> R w is a conjugation if and only if there exists a coupling function go : X x W —> R uniquely determined by c such that c is the conjugation c(w) of (0.47) associated to yo. Hence the correspondence c --> ço between conjugations and coupling functions is one-to-one. R + = 10, -Hoc], it is also natural For two sets X and W and a function f : X to consider another concept of "conjugate function" f M(P) : W R+ associated to a coupling function ço :xxvv->k +, namely

1 fmw( w ) = sup kx , w) >.< f( x ) I

(w

E

W),

(0.51)

where a/0 = +Do, a/±oo = 0 (a E W+ ), and x is the "lower multiplication" on R + 1205 1 , that is, the extension of the usual multiplication x by the convention

-Foo x O = O x +co = 0.

(0.52)

An equivalent axiomatic approach to the conjugates (0.51) was given in [187], —x —w R ± that satisfy considering mappings M : R ±

(inf fi ) m = sup fim ier iE/

1 (f a) m = fm x — • a

(0.53)

({,fihEr g R'±'),

(f

E

Te+ , a

E

7R+ ),

(0.54)

where X is the "upper multiplication" on R + [205], i.e., the extension of the usual multiplication x by the convention -Hoc X 0 = 0 X -koo = +oo.

(0.55)

Actually, one can develop [187], [190] a more general theory of conjugations, called 0-conjugations, for functions with values in extensions of complete ordered groups (FI, o), which encompasses, as particular cases, (0.47) (for (R, ±)) and (0.51) (for (R + \{0 } , x)). We will present this approach briefly in the Notes and Remarks to Chapter 8. By the definition (0.45) of dualities for functions, condition (0.49) means that every conjugation c : Rx R w is a duality. There exist also dualities A : R x —> Rw

18


that are not conjugations but satisfy (instead of (0.50)) another "second condition". For example, if X and W are two sets, a mapping A : R x —> R w is called [182] a v-duality (or a max-duality) if it is a duality (in the sense (0.45)) satisfying (f y d) ° =- f° A —d

(f G R x , d G R),

(0.56)

where v and A stand for (pointwise) sup and inf in R x and R w , respectively. It turns out [182] that a mapping A : R x —> R w is a v-duality if and only if there exists a coupling function i,lr :XxW—> R, uniquely determined by A such that f ° (w) = sup flif(x, w)

A

—f (x)}

(f e R x , w e W);

(0.57)

rEX

hence the correspondence A —> 1/f between v-dualities and coupling functions is oneto-one. In the particular case where X is a locally convex space, X* is the conjugate space of X, and tif : X x X* —> R is "the natural coupling function" defined by *(x, w) = w(x)

(x c X, w E X*),

(0.58)

formula (0.57) becomes f ( w ) = sup [w(x)

A

—f (x)}

(f E R X , w E X * ).

(0.59)

xEX

The conjugation (0.59) was introduced by Flachs and Pollatschek [88], as a tool for the study of duality for optimization problems involving minimum or maximum operations. The problem of determining the dual A' : R w —> R x (in the sense (0.42)) of a v-duality A : R x —> R w leads to a different class of dualities, called "1-dualities" [182], satisfying another "second condition" that involves two new binary operations on T? denoted by 1 and T, respectively. Since they are more technical, we will give them only in Chapter 9. A unified approach to conjugations, v-dualities, and 1-dualities from R x into — w R , where X and W are two sets, has been developed in [189], [191], namely, that of dualities associated to a binary operation * on R, with the property that for any index set I we have (inf bi )* c = inf (bi * c) iEt

R, c E R);

iEl

these are the dualities A : R x

(0.60)

R w satisfying the "second condition"

(f * d) ° = f° Td

(f E R x , d E R),

(0.61)

where is the binary operation on R defined by a T, ( c = —(—a * c)

(a, c E R).

(0.62)

0.7 Abstract Subdifferentials

19

This general concept encompasses the above-mentioned special cases, taking * = * = v, and * = 1, respectively. Finally, let us mention that the dualities from R x into R w , associated to a binary operation * on R satisfying (0.60), have been extended [189] to the case of functions with values in extensions of complete ordered groups so as to encompass also the "o-conjugations" for such functions, mentioned after (0.55). We will present this approach briefly in the Notes and Remarks to Chapter 9.

0.7 Abstract Subdtfferentials Another main concept of (usual) convex analysis, with applications in optimization theory and other fields, is that of a subdifferential. We recall that if X = R", or, more generally, if X is a locally convex space with conjugate space X*, and if f : X —> R and X0 E X are such that f (xo) E R, the subdifferential of f at xo is the subset af (x0) of X* defined by

af (x0) =

{WO E

X * I WO(X)

WO(X0)

f (x) — f(x0) (x

e X)).

(0.63)

Balder [19] has extended this concept to the case where X and X* are replaced by two arbitrary sets, and the bilinear function (x, w) G X x X* —> w(x) G R occurring in (0.63) is replaced by an arbitrary coupling function ço : X x W > R. Namely, for f : X —> R and xo e X such that f (xo ) E R, the subdifferential off at xo with respect to ço, or, briefly, the ça-subdifferential of f at xo, is the subset aç° f (x0) of W defined by —

a`P f (x0) =

two

E

W I ÇO(X,

f (x) — f(x 0 ) (x

WO) — OXO, WO)

E

X)}. (0.64)

Martfnez-Legaz [174] has observed that this concept can be extended also to coupling functions ço : X x W R as follows: a`P f (xo) =

{ WO

E

W I (p(x0, wo) e R,

(x, wo) — (x0, WO)

f (x) — f (xo) (x

E

X)).

(0.65)

The generalization (0.65) encompasses, as particular cases, by taking a suitable set W and a suitable coupling function ço, various other subdifferentials different from (0.63), for example, those in which the coupling function ça takes only the values 0 and —oc (related to "quasi-subdifferentials"). By the one-to-one correspondence between coupling functions ço : X x W R and conjugations c((p) : R x R w (see Section 0.6), we will also call f (x0) of (0.65) the subdifferential of f at x o with respect to the conjugation c(p) (or, briefly, the e(ço)-subdifferential of f at x 0 ), and we will also denote it by acm f (x0). This concept has been extended hi [188] to the case where the conjugation c(p) is replaced by an arbitrary duality A : R x —> R w , in such a way that the subdifferential a° f (x0) of f at xo with respect to A preserves some of the main properties of 8c (0 (x () ) (and of af (x0) of (0.63)). The general a° f (x () ) is defined by using the

20


fact that the representation of conjugations (0.49), (0.50) in the form of FenchelMoreau conjugations (0.47) can be extended to arbitrary dualities A : R x —> R w , yielding a general "Fenchel—Young inequality" with respect to A. Since this definition of a° f (x0) is more technical, we will give it only in Chapter 10. We recall that if X = , or, more generally, if X is a locally convex space with conjugate space X*, and if f : X —> R and xo E X are such that f(x 0 ) E R, then, for each e 0, the e-subdifferential of f at x o is the subset as f (x0) of X* defined by

as f (x0) = two

E

f (x) — f(x 0 ) + E (x e X)}. (0.66)

X * I wo(x) — wo(xo)

Clearly in the particular case E = 0 we have ao f (x0) = af (xo) , the (usual) subdifferential of (0.63). Similarly to the above, one can define the abstract E-subdifferentials 0) for any sets X, W, any coupling function f (x0) and 8 f(x0) (E af f (x0) = acm E x :XxW—> R, and any duality A : R —> R w .

0.8

Some Applications of Abstract Convex Analysis to Optimization Theory 0.8a

Applications to Abstract Lagrangian Duality

Let us first recall the definition of Lagrangian dual problems relative to perturbations. Let F be a locally convex space, G a (nonempty) subset of F called the constraint set, and h : F T? a function called the objective function, and let us consider the (constrained, global, scalar) "primal" infimization problem (P)

=

(

a =

PG,/)

=

inf h(G).

(0.67)

In order to define a dual problem to (P), let us assume that (P) is embedded into a family of "perturbed" (or "parametrized") infimization problems, as follows: Let X be a locally convex space called perturbation space, and let p = pG,h : F x X —> be a function called perturbation function, such that

p (y,

0) =

f h(y)

if y

E

G,

-Foo

if y

E

F\G,

(0.68)

and let us consider the infimization problems (Px ) =

h)

f (x)

= fG,h(x) =

inf p(Y, x) yEF

(X E

X).

(0.69)

By (0.67)—(0.69) we have a = inf h(y) = inf p(y, 0) = yEG

yEF

(0.70)

0.8 Some Applications of Abstract Convex Analysis to Optimization Theory

21

so (P) = (P° ) is indeed "embedded" into the family (Px )xEx. The function f : X •--> R occurring in (0.69), called the optimal value function (or the "primal" function, or the "marginal" function), shows how the optimal value f (x) of problem (Px) varies when the parameter x varies. The properties of the functions h XG) p and f are intimately related, where xG denotes the indicator function of the set G, defined by if y E G,

{0

(0.71)

XG(Y) =

if y e F \ G.

The Lagrangian dual problem to (P) relative to the perturbation pair (X, p), in the sense of Rockafellar [236], is, by definition, the (unconstrained) supremization problem (Q)

(QG, h

p

pG'h

(0.72)

= sup tk(X * ),

where X* is the conjugate space of X, endowed with the weak* topology and where the dual objective function p, : X* —> R is ,u(w) =-

inf

(y,x)EFxX

(w E

{p(y, , x) — w(x)}

x*).

(0.73)

F x X* —> R, or, briefly, the Lagrangian (for the above primal-dual pair of optimization problems {(P), (Q)}) by

Defining the Lagrangian function L =

LG,h :

L(y, w) = inf {p(y, , x) — w(x)}

(y E F, w E X * ),

xeX

(0.74)

one can also express the dual objective function ,u of (0.73) and the optimal value p of the dual problem (Q) of (0.72), respectively, in the form ,u(w) = inf L(y, w)

(w c X*),

yEF

p = sup inf L(y, w).

(0.75) (0.76)

weX* yEF

For suitable choices of the perturbational pair (X, p), this definition yields various Lagrangian dual problems to the primal problem (P) of (0.67). For example, let us consider the inequality constrained mathematical programming problem (P)

a=

where u : —> R and h : fri on R"; this is the particular case "

F =- R",

inf

yER" u (y)

(0.77)

h(y),

R are arbitrary and

G = ly e R n u(Y)

is the usual partial order

01,

(0.78)

22


of (0.67). Then, taking X = Rm and p : R" x R 1 —> R defined by h(y)

if y c R', x e

+co

if y e R" ,

u(y)

x,

, u(y)

x,

(0.79)

p(y, x) = x E

which satisfies (0.68), the function f of (0.69) becomes f (x) =

inf

h(y)

(x

yER n

e Ir").

(0.80)

u(Y)“

1

Furthermore one can show (see the proof after (0.113) below) that the function L of (0.74) becomes the usual Lagrangian:

L(y, w)

1

(Y) — w(u(Y))

+co

if y

E

R", w c (RI)*, w

if y

E

dom h , w

E

0,

(1r)* , w

0,

(0.81)

if y dom h,

where dom h = fy e F I h(y) < -koo} and where w 0 means that w(x) 0 for all x E , x 0. Note that by (0.81) and (0.75), for h 0 +c>o we now have inf (h(y) — wu(y)}

if w

E

(R 1")* , w ( 0,

yE Rn

(0.82)

,u(w) = {

if w

-00

E

(e)*, w

0.

The above primal-dual pairs of optimization problems ((P), (Q)} have been successfully studied with the aid of (usual) convex analysis. Indeed, by (0.73) and the definition (0.46) of the Fenchel conjugate one obtains, for the function f of (0.69), (0.83)

= -f*,

whence, by (0.72) and again (0.46), fi = sup (—f*)(X*) = sup inf ( f (x) — w(x)} = f

(0),

(0.84)

IDEX* xEX

where f** = (f*)* . These formulas permit us to use the well-developed machinery of Fenchel conjugations in order to study the dual problems (0.72), (0.73). For example, since f f** (for any function f), from (0.70) and (0.84) we obtain the so-called "duality inequality"

a

, 8.

(0.85)


23

Also, since for any function f : X —> R we have (e.g., see [236], thm. 5)

= fui,

(0.86)

where f.,, denotes the closed convex hull of f (i.e., the greatest closed convex function f ; recall that a function 0- : X —> k is said to be "closed" if either ik is lower semicontinuous, nowhere having the value —co, or 0- is the constant function —oo), from (0.84) we obtain

,8 =

,f, (0 ),

(0.87)

with f of (0.69). Formulas (0.83) and (0.84) have been called by Rockafellar ([236], p. 19) "the central theorem about dual problems". For, let us recall that the most useful cases are when weak duality holds (i.e., a = ,8), or when strong duality holds (that is, a = p and there exists an optimal solution of the dual problem (Q), i.e., a WO G X* such that ,u(wo) = sup ,u(X*) = ,8), since then often the optimal solutions of the primal problem (P) (i.e., the elements go E G such that h(go ) = inf h(G)) can be found via the dual problem (Q) (e.g., see [236], pp. 4-5). Now, from (0.70) and (0.87) we see that weak duality a = ,8 is equivalent to the "stability" relation f(0) = ,f.,(0). Hence, in particular, when G and h of (0.67) and p of (0.68) (whence also f of (0.69)) are convex and a = f (0) is finite, we have a = 13 if and only if f is finite and lower semicontinuous at 0 (e.g., see [68], ch. 3, prop. 2.1). Also it is known (e.g., see [236], thm. 16, or [68], ch. 3, prop. 2.2) that strong duality holds if and only if the subdifferential af (0) off at 0 is nonempty, and then af (0) coincides with the set of all optimal solutions WO of the dual problem (Q). Since the above classical approach to solving a primal problem (P) via a dual problem (Q) becomes ineffective in the case when a 0 p (at least, when the "duality gap" a — 18 > 0 is large), many modified Lagrangians have been introduced. These generate other dual problems (Q') to a primal problem (P) (e.g., of (0.67) or (0.77)), and in a parallel way to the above, various results on (weak or strong) duality, stability, and the likes have been proved for the new primal-dual pair {(P), (V)}, having both theoretical and numerical applications. Abstract convex analysis leads to a unified theory of duality results for primal-dual pairs of optimization problems {(P), (Q)}, encompassing, as particular cases, known and new results about the above mentioned modified Lagrangians. To describe this unified theory, let F be an arbitrary set, G a (nonempty) subset of F and h : F IR, and let us consider the primal problem (P) = (PG,h) of (0.67). Furthermore, let X be a set (called set of perturbations) and p = PG,h : F x X —> R a function (called perturbation function), for which there exists an element xo e X such that

p (y,

x0) =

f h(y)

if y e G,

1 +cc

if y e F \ G,

(0.88)

24


and let us consider the infimization problems (0.88). By (0.67), (0.88), and (0.69), we have

(EX)

=

(Pb) of (0.69) with

a = inf lz(y) = inf p(y, xo) = yEG

p of

(0.89)

yEF

so (P) = (Px") is "embedded" into the family of perturbed infimization problems (Px )xEx. To define a corresponding dual problem, let W be a set and q) : X xW —> R a coupling function. Replacing in (0.83) and (0.84) f* by fc (0 --(p(x0, •) (with fc (`P) of (0.47), for f of (0.69)), we obtain a dual objective function p, : W —> R, namely, bt(w) = — f` 4) (w)

=

inf (y,x)EFxX

—(p(x, w)) ±(p(xo, w)

ço(xo, w) = ,ic rElfx {f (x)

tp(y, , x)

w))

(21) E

49 (xo, w)

W), (0.90)

and a dual problem (Q)

p

(QG, 11

pG,h

sup [t (w)

(0.91)

(with p, of (0.90)), which may be called the (abstract) Lagrangian dual problem to (P) relative to the perturbational triple (X, p, x 0 ). We can associate to this primal-dual pair [(P), (Q)) a Lagrangian L = LG,h : F x W —> R defined by L(y, w) = inf {P(y, x) X EX

— 40 (x, w))

(y

So(xo, w)

E

F, w

E

W), (0.92)

which satisfies ,u,(w) = inf L(y, w) yEF

(w E W),

,8 = sup inf L(y, w). tvEw yEF

(0.93)

(0.94)

Clearly in the particular case where F and X are locally convex spaces, W = X", xo =- 0, and ya : X x X* —> R is the "natural coupling function" (0.58), formulas (0.88)—(0.94) reduce to (0.68), (0.70), and (0.72)—(0.76), respectively. The first versions of the preceding abstract approach were developed independently by several authors. Balder [19], considering only coupling functions („o : X x W R, has introduced essentially the above ,u, and ,8, but instead of the additive term ço(xo, w) in (0.92), he has assumed that (p(xo, w) = 0 for all w e W. Dolecki and Kurcyusz ([62], [63]; see also Kurcyusz [149]) have considered only the case where W C R X , W±R Ç W and go is that of (0.58), and only perturbations of (P) of the special form f (x) =

inf h(y) yEF (x)

(x

E

X),

(0.95)


25

where F : X —> 2 1' is a multifunction such that for some X0 E X, (0.96)

r(x 0 ) = G.

Clearly (0.95) is the particular case of (0.69), in which h(y) p(y, x) =

if y E T(x), (0.97)

+oo

if y EF\F (x),

and then (0.97) and (0.96) yield (0.88). Lindberg [159] has used a different conjugation with respect to ço (a modified version of (0.47)) and has assumed that ço (xo, w) = 0 for all w E W. The approach (0.88)—(0.94) for W c R x and ça = 0- of (0.58) (with X* replaced by W) which is essentially equivalent to the general case of a pair (W, (p ), R is an arbitrary coupling function, where W is an arbitrary set and ço : X x W was given in [266]. The theory of the Fenchel—Moreau conjugations (0.47), developed in abstract convex analysis, permits us to study the primal-dual pairs [(P), (Q)}, where (Q) is the abstract "Lagrangian dual problem" (0.90), (0.91) to the primal problem (P). For example, by (0.90) and the rules of computation with and + (e.g., see [206]), we have ,a(w) ( If (xo)

—(10(xo, w)} ± (P(xo, w)

f (xo)

{ — ço(xo, w) ± Oxo, w)} (w E W),

f(x0)

(0.98)

whence, by (0.91) and (0.89), we obtain that the duality inequality (0.85) remains valid also in the general case. Note that by (0.91), (0.90) and the expression of f c (9* (0/ = (fc ((P) )(*PY (e.g., see (8.59)), we have

fi = su p (W) = sup

{0X0, w) ± — f e(g)) (W))

f c0Y (X0),

(0.99)

WEW

which, together with (0.89) and f fc (0` (0 ' , yields again (0.85). Moreover, for any set W and any ço :XxW—>R, we have (see Theorem 8.8) P)' f c(ç'')c(` = fco(F)

(f E R x )

(0.100)

wEW,dER).

(0.101)

(with f,„(F.) of (0.35)), where

Thus, by (0.99) and (0.100), for f of (0.69) we have

= fc0(T)(xo),

(0.102)

whence, by (0.89), we see that weak duality a = /3 is equivalent to the "stability" relation f (x 0 ) = fco (F)(x0). Note that for W = X* and cp = 0- of (0.58), formula

26


(0.102) is equivalent to (0.86) (by Theorem 3.7). Also one can show that when a E R, strong duality holds if and only if aço f (x 0 ) of (0.65) (for f of (0.69)) is nonempty, and then f (x0) coincides with the set of all optimal solutions 14 of the dual problem fi, we have strong (Q). Indeed, by (0.89), (0.91) and the duality inequality a duality if and only if there exists wo E W such that f (xo) = I-4w 0 ) (and then wo is an optimal solution of (Q)), i.e. (by (0.90)), such that

f(x 0 ) = inf {f(x) + xEx

w0)} + ço(xo, w0),

(0.103)

which, by f(x0) E R and (0.65), is equivalent to wo E aço f (x0). In a similar way, using the results of abstract convex analysis, many other results on duality (e.g., on saddle points of Lagrangians) can be extended from locally convex spaces F, X, perturbations (0.68), (0.69), and dual problems (0.72), (0.73) to arbitrary sets F, X, perturbations (0.88), (0.69), and dual problems (0.91), (0.90) (with arbitrary (W, ço)). In turn the general theory obtained in this way can be applied not only to convex optimization but also to a large number of other (known and new) cases. Indeed, for example, we will show that the general Lagrangian (0.92) encompasses, as particular cases, various modified Lagrangians. Let us consider the equality-constrained primal problem a =

(P)

)

inf h(y), 'E R"

(0.104)

u (Y)=0

where u : Rn —> Rtn and h : Rn —> R are arbitrary; this is the particular case F = R", G = fy e R" I u(y) = 0} of (0.67). Taking X = , xo = 0, and p: x R"' —> R defined by if x = u(y),

h(y)

(0.105)

p (y, x ) = {

+oo

if x 0

we have (0.88). Also, taking W = (R'")* and ço : Ir x (Ir)* —> R defined by ço (x, (I)) = (I)(x

)

(x E R", (I) E WY),

(0.106)

formula (0.92) becomes the usual Lagrangian L(y, (1)) = inf {h(y) ± Xty/ERnium=x)(Y) — R

(1) (x)1

"

(y E

= h(Y) — (1) (u(Y))

, 14) E (Rm)*).

(0.107)

However, taking other pairs (W, ço : x W —> R), one can obtain "better" Lagrangians. For example, let W = (Ir)* x R, and define ço : x ((lr)* x R) --> R by 40(x, (4) , r)) = (1) (x) + rIlx11 2

(x E

", ((I), r) E (e)* x R),

(0.108)


27

where II • is the Euclidean norm on R". Then the Lagrangian (0.92) becomes L(y, (c1 ) ,

(y E R", (I) E (R m )

r)) = h(Y) — (1) (u(Y))+ rIlu(Y)11 2

which induces, for any fixed r

E

r E R), (0.109)

R, the Lagrangian

L r (y, cD) = h(Y) — cl ) (u(Y)) + rIlu(Y) 1 2

(y E R", (I) E (R m )*). (0.110)

L r , whence by (0.94), ,8 By (0.107) and (0.110), for any r > 0 we have L Thus L, is "better" than a— supi„ w infyE F r (Y, w) = Sr and hence a — ,8 L in the sense that it yields a smaller "duality gap." Since L,. of (0.110) is obtained by adding to the usual Lagrangian L (of (0.107)) the term rIlu(Y)11 2 , it is called an augmented Lagrangian. Besides L r , introduced by Hestenes [115] and Powell [227], one can define [159] a more general class of augmented Lagrangians simply by replacing in (0.108) the function x > IIx11 2 by an arbitrary convex function on Rm. For the same primal problem (P) of (0.104), Dolecki and Kurcyusz [63] have x R ±) observed that if we take W = Rtm x R ± and ço :Rtm x R defined by ,

Ox, (z, r)) = — nix — z11 2

(x, z c R" 1 , r G 14),

(0.111)

then we obtain the modified Lagrangian L(y, z) = h(Y) + rIlu(Y)

(y G R", (z, r)

zII 2

E

R`" x R +),

(0.112)

which, too, yields more effective computational techniques than the usual Lagrangian (0.107). Let us now return to the inequality-constrained primal problem (P) of (0.77), where u : R" —> Rf" and h : R" —> R are arbitrary. For X = xo = 0, and p of (0.79) we have (0.88). Also, taking W =- (1r)* and an arbitrary cp : R"' x (R"')* —> R, formula (0.92) becomes the Lagrangian L(y, w) = inf [h(y) ± X{y'ERnIticyux}(Y) xER", = {h(y)

inf [—ço(x, w)]) + ço(0, w) xER "

w)) ± 40(0, w) (y

E

R", w E (1r)*).

u(Y)0 -

(0.113) For go = 0- of (0.58) this yields the usual Lagrangian (0.81). Indeed, if y c R", x E Rm, u(y) x, and w E (In*, w 0, then w(x) — w(u(y)) w(x — u(y)) ( 0, whence, by (0.113) (for ço = of (0.58)), L(y, w) = h(y)

inf [—w(x)] = h(y) — w(u(y)).

xER"?

On the other hand, if w E (1?`")*, w 0, then there exists x' \ {0}, x' 0, such that w(xi) > 0. Then, for t > 0 sufficiently large and x = u(y) + tx', we have u(y) x and —w(x) = —w(u(y)) — tw(x') < 0, whence the inf in (0.113) is —oo.

28


Thus L(y, w) = —oo whenever h(y) < +ox. Finally, if y dom h, then (0.113) yields L(y, w) = ±oo for all w e (Ir")*, which proves (0.81). For other pairs (W, ç : Rf" x W —> R), formula (0.113) yields various modified Lagrangians. For example, Lindberg [159] has observed that for W =- (R )* x R and ya of (0.108), from (0.113) one obtains the augmented Lagrangian of Rockafellar [237]. The case of the general mathematical programming problem, i.e., with simultaneous inequality constraints u(y) 0, where u : R" RP, and equality constraints v(y) = 0, where y : R" —> R", can be treated similarly. A large number of other examples of modified Lagrangians, introduced and studied by several authors, which fit into the above general framework can be found, for example, in the papers of Balder [19], Dolecki and Kurcyusz [63], and Lindberg [159].

0.8b

Applications to Abstract Surrogate Duality

The Lagrangian dual problem (0.82), (0.72) (with X = Rm) to the mathematical programming problem (P) of (0.77) is especially useful in the case of "convex programming," i.e., when u : R" —> R`" and h : R" —> R are convex. Since there are many nonconvex programming problems (0.77), other concepts of dual problems to (0.77) have also been developed. We will recall now the surrogate dual problem to (0.77) which has been successfully applied for "quasi-convex programming," for the case where u : R" —> R" 1 is convex and h : R" —> R is quasi-convex (such problems often occur, e.g., in mathematical economics) and for "0-1 integer programming." For the beginnings of this theory, see Glover [99], [100], Luenberger [163], Greenberg and Pierskalla [103], and the references therein. Let us consider again the primal problem (P) of (0.77), where u : R" —> f? " and h : R" —> R are arbitrary. The (usual) surrogate dual problem to (P) is, by definition, the supremization problem (Q

(Q?,h

/3s =

)

)

=- sup ,u, ( (R " )*),

(0.114)

where the surrogate dual objective function is

I

inf

h(y)

if w

E

(Rn)* , w

0,

VER'

(0.115)

tv(u(Y))?-0

inf h(Rn)

if w

E

(R m )* , w % 0,

and where the subscript s stands for "surrogate." The main difference between ,u of (0.82) and p of (0.115) is that in (0.82) the "penalty term" —wu is added to the objective function h in order to "compensate" that for each w e (Ri")* , w 0, (0.82) is an unconstrained infimization problem (i.e., the constraint set of (0.82) is the whole space R" instead of the initial constraint set G = {y E u (y) 0)), while in (0.115) wu is used to form the so-called "surrogate constraint sets" ly

E

frz w(u(y))

0)

(w e (e)* , w

0),

(0.116)


that is, the constraint sets of the infimization problems (0.115) (for w the sets (0.116) contain the initial constraint set G = {y E R 11 I u(y)

=

01

inf yER" u(y). 2 F is any multifunction, then the dual problem (QE) = (Q

/3E =

h)

G.h

= sup /L E (W),

(0.119)

where [LE (w)

=

ho

= inf h(EG.w)

(0.120)

E W),

is called a surrogate dual (namely, the E-surrogate dual) problem to (P); the sets EG,„, C F are called surrogate constraint sets. If G C EG , „„ then inf h(G) inf h(E G ,,,), whence a

(0.121)

PE.

If a = p E , we say that weak surrogate duality holds; if a = /3 E and there exists WO E W such that p, E (w o ) = /3 E , we say that strong surrogate duality holds. For the primal problem (P) of (0.77) (which is the particular case (0.78) of (0.67)), the above scheme encompasses the usual surrogate dual problem (0.114), (0.115), as a particular case, by taking W = (Ri")* and

=

I

fy E

R"

R rl I w(u(y))

0}

if w E (R' 1")*, w

0,

if w e (Rm)* , w

0;

(0.122)

indeed, then (0.120) reduces to (0.115). Let us note that the above general concept of surrogate duality is useful even for some convex primal problems for which it is known that strong (and hence also weak) Lagrangian duality holds, for example, the problem of best approximation by

30


elements of convex sets. Indeed, if F = (F , II • 11) is a normed linear space, G a convex subset of F, yo E F \ G (where G is the closure of G), and (y

h(y) =11Yo — Y11

E

F)

(0.123)

(which is a finite continuous convex function on F), then (P) of (0.67) becomes a = inf IIYo

(P)

geG

(0.124)

g II = dist (yo, G),

for which it is known (e.g., see [268], formula (16)) that dist (yo, G) = max

wEF*

inf

yEF

HY°

YII

(0.125)

sup w(G)

(where max denotes a sup which is attained for some WO E F*). By (0.123), formula (0.125) may be regarded as one of strong surrogate duality for problem (P) above, by taking W = F* and EG,w = ty E

F j w(Y)

sup w(G)}

(w

E

F*).

(0.126)

Geometrically, for each w E F* \ EG, v, is a closed half-space containing G and supporting G (i.e., such that its boundary is a support hyperplane of G), so formula (0.125) means that the distance from yo to G is equal to the maximum of the distances from yo to the closed half-spaces containing G and supporting G. This result of best approximation theory provides another motivation for studying surrogate duality. Abstract convex analysis provides the tools (appropriate concepts of conjugations, subdifferentials, etc.) for the study of surrogate dual problems (0.119), (0.120), and of more general "perturbational surrogate dual problems" to (P) of (0.67) in a parallel way to that of Section 0.8a. Here we will present another application of abstract convex analysis, namely to the characterization of surrogate dual problems (0.119), (0.120) among all dual problems to (P) of (0.67) of the form (Q)

( QG, h

sG,h

sup X(W),

(0.127)

where W is an arbitrary set and X = À G ' h : W —> R is any function. To this end we will regard the objective function X : W T? of the dual problem (0.127) as a function not only of the dual variable w e W but also of the primal constraint set G and the primal objective function h. The following theorem gives characterizations of the surrogate dual objective functions 14'h of (0.120) among all functions of three variables G, h, and w (where F and W are fixed). Theorem 0.1 ([184], thm. 3.1; [185], thm. 3.2) Let F and W be two sets. Then, for a function À = À . -(.) : the following statements are equivalent:

x R F x W —> —R,


1 0.

31

There exists a multifunction E : 2 F x W --> 2' such that X = p, E , i.e., such

that Gh

G,h

(W) =

LE

(G

(D)

2F , h

E

R F, w

E

E

W),

(0.128)

kF

(0.129)

with /1,E of (0.120).

20 .

For any index set I we have X

G,inf h,

=

inf X G ' h i iEi

A, G,hvd

3".

(G

E 2 F , (hiLE1 C

d

(G

E

2 F ,hee,dE R),

(0.130)

d

(G 2 F , hey, de R).

(0.131)

We have (0.129) (for any index set I), (0.130), and

(G e 2F , yeF, 'WE W),

E (0, +00}

(0.132)

where x {y} is the indicator function (see (0.71)) of the singleton {y}. Moreover, in these cases E of 1° is uniquely determined, namely EGAD = ( y E F I X

(G

vI (W) = 0)

E 2, w E

W).

(0.133)

Proof The implication 1°

2' is immediate from the definition (0.120) of it E . Furthermore, applying (0.120) to h = x{ y }, where y e F, we obtain G,Xfv)

/LE

{ 0+00 =

if y

G EG .w ,

if y

G

(0.134)

inf X{MEG.w) =

F \ EG,w,

which proves that 1° implies (0.133) (inCidentally, it also proves the implication 30

) .

3". Note that, obviously, x (y) v d = x{ y } ± d

(y

E

F, d

(0.135)

0).

Hence, if 20 holds, then

X G 'xhl v d = ),G, xf ,,vd _ ),G,x, o +d _ ),G,xf,}

d

(G

E 2F ,

y

E

F, d

0),

which implies (0.132). 1'. Note that for any h e R F we have h =-

inf

(y,d)cEpi

h

{X{y}

d},

(0.136)

32


where Epi h = ((p, d) EFxRI h(y) {

cil, the epigraph of h. Indeed,

0+ d =d

if y' = y,

+Do + d = +oo

if

X{y)(Y i ) + d =

y,

whence (0.136) follows. Furthermore, let us recall that a function T : R F —> R is called [185] a strong niveloid if for any index set /, we have T(inf h i ) = inf T (h i )

((h i L EI c R F),

(0.137)

iEi

iEl

T(h ± d) = T(h) ± d

(h

E

RF, d

E

(0.138)

R).

By (0.136), for any strong niveloid T : –R- F —> T? we have T(h) = T(

inf (x {y} ± d)) =

(y,d)cEpi h

inf

(y,d)cEpi h

inf

(y,d)cEpi h

(h E R F ).

d)

(X1y1)

T(x {y} ± d)

(0.139)

Hence, if S, T : R F —> R are two strong niveloids satisfying

s(x t y } )

(xtyl

(y

)

E

(0.140)

F),

then, by (0.139), S = T. Assume now that 3 0 holds, and define E : 2' x W —> 2 F by (0.133). For each G G 2F and w e W, define two functions S = SG,w, T = T T.?, by : S(h) = SG, w (h) =

?J1 (w),

T(h) _ Tw(h)

14,h (w)

(h

e R F ).

(0.141) Then, by (0.129), (0.130), and (0.120), S and T are strong niveloids. Also, by (0.133) and (0.132), we have

XG, X(1)

(w)

10

if y

E EG,w,

(0.142) if y e F \EG,w•

By (0.142) and (0.134), the strong niveloids (0.141) satisfy (0.140), and hence, by LII the above observation, we must have S = T (for all G, w), which is F. (Here and throughout the sequel the symbol El stands for: end of proof). One can also give the following proof (see [184]) of the implication 2° = 1 ° above:


Assume 2°, and for each G

2 F define a mapping CG :h e R F

E

33

h`6 e R w

by (h E R F , W E W).

hc" (w) = —À G (w)

(0.143)

Then, by (0.129)—(0.131) and Proposition 8.1, CG is simultaneously a conjugation (in the sense of (0.49), (0.50)) and a v-duality (in the sense of (0.45), (0.56)). Hence, by Theorem 9.10, there exists a (unique) subset QG of F x W such that

inf

he" (w) = —

(h

h(y)

E

yEF (Y,w)EQG

For each G

E 2F

and w

E

W define

re,

WE

W).

(0.144)

EG, w C F by

EG,w = {.)) E F (Y w)

Qc}.

(0.145)

Then, by (0.143)—(0.145), we obtain k G,h (w

j

)

_V G

( w

)=

inf

yuF (Y,w)EQG

h(y) = inf h(EG,w),

which proves (0.128). Thus 2' = 1'. Let us observe that, fixing a subset G of F in (0.128)—(0.133), one obtains characterizations of the surrogate dual objective functions AV of (0.120) (with fixed F, G and W) as functions of two variables w and h. Using again some tools of abstract convex analysis, one can also give corresponding characterizations of Lagrangian dual objective functions and of Lagrangians (see Section 0.8a) as functions of the dual variables and of the primal parameters (see [185], [1861, and the references therein). These characterizations and, respectively, Theorem 0.1, yield axiomatic approaches to the study of Lagrangian dual and surrogate dual optimization problems. In the above we have given only a few examples of applications of abstract convex analysis to optimization theory. In Section'8.5 we will prove some duality results for the "abstract d.c. optimization problem" (P)

a = inf { f (x) xex

—h(x)},

(0.146)

where X is an arbitrary set and f, h E R X , which have applications, in particular, to convex maximization and reverse convex minimization. Some other hints and references on the applicability of various parts of abstract convex analysis to optimization theory (to quasi-convex optimization, fractional programming, combinatorial optimization, etc.) will be also mentioned briefly in several chapters and in the Notes and Remarks.

Chapter One

Abstract Convexity of Elements of a Complete Lattice In the present chapter we will study abstract convexity in the framework of arbitrary complete lattices. For any complete lattice E = (E, we will denote the greatest (resp. the least) element of E by -Hoc or, if necessary, by +oo E (resp. by —oc or —oo E ), and the lattice operations in E by sup or sup E or, sometimes, y or v E (resp. inf or inf E or A or A E ). We will denote by max (resp. min) a sup (resp. an inf) that is attained. We will adopt the usual conventions inf = ±oo,

sup 0 = —oo,

(1.1)

where 0 denotes the empty set.

The Main (Supremal) Approach: M-Convexity of Elements of a Complete Lattice E, Where M C E

1.1

Definition 1.1 Let E be a complete lattice and M c E. An element x E E is said to be convex with respect to the set M, or, briefly, M-convex, if

x = sup fin e Mini

xl;

(1.2)

, in for the right-hand in the subsequent chapters we will also use the notation supm side of (1.2). We will denote by C(M) the set of all M-convex elements of E:

C(M) =

Elx = sup{in e Mim

x}I.

(1.3)

Remark 1.1 (a) If M = 0, then (in e Mini xl = 0 for all x e E, and hence, by (1.3) and (1.1), C(0) = [—cc). In the sequel, in order to avoid the separate 34

1.1 M Convexity of Elements of a Complete Lattice E, Where .A4 C E -

35

treatment of this trivial case, we will assume, without any special mention, that

(1.4)

M 0 0;

for example, (1.7) below will be proved only under assumption (1.4). Note also that if M = E, then, by (1.3), C(E) = E; however, we will not exclude this case (i.e., E). we will not assume that M (b) We have IM E

MI in

(x

x) 0 0

E

C(M) \ (—cc)).

(1.5)

Indeed, if x e C(M) and (in e M1m x) = 0, then, by (1.2) and (1.1), we x} = sup 0 = —oc. get x = sup {m e Mlm (c) We have x E C(M) if and only if there exists a subset M x of M such that X = sup M x .

(1.6)

Indeed, if we have (1.6) for some subset M x of M, then M x c (m E -A4 I M sup {m E M 1m XI X, so (1.2) holds. Conversely, X}, whence x = sup M x x) c M we have (1.6). if (1.2) holds, then for M x =- (rn e M 1m (d) In general, the sup in (1.2) cannot be replaced by max (i.e., there need not exist = MO). MO E M such that x = sup (m E MIM Propositon 1.1 Let E be a complete lattice, and let (0 0).M

M C(M U {—oc}) = C(M),

M 1 c M2

C

C

E. We have

C(M),

(1.7)

C(.1vi U 1-Foo}) = C(M) U (-Fool,

(1.8)

C(Mi) c C(A42).

(1.9)

Proof If m E .A4, then m = max {m' M I M' M), SO M E C(M). Next observe that for any x E E we have sup {m E M U {-00} 1m x) =max (sup {m E M I n x}, —oc) = sup {m e M 1 m x}, whence, by (1.3), we obtain the first equality in (1.8). The proof of the second equality in (1.8) is similar. Finally, if M1 C M2 and x E C(M ), then x = sup (m E M1 I M -V) sup 1M E M2 I M x x, whence = sup {m E M2 1 in x), SO X E C(2)• Remark 1.2 (a) By Remark 1.1(d), in general, the inclusion (1.7) is strict (i.e., in general, M is a proper subset of C(M)). (b) Formula (1.8) shows that in the study of M-convexity it would be no loss of generality to assume that —00 E M. However, we will not make this assumption, since there are many natural examples in which —oc M (e.g., see Sections 2.2a and 3.2a). (c) Formula (1.8) shows that C(M U { -F oc}) = C(M) if and only if +00 E C(M).

36


Proposition 1.2 (a) C ( /VI) is closed for sup; that is, for any index set I we have the implication E C(M)

(i E /) = sup xi E C(M). iE1

(1.10)

In particular (for I = 0), there holds —00 E C(M).

(b) We have +oo e C(M) if and only if sup M = +oo. Proof (a) If I 0 0 and x E C (.M ) (i E I), then

x, = sup {m e Mlm

xi }

sup fm e Mlm

sup xi } jE I

whence sup xi

sup fm e Mlm

iel

sup x i )

sup x j .

je!

jEl

Hence suPia x = sup {m e M lin sup,,/ xi}, so suPi E / xi E C(M). This proves (1.10) for I 0 0. Furthermore, by (1.7) (for M replaced by M U (—Do)) and (1.8), we have —oc e M U (— oc ) C C(M U Fool) = C ( /VI). This proves (1.11) (i.e., (1.10) for/ = 0). (b) By (m E MIM +00) = M, we have +00 E C(M) if and only if +00 suP{m E MIM +00} = sup M. Definition 1.2 Let E be a complete lattice, and let M c E.

(a) For each x e E, the set of all minorants of x belonging to M, that is, the set

U(x) = fm

EMIM

x),

(1.12)

is called the carrier of x (in M). (b) A subset U of M is called a carrier, if there exists an element x e X (not necessarily unique) such that U = U(x) = im e Mlm Proposition 1.3

Let E be a complete lattice, and let M C E. A subset U of M is a carrier if and only if it is the carrier of sup U: U = {M EMIM

sup U}.

(1.13)

x}, Proof Assume that U c M is a carrier, say, U = U(x) = fin e M I m but U does not satisfy (1.13). Then, since the inclusion c in (1.13) is obvious, we have U e Mlm sup U), so there exists mo e M, mo U (i.e., mo X) , such that mo sup U = sup fm e M lm x) x, which is impossible.

1.1 M-Convexity of Elements of a Complete Lattice E, Where M c E

37

Conversely, if U C M satisfies (1.13), then for x = sup U E X we have sup U } = U(sup U) = U (x), so U is a carrier. U = {m eMlm Remark 1.3 (a) For any carrier U C M, sup U is the least element x e X x}, then such that U = U (x). Indeed, if x v X, U = U(x) = {in eMlm x) x. supU = sup{m e Mlm

(b) As has been observed in the proof of Proposition 1.3, the inclusion c in (1.13) is obvious (for any U C M). Therefore a subset U of M is not a carrier if and only if there exists m o E M such that

Mo g U,

mo

(1.14)

sup U.

Definition 1.3 Let E be a complete lattice, and let M c E. For any x E E the M-convex hull of x is the element com x e E defined by

com x = sup {h

E C(M) I h

(1.15)

x}.

Remark 1.4 (a) By (1.11), we have —00 E

th

E

C(M) 1 h

x} 0 0

(x e E).

(1.16)

E),

(1.17)

(b) By (1.15) and (1.8), we have

como_ 00 } x = co.A4 x and, if and only if +oo

E C(A/i)

(X E

(or, equivalently, sup M = +oo), then also

COmu{± 00 } X = COm X

(x

E

(1.18)

E).

Hence one can adjoin {—oc) to M (i.e., replace M by M i = M U (—oc)), and if and only if +00 E C(M), then one can adjoin {—cc, -Foc} to M, without modifying C(M) and com. (c) If +co C(M) (or, equivalently, sup M < +oo), then, by (1.15) and (1.8), we still have com ui+,01 x = sup th

E C(M)U(+00}

1h

x) = com x

(x

E

1), (1.19)

E \H-

but, by (1.15), (1.18), and (1.23) (or (1.22)) below,

com ut+,01 (+0o) = sup th

E C(M) U

(+00) I h

+0.0} = +oo co.A4 (+oo).

(1.20) Proposition 1.4 (a) For any x

E

Ewe have CO.A4 X E

C(M),

(1.21)

38


and com x is the greatest .A4-convex minorant of x:

com x = max (h e C(M) I h

x}.

(1.22)

(b) C(M) is the set of all fixed points of the mapping com : E —> E:

C(.1l4) = (x

E

E I co.A4 x

x).

(1.23)

Proof (a) By (1.15) and (1.10), we have (1.21) and hence (1.22).

(b) Formula (1.23) follows from (1.22).

Corollary 1.1 (a) We have (in E M),

com (—co) = —oc.

(1.24)

(1.25)

(b) We have com (+oc) = +oc if and only if sup M = +oo. (c) There holds

C(M)

= tCOm

E

XIX

E}.

(1.26)

Proof (a) These statements follow from (1.7), (1.11), and (1.23).

(b) By (1.23), we have com (+oo) = +oo if and only if +00 E C(M), which, by Proposition 1.2(b), is equivalent to sup M = +oo. (c) If x e C(M), then, by (1.23), x = com x e {co.A4 e E}, which proves the inclusion c in (1.26). On the other hand, the inclusion D in (1.26) holds by (1.21). LI

Definition 1.4 A hull operator on a complete lattice E is an isotone, contractive and idempotent mapping u E —> E, i.e., such that for any x, E E we have :

x

.77 = u(x) u(x)

u(Y),

x,

(1.27) (1.28)

u(u(x)) = u(x).

(1.29)

Theorem 1.1 The mapping com : E —> E is a hull operator on E. Proof We have to show that for any x,

x

X

.7"

co.A4 x CO.A4 X

E

E,

com X,

(1.30) (1.31)

39

1.1 M-Convexity of Elements of a Complete Lattice E, Where M c E

(1.32)

CO m COm X = CO.A4 X.

Now (1.30) and (1.31) are obvious from the definition (1.15) of com x. Finally, for y ,--- com x we have, by (1.21), y E C(M), and hence, by (1.23), com y = y, which El proves (1.32).

Theorem 1.2 For any x E E we have com x = sup fm e .A4 I n't, xl.

(1.33)

Proof: By (1.21), (1.3), (1.31), (1.7), and (1.15), we have com x = sup (in

E M I in •- E be a hull operator on E (see Definition 1.4). We will say that an element x E E is convex with respect to u, or, briefly, u-convex, if x is a fixed point of u, that is, if x = u(x).

(1.67)

We will denote by C(u) the set of all u-convex elements of E: C(u) = { x E Elx = u(x)}.

(1.68)

Remark 1.14 For u = coB, where B is any convexity system in E, we have, by (1.68) and (1.62), C(c08) = B.

(1.69)

Theorem 1.5 Let E be a complete lattice. (a) For any isotone contractive mapping (and hence, in particular, for any hull operator) u : E —> E, C(u) is a convexity system. (b) If u : E —> E is a hull operator, then for the convex hull operator coc () (see (1.61)) we have COC( u

) = u.

(1.70)

Proof (a) Assume that u : E —> E satisfies (1.27) and (1.28), and let / 0 0, {x1}iE1 C(u). Then, by xi supi , / x i (i E I) and (1.27), we have u(x i ) u(sup i , / xj ) (i E I), whence sup u (x i ) iE/

u (sup xi ); ier

(1.71)


48

note that this holds for any (xi lie/ c E. On the other hand, by ki lie/ C C(U) and (1.68), we have xi = u(x i ) (i E I), whence, by (1.28) (with x = suptEi u (x i )), we obtain sup u(x i ). ie/

u(sup xi ) = u(sup u(x i )) iei

Hence, by xi = u (x i ) (i

E

(1.72)

I), (1.71), and (1.72),

sup xi = sup u(x i ) = u (sup xi ), jet iel jet and thus, by (1.68), sup / X

E C(U).

Also, by (1.28) for x = —oo and (1.68),

—00 E

(b) Assume now that u : E (1.28), we have

E is a hull operator. Then, by (1.29), (1.68), and

G(x) G (17 G C(U) I h

x}

(x G E),

(1.73)

whence, by (1.61) for the convexity system B = u(x)

max 1ft e C(u) I h

xl = coc(u) x

(x G E).

On the other hand, by (1.68) and (1.27), for each h G C(u) with h h = u(h) u(x), whence max {h.

E C(U)

Ih

x}

u(x)

E

E),

(1.74) x we have

(1.75)

which, together with (1.74), yields (1.70).

111

Corollary 1.4 Let E be a complete lattice. Then u

C(u)

(1.76)

is a one-to-one mapping of the family H(E) of all hull operators u : E --> E onto the family .F(E) of all convexity systems B in E, with inverse B

Proof By Theorem 1.5(a), C(u) E .F(E) (u 7-t(E) and C(u1) = C(u2), then, by (1.70), u1

COC(ui)

(1.77)

cos . E

H(E)). Furthermore, if u1,142

COC(u2) = U25

which proves that the mapping (1.76) is one-to-one. Finally, if B E .F(E), then, by (1.69), we have B = C(u) for u = COB E H(E), which proves that (1.76) maps

1.4 Another Equivalent Approach: Convexity with Respect to a Hull Operator

49

R(E) onto ,F(E). Also, by (1.70) and (1.69), the compositions of (1.76) and (1.77) are u ----> C(u) ----> coc(,) = u and B -± co 6---> C(co8) = B, so they are inverse to 0 each other.

Remark 1.15 By Corollary 1.4, the theories of convexity with respect to hull operators u : E ---> E and convexity systems B e 2 E are equivalent, and hence, by Remark 1.12(c), they are both equivalent to the theory of M-convexity (which is more versatile). In the sequel it will be convenient to use also convexity with respect to a hull operator u.

Chapter Two

Abstract Convexity of Subsets of a Set We recall that if X is any set, 2x denotes the family of all subsets of X. For an arbitrary set X, we will consider the complete lattice E = (E, = (2x , of all subsets of X ordered by containment (i.e., for GI, G2 E 2X , GI G2 if and only if G i D G2). Then clearly the greatest and least element of E and the lattice operations in E are, respectively, +00 , 0,

+00

= X,

sup „

fl

= U'

(2.1)

and the conventions (1.1) become

°

no, x.

(2.2)

2.1 M-Convexity of Subsets of a Set X, Where M C 2 x In this section we will restate some of the concepts and results of Chapter 1 for the complete lattice E = (2X, D), using (2.1) and (2.2), and for a suitable infimal generator of E, and we will give some further facts in E = (2X, D). To this end, let us first explain some notational matters, which seem unavoidable, since we want to follow the tradition of denoting elements of sets (in particular, elements of a complete lattice E) by lowercase letters and sets by capital letters. In this chapter the role of the elements x E E of Chapter 1 is played by the sets G E 2 X , that is, by the subsets G of X, where X is an arbitrary set (whose elements are now denoted by x, which, however, will lead to no confusion with the elements x E E of Chapter 1). Also the role of the subset M of E of Chapter 1 is played now by a subset M of 2, that is, by a family M of subsets M of X (the elements of such a set M C X are now denoted by m, which, however, will lead to no confusion with the elements ni e M of Chapter 1). For E = (2 X , D), Definition 1.1 yields 50

2.1 .A4-Convexity of Subsets of a Set X, Where .A4 c 2 x

51

Definition 2.1 Let X be a set and M c 2x , a family of subsets of X. A set G C X is said to be convex with respect to the family M, or, briefly, M-convex, if

G = n t„,,

E

.M I G

(2.3)

MI

We will denote by C(M) the family of all M-convex subsets of X: C(M) = {G c XIG = n{M E ./vLIG

M}}.

(2.4)

By Proposition 1.2, there holds: Proposition 2.1 Let X be a set and M c 2 X . Then (a) C(M) is closed for intersections, that is, for any index set I we have the implication

Gi

E

C(A4) (i

E

nGi E

(2.5)

C(M).

iE

In particular (for I = 0), there holds

X (b) We have 0

E

E

(2.6)

C(M).

C(M) if and only if n mE.A4 M = 0.

By Definition 1.2(a), the carrier of any G

E

2 X is the family

U(G) = {MEMIGcM}.

(2.7)

Definition 1.3 yields now: Definition 2.2 Let X be a set and M c 2. For any G c X, the M-convex hull of G is the set com G c X defined by com G = n {,

E

C(M) I G ç H).

(2.8)

Remark 2.1 By (1.8), Remark 1.4(b), and formula (2.1), we have C(M U {X}) = C(M), C(M U (0 }) (2.9) = C(./14) U {0}, comu{x) G = com G

(G c X),

and, if (and only if) 0 E C(M) (or, equivalently, n mEm M = 0), then also

C(M U {0}) = C(M),

comutol G = com G

(G c X).

(2.10)

Abstract Convexity of Subsets of a Set

52

Hence one can adjoin (X) to M (i.e., replace M by M i = M U ( X)), and if (and only if) n mEm M = 0, then one can adjoin (X, 0) to M. without modifying C(M) and com . Also, by Remark 1.4(c) and formula (2.1), if 0 Ø C(M), then CO.A4u{0) G

I

=

com G

if G 0,

(2.11)

G=ø.

0 co.A4 0

By Proposition I.4(a), co m G is the smallest M-convex set containing G, and, by Proposition 1.4(b), C(M) is the invariancy class of the mapping com : 2 x , that is, we have

{G e 2x I com G = G}.

C(M)

(2.12)

Applying Definition 1.4 to E = (2 X , D), we arrive at: Definition 2.3 Let X be a set. A hull operator u on (2 X , D) is a mapping u : 2x ---> 2x satisfying, for any G, '6' c X,

6" - c G

u(6) c u(G),

(2.13)

G c u(G),

(2.14)

u(u(G)) = u(G).

(2.15)

Theorems 1.1 and 1.2 become now, respectively: Theorem 2.1 The mapping com : 2 x —> 2x is a hull operator on (2X, D); that is, for any G, G c X we have

c G

com G c com G,

G c com G,

(2.16) (2.17)

com com G = com G.

(2.18)

Theorem 2.2 For any G c X we have

com G =

G

M

GI

C

(2.19)

Remark 1.5(b) yields now com 0

=n G n GEC(m)

mEm

which, together with (2.12), implies again Proposition 2.1(b).

(2.20)

2.1 M-Convexity

of Subsets of a Set X, Where M c 2 )(

53

By Corollary 1.2, C(M) is the smallest family of subsets of X, closed for intersections and containing M. For the complete lattice E = (2X, D), where X is a set, the set Y of all singletons, i.e., the set Y = {{x} Ix

(2.21)

E X}(C 2 X ),

is an infimal generator, since for any G c X we have (2.22)

G Ut x} • xEG

For this infimal generator Y, if we identify each collection of singletons S with UxEstx}, then (1.45) is the identical mapping of 2 x onto itself. Theorem 1.3 now yields the following result (which can be also obtained from (2.19) and (2.4)): Theorem 2.3 Let X be a set and M c 2 X . Then com G , (xeXi /3ME.M,GcM,xeX\MI

{x e XIG fl (X \ M)

0 (M e M, x e X \M))

C(M) = {G c XIV x g G, ]M

E

M, G c M, x

E

(G c X), (2.23)

(2.24)

X \ M}.

For Y of (2.21), Remark 1.9(a) yields: Remark 2.2 We say that a set M GcM,

EM

separates G

XE

X

\

C

M.

X from x

E

X if

(2.25)

Then, by (2.24), a set G C X is M-convex if and only if for each x igt G, G can he separated from x by a set M E M. Thus M-convexity is an "outer" approach to abstract convexity of subsets G of X, in the sense that it is characterized by a property of the elements x E X \ G. For E = (2x , D), Definition 1.7 and Remark 1.10(a) yield, respectively, Definition 2.4 and Remark 2.3(a) below. Definition 2.4 Let X be a set. A family B C 2 X is called a convexity system if it is closed for intersections, that is, if for any index set I we have the implication B i E B (i E I)=

nB iEr

i G B.

The sets belonging to B are called 13-convex subsets of X.

(2.26)


54

Remark 2.3 (a) By (2.26) for I = 0 and (2.2), for any convexity system B we have (2.27)

X E B.

(b) All results of Section 1.3 can be applied, in particular, to convexity systems B in E = (2 X , D). For example, note that by Theorem 1.4, the family C(B) of all B-convex subsets of X (in the sense of Definition 2.1, with .A4 = B), i.e., the family

C(B) = 1G E 2 X I G = nt/3 E BIG g Bi}, coincides with B and that, by (1.59) (or (2.19) for M = B), we have

co5 G = DBEB,G, B)

(G c X).

(2.28)

Also, by Remark 1.12(c), the theories of convexity systems B c 2x and M-convexity (where M c 2x ) are equivalent. (c) For topological spaces X some theories of convexity systems B c 2 x that take into account the topology of X have been also developed (see the Notes and Remarks). Definition 1.8 becomes now:

Definition 2.5 Let X be a set and B c 2x a convexity system. A family M c 2x is called a base of B, if M c B = C(M).

(2.29)

In view of Remark 1.13(c), let us consider now the problems of minimal bases and of the smallest base of B (in the sense of inclusion).

Theorem 2.4 Let X be a set and B c 2" a convexity system. If .A4 is a minimal base of B (i.e., a base of B, containing no proper subfamily .1t71 that is a base of B), then .114 is the smallest base of B. Proof Assume that a base M of B is not the smallest base of B, so there exists a base Ar of B such that M g N. Let M E M \ N. Then, since is a base of B, there exists a subfamily IN/ lid of Ar such that M = n,E, Ni . But, since M is a base of B, there exists a subfamily {M ij j of M, such that Ni = niEJ Mij (i E I). Hence

m=nNi=nn Mij . iEl

(2.30)

iE l jEJ

We claim that

IA;

.A4 \ {M}.

(2.31)

2.1 M Convexity of Subsets of a Set X, Where M c 2x -

55

Indeed, assume a contrario, that M = Mio ,i„ for some io E I, io E J. Then, we obtain Nio =fl' EJ M i01 C Mi 0 10 = M, whence M = niE, N c Ni, c M, so M = Nio E Ar, in contradiction with M E M .A.r; this proves the claim (2.31). Hence, since M is a base of B, from (2.30) and (2.31) it follows that M \ {M} is a III base of B, so M is not a minimal base of B. Definition 2.6 Let X be a set and B C 2x a convexity system. A subset M of X

is called (a) a B-semispace at Z E X if M is a maximal (with respect to inclusion) convex subset of X \ {z}; (b) a B-semispace if there exists an element z E X such that M is a B-semispace at z. Theorem 2.5

Let X be a set and B C 2" a convexity system. A base M of B is the smallest base of B if and only i f M consists of all B-semi-spaces.

Ar c B be an arbitrary base of B. If M0 is a B-semispace at z (i.e., a maximal convex subset of X\ {z}), then there exists a subfamily {N, } i , / of AI such that Proof Let

Nio . We claim = niE, Ni . Then, since Z g M 0 , there exists io e I such that z that Mz = Ni „. Indeed, if Mz Nio, then Mz C Ni, c X \ {z}, in contradiction with the maximality of Mz ; this proves the claim. Thus Mz = Ni() E A I which proves that each B-semispace belongs to every base Ar of B, and hence also to the smallest base of B (if it exists). Conversely, assume now that .A4 is the smallest base of B. If we had X E M, then by (2.2), M \ {X} would be a base of B, with M \ {X} c .A4, M \ {X} 0 M, in contradiction with the minimality of .A4; thus X g M. Let M E M. If M is not a B-semispace (i.e., M is not a maximal convex subset of X \ (Z), for any z E X), then for each z E X \M there exists a convex set Bz cX\fz} (so z g B z ) with M c B z (i.e., M C B z , M B z ); hence, M = nizEx\m B z . But, since M is a base of B and B z E B, there exists a subfamily {Mz,}zEx\m,,E/ of .A4 such that Bz = R E / Mzi (z E X \ M). Hence M

=

n B0= n fl

zEX\M

M0 .

(2.32)

zEX\M iEl

We claim that {Mz i }zEX\M,iEl

g_ M

(2.33)

Indeed, assume, a contrario, that M = Mzoio for some Zo E X \ M, i0 E I. Then, we obtain Bz„ = rl i El M Z0/c- *0/0• , M, whence M = n zExvwBz c _B z o c_m , so M , B zo , in contradiction with our choice of B zo ; this proves the claim (2.33). Hence, since M is a base of B, from (2.32) and (2.33) it follows that M \ {M} is a base of B, which contradicts our choice of M. Thus each M E M is a B-semispace.


56

Corollary 2.1 Let X be a set. A convexity system B c 2x has a (unique) smallest base .11/1 c B if and only if the family of all B-semispaces is a base of B. Remark 2.4 (a) In other words, Corollary 2.1 means that B has a smallest base if and only if each B E B is an intersection of a family of B-semispaces. (b) In Section 2.2 we will see many examples of convexity systems B for which there exists the smallest base B, and some natural examples showing that B-semispaces (and hence the smallest base of B) need not exist. Applying Definition 1.9 to E = (2X, D), we arrive at: Definition 2.7 Let X be a set and u : 2x ---> 2x a hull operator on (2 x , D) (see Definition 2.3). A subset G of X is said to be convex with respect to u, or, briefly, u-convex, if G = u(G).

(2.34)

We will denote by C(u) the family of all u-convex subsets of X: C(u) = {G Ç XIG= u(G)j.

(2.35)

Remark 2.5 All results of Section 1.4 can be applied, in particular, to E = (2x , 3). We omit the formulations of the results on C(u) (of (2.35)) obtained in this way, mentioning here only that the theories of convexity with respect to hull operators u : 2x —> 2' and M-convexity (where M c 2 x ) are equivalent, and that in many cases it is also convenient to use C(u) of (2.35) (e.g., see Sections 4.8, 6.1, and 8.6-8.10).

2.2 Some Particular Cases 2.2a

Convex Subsets of a Linear Space X

In the sequel by "linear space" we will understand a real linear space. (1) We will now show that if X is a linear space, then the convex subsets of X (in the usual sense) are nothing else than the M-convex subsets of X (in the sense of Definition 2.1) for a suitable choice of M c 2 x . We recall that a subset M of a linear space X is called a semispace at Z E X if M is a maximal (with respect to inclusion) convex subset of X \ [z (i.e., if M is a B-semispace at z, in the sense of Definition 2.6(a), with B being the convexity system consisting of all convex subsets of X, in the usual sense). A set M c X is called a semispace if there exists an element z E X (clearly, unique) such that M is a semispace at Z. It is well-known (e.g., see [148], p. 188, thm. 1) that if G c X is convex in the usual sense (i.e., if g i , g2 E G and 0 0}

(2.37)

(however, this result is true only for real linear spaces, but it is not true for linear spaces of dimension 2 over other ordered fields [131]). In particular, for X = R" one can show (e.g., see [148], p. 189) that a subset M of R" is a semispace at 0 if and only if there exists a basis {x,}7 of R" such that

M=

R" Icr) p (x )(x) > 01,

fx E

(2.38)

where {t,}? are the "coordinate functionais" associated to the basis (x,}? (i.e., x = (D,Goxi for each x E R") and where p(x) is the first index such that ■21:, p(x) (x) 0. In other words, one can take, in the above-mentioned result, ,F = {(1) 1 , .. . 43,} and (I)„. The representation of a semispace at any z c X is obtained DI (I)2 ... from that of a semispace at 0, by translation. (b) For X = R" one can show (see [266], lmm. 1.1) that a subset M of R" is a semispace at Z e IV' if and only if there exists a linear isomorphism u of R" onto R" such that (

M = [x

E

(2.39)

R" u(x) R. (1) We recall that a subset M of a locally convex space X is called a closed half-space if there exist (13 E X* \ [0) and d E R such that M = (X E X

I 143 (X)

( dI.

(2.42)

By the well-known "strict separation theorem" (e.g., see [148], p. 245, thm. 1), if G is a closed convex subset of X and x E X \ G, then there exists (13. E X* \ {0} such that sup .213(G)
d}.

(2.46)

Thus for M of (2.42) the separation condition (2.25) means that the closed hyperplane (2.45) bounding M separates closedly G from the singleton {x}. (b) In the above we have assumed that X is a locally convex space (rather than a more general topological linear space) in order to make sure that X* 0 {0 } and that the strict separation theorem is valid (see (2.43)). Such a remark remains valid also for the subsequent examples, in which X is a locally convex space. (2) Assume now again that X is a locally convex space, and choose M c 2x to be the family of all closed half-spaces containing 0 as an interior point, that is, the family of all M of the form (2.42), with d > O. Replacing here (1). c X* \ {0 } by d(1), we see that M is the family of all closed half-spaces of the form M = fx e X I

(x)

1),

(2.47)

where (1) e X* \ ( 0). Then, by (2.4) and the "bipolar theorem" (e.g., see [148], p. 248, we have G E C(M) if and only if G is a closed convex subset of X containing thm.5), O. Hence for any subset G of X we have com G =

(G

U { 0 } ).

(2.48)

Remark 2.8 (a) By Remark 2.1, one can replace in this example M by M1 = M U (X). However, note that now 0 ig C(M) (also clearly n mEm M = (0) 0 0), and one cannot replace M by M2 = .A4 U (X, 0); in fact, C(M2) = C(M) U { 0 } and com, 0 = 0, but com 0 = :){0} = {0} 0 0. (b) One can replace the family M of the above example by the larger family Mo of all closed half-spaces containing 0 (not necessarily as an interior point), i.e., of all M of the form (2.42), with d 0, while keeping como = com of (2.48); one can say that M and Mo are equivalent families of subsets of X (in symbols, M MO). (3) Let X be a locally convex space, and let us choose M C 2 X to be the family of all closed half-spaces containing 0 as a boundary point, that is, the family of all M of the form (2.42), with d = 0; these are called homogeneous closed halfspaces. Then, by (2.4), each G G C(M) is a closed convex cone with vertex 0 (i.e., G is closed, G+G c G, and XG C G for all x 0). The converse is

60


also true, since if G is such a cone, then, applying the strict separation theorem to G and any x G X \ G, for any (1) E X* \ {0} satisfying (2.43) we must have sup D(G) = sup 44) (XG) sup (NG) < (I)(x) 0), whence (by 0 E G) sup 'D(G) = 0 < (I)(x), i.e., (2.25) for M = fx E X I (I)(x) E A4; thus, by (2.24), G E C(M). Hence for any set G C X we have co m G = cone G,

(2.49)

where cone G denotes the closed convex cone with vertex 0, spanned by G. (4) Let X be a locally convex space, and let us choose A4 c 2 x to be the family of all closed half-spaces that do not contain 0, i.e., the family of all M of the form (2.42), with d < 0; replacing here (DE X* \ {0} by —d(I), we see that A4 is the family of all closed half-spaces of the form M=

(X E

X

I (1)(X)

—1),

(2.50)

where (1) e X* \ (0), or, equivalently, replacing (1) by —(1) in (2.50) (or (I) by c/(1) in (2.42), with d < 0), .A4 is the family of all closed half-spaces of the form M = {x E XI(I)(x)

11,

(2.51)

where (D e X* \ {0}. For X = Rn characterizations of C(M) and expressions of com have been given by Tind ([2881, [289]), Ruys and Weddepohl ([242 ] and the references therein), and, for more general spaces, by El Qortobi [72] and Barbara and Crouzeix [201, among others. For example, one can show (see El Qortobi [72], ch. 3, thm. 1.1) that if G is a subset of a locally convex space X such that 0 g c—o- G, then G.

com G =

(2.52)

x?,1

The sets belonging to C(M) are also called reverse closed (Tind [2891) or f3-closed (Araoz [3]). 2.2c

Evenly Convex Subsets of a Locally Convex Space X

(1) We recall that a subset M of a locally convex space X is called: (a) an open half-space if there exist (I) e X* \ {0} and d E R such that M=

c X I I)(x) < d);

(2.53)

(b) a half-space if it is either a closed half-space or an open half-space. Following Fenchel [84], a subset G of X is said to be evenly convex if it is the intersection of a family of open half-spaces. Since every closed half-space is the intersection of all open half-spaces containing it, one can equivalently define an evenly convex set as an intersection of (closed or open) half-spaces. Now let X be a locally convex space, and let .A4 be the family of all open half-spaces (2.53) in X. Then, by the above definition of evenly convex sets and by Remark 1.1(c)

2.2 Some Particular Cases

61

(in E = (2X, D)), a subset G of X is M-convex if and only if it is evenly convex. Hence for any set G c X we have com G = eco G,

(2.54)

where eco G denotes the "evenly convex hull" of G, that is, the smallest evenly convex subset of X containing G. Also the C(M)-semispaces in X are just the sets M c M, and hence, by Theorem 2.5, M is the smallest base of B = C(M). Another larger base of C(M) is the family M of all (closed or open) half-spaces. Remark 2.9 We recall that following Klee [142], if Gi and G2 are two subsets of X, then a closed hyperplane (2.45) is said to separate openly G1 from G2 if

G1 c Ix c X I (I) (x) < d), G2 C {x c X I (I)(x)

d).

(2.55)

Thus for M of (2.53) the separation condition (2.25) occurring in the characterization of evenly convex sets obtained from Remark 2.2 means that the closed hyperplane (2.45) bounding M separates openly G from the singleton {x). (2) Let X be a locally convex space, and let M be the family of all open half-spaces containing 0, that is, the family of all M of the form (2.53), with d > 0, or, what is the same thing, the family of all M of the form M = ix c X I (t(x)
d) = {x E X I — (I)(x) < —d)). Then G e C(.114) if and only if G is evenly coaffine. Also the C(A4)-semispaces are just the sets M e M, and hence M is the smallest base of C(M). Since evenly coaffine sets will be used in the sequel, let us mention here some properties of such sets. Remark 2.10 (a) G is evenly coaffine if and only if it is the complement of the union of a family of closed hyperplanes {Hi ) jel (i.e., G = X \ U Ez Hi). (b) By Remark 2.2, a set G is evenly coaffine if and only if for each x V G there exists a closed hyperplane H with G C X \ H, x e X \ (X \ H), that is, with G

n H = 0,

x

G

H.

(2.64)


63

Proposition 2.2 (a) Every evenly convex set is evenly coaffine. (b) Every connected evenly coaffine set is evenly convex. Proof (a) If G is an evenly convex set and x V G, then there exists an open half-space, say, D = {y c X I (1)(y) < d) (where (1) E X * \ {0), d E R) such d. Then for that G c D, x c X \ D, i.e., such that (I) (g) < d (g G G), 't(x) the closed hyperplane H = {y c X I (I)(y) = (I)(x)} we have (2.64), so G is evenly

coaffine. (b) Let G be a connected evenly coaffine set, and for any x G let H be a closed hyperplane satisfying (2.64). Then, since G is connected and G n H = 0, G must lie in one of the two open half-spaces determined by H, say, D, and by (2.64), we Li have x E H c X \ D. Therefore G is evenly convex. In particular, a convex set that is not evenly convex (e.g., the union of an open square with one of its vertices, in X = R 2 ) is never evenly coaffine. Also a set of the form X \ B, where B contains no closed hyperplane, is never evenly coaffine; indeed, by Remark 2.10(a), the complement of a proper evenly coaffine set G must contain a closed hyperplane. Corollary 2.2 For a subset G of a locally convex space X, the following statements are equivalent: L. G is convex and evenly coaffine. 2'. G is connected and evenly coaffine. 3". G is evenly convex. Proof The implication 1° follow from Proposition 2.2.

2° is obvious. The implications 2°

3°

1' LII

2.2f Spherically Convex Subsets of a Metric Space X A subset G of a metric space X = (X, p) is said to be spherically convex if it is an intersection of a family of closed balls, that is, of subsets of X of the form Ma,d =

{x

G X I p(a, x)

d},

(2.65)

where a G X, d 0. Thus, if .A4 is the family of all closed balls (2.65) in a metric space X, then G E C(M) if and only if G is spherically convex. We recall that a normed linear space X is said to have Mazur's intersection property if every bounded closed convex set in X is the intersection of a family of closed balls; for example, the Euclidean space and the spaces LP (0, 1), with 1 < p < +oo, have this property. (For some characterizations of the normed linear spaces having Mazur's intersection property, see e.g., Giles [981, pp. 219-235 and the references therein.) Let us note that by the above, a normed linear space X has Mazur's intersection


64

property if and only if the M-convex sets (where M is the family of all closed balls in X) are precisely X and the bounded closed convex subsets of X.

2.2g

Closed Subsets of a Topological Space X

Let X be a topological space and M a base (an intersectional base) for the closed subsets of X. Then, by Remark 1.1(c) (in E = (2X, D)), a subset G of X is M-convex if and only if it is closed. Hence for any set G C X,

com G = G.

(2.66)

Note that, when X is a topological linear space, there exist no C(M)-semispaces, and hence, by Corollary 2.1 and Theorem 2.4, C(M) has no minimal base. Remark 2.11

Besides C(M), for any family M C 2 x one can consider the family of complements of the M-convex sets:

D(M) = {X\GIG E C(M)} = {F c XIX \F CC(M)}.

(2.67)

Then, by (2.5), we have the implication

Fi c 1)(M) (i

E

i c D(M), I) =F

(2.68)

iel

in particular (for I = 0),0 e D(M). For some of the families M c 2x considered in the above, the family V(M) is useful in applications (e.g., the family of all complements of convex sets in a linear space, the family of all complements of closed convex sets in a locally convex space, the family of all open sets in a topological space).

2.2h

Order Ideals and Order Convex Subsets of a Poset X

As in Section 0.1a, a partially ordered set will be called a poset. We recall that a subset G of a poset X = (X, is called (a)

an order ideal (or a downward set, or a down set) if

{xEXIx ,g)cG (b)

(g E G);

(2.69)

an order convex set, if {X E

X Igi

x

g2}

G

(gi,

g2 E

G, gt

g2)

(2.70)

(the set x E X I gi x g2) is called an order segment). Clearly every order ideal is order convex. The empty set 0 is an order ideal (and order convex). {


(1) Let X be a poset, and let M

65

C 2 x be the family of all subsets M of X of the

form M = Ma =1.xcXla% x),

(2.71)

where a c X. Note that 0 = M_ 00 G M. Proposition 2.3 (a) We have G E C(M) if and only if G is an order ideal in X. (b) For any subset G of X we have

com G = Ix c X 13 g c G, x g}.

(2.72)

Proof (a) Assume that 00GEC(M), i.e., that there exists a subset A of X

satisfying G= n Ma = {xcX1a%x(acA)),

(2.73)

aEA

and let g c G, x c X, x ,, g. If A = 0, then, by (2.73) and (2.2), G = X, so 0 and x g G, then, by (2.73), there exists ao c A x c G. On the other hand, if A g, whence, again by (2.73), g g G, a contradiction. Thus x c G, such that ao -< x which proves that G is an order ideal. Conversely, assume now that 0 G c X is an order ideal, and let us show that then for each a g G there exists M c M satisfying (2.25) (with x = a), which, by (2.24), will prove that G c C(M). To this end, since a g Ma (a c X), it will be enough to show that G c Ma

(a c X \ G).

(2.74)

Assume, a contrario, that 0 0 G g Ma for some a c X \ G, so there exists gcG,ggM a = { x EXIa%x}, that is, a -., g. Then, since G is an order ideal, we obtain a c G, in contradiction with a c X \G. Thus (2.74) holds, which proves that G c C(M). (b) If a e X is not in the right-hand side of (2.72), that is, if a % g (g E G), then, by (2.71), we have G c Ma and a g Ma , whence, by (2.23), a g com G. Conversely, assume now that x e X is in the right-hand side of (2.72), i.e., that _ there exists g E G with x g, and let G any order ideal in X such that G c G. Then, since g c Gc G-', x g, and since 6 - is an order ideal, we must have x c G. Thus x belongs to the intersection of all order ideals containing G, whence, by (2.8) and part (a) above, x c com G. Ill Proposition 2.4 For each a c X there exists a unique C(A4)-semispace at a, namely M a (of (2.71)). Proof Let a c X, and assume, a contrario, that Ma is not a C(M)-semispace at a, i.e., that there exists a CM) -convex set C c X \ {a} such that Ma C C. Let


66

Ma . Then, by (2.71), we have a Z e C, and hence, since C is an order ideal (by Proposition 2.3(a)), we obtain a c C, in contradiction with C c X \ {a}. Thus Ma is a C(A4)-semispace at a. Assume now, a contrario, that Ma is not the unique maximal (i.e., not the largest) C(M)-convex subset of X \ (a), so there exists a C(M)-convex set C c X \{a} such that C g Ma . Let Z e C \ Ma . Then, as above, we obtain that a E C, in contradiction with C c X \ (a). Thus Ma is the unique C(M)-semispace at a. ZG C \

Remark 2.12 (a) By Proposition 2.4 and Theorem 2.5, M is the smallest base of C(M). (b) Defining upward sets by the condition {xEXIx?-g}cG

(g

E

(2.75)

G)

and taking the family M c 2' of all sets of the form

/1/1 1 = {x

(2.76)

Xlx %a},

where a E X, one obtains corresponding propositions on the new C(M) (simply by applying Propositions 2.3 and 2.4 to the poset X - = (X,

(2) Let X be a poset, and let .A4 = {M, a

(2.77)

E X} U (1171a la E X),

where, for each a G X, Ma and fia are the sets (2.71) and (2.76), respectively. Proposition 2.5 (a) We have G E C(M) if and only if G is an order convex subset of X. (b) For any subset G of X we have com G = {x E X 13gi, g2 E G, g i

x

g2 } .

(2.78)

Proof The proof is similar to the above proof of Proposition 2.3 (see [183],

prop. 4.2).

Proposition 2.6 (a) If a E X is nonmaximal (with respect to then fia (of (2.76)) is a semispace at a. (b) If a G X is maximal, then X (a) = M a (of (2.71)) is the unique C(M)semispace at a. (c) If a G X is nonminimal, then M a is a C(M)-semispace in a. (d) If a G X is minimal, then X (a) = fia is the unique C(A4)-semispace ara.


67

(e) For each a G X there exist at least one and at most two C(M)-semispaces at - a and/or Ma . a, namely M Proof (a) Let a c X be nonmaximal and assume, a contrario, that [4, is not a C(M) -semis pace at a, i.e., that there exists a C(M)-convex set C c X \ {a} such

a. On the other hand, that ii, c C. Let x e C \ ;4, . Then, by (2.76), we have x since a E X is nonmaximal, there exists Z E X with a < z, so z E ils'a C C. Then, since C is order convex (by Proposition 2.5(a)) and x,z G C, x a < z, we obtain a c C, in contradiction with C c X \ {a}. (b) Assume that a c X is maximal, so X \fa} = Ma , and let us show that in this case X \fa} is order convex (and hence, by Proposition 2.5(a) and Definition 2.6(a), z, and assume, a C(M)-semispace at a). Let x,z E X\ {a}, yEX,xya contrario, that y V X \ (a], i.e., y = a, so x a z. Then either a < Z, in contradiction with the maximality of a, or a = z, in contradiction with z c X \la]. Thus y E X \ {a}. (e) and (d) follow from (a) and (b), respectively, applied to X endowed with the reverse order (i.e., to the poset X - = (X, ,)). (e) Let a c X. Then, by (a)-(d), either Ma or Ma is a C(M)-semispace at a. Now let C c X \ {a} be a C(M)-semispace at a. If there exists no element z e C such that a < Z, then C c Ma (of (2.71)), and hence, since C is a C(M)-semispace at a and Ma C X \ {a} is order convex, we obtain C = Ma . Similarly, if there exists no element x E C such that x < a, then C = ftia (note also that if both situations occur simultaneously, i.e., if a is incomparable with all elements of C, then C = Ma = Ma ). Finally, it remains to observe that since C is order convex and a V C, there cannot exist two elements x,Z E C such that x < a < Z, so at least one of the above situations must occur. D Remark 2.13 (a) If a is maximal and nonminimal, then, by Proposition 2.6, X \(a) = Ma is the unique C(M)-semispace at a, and hence fi a (C X \ la} = Ma) is not a C(M)-semispace at a. Similarly, if a is minimal and nonmaximal, a dual statement holds. (b) If a is both nonmaximal and nonminimal, then, by Proposition 2.6, there are exactly two semispaces at a, namely Ma and Me„ with ma g ma and ma g ma . (c) If a is both maximal and minimal (i.e., incomparable with all x 0 a), then X \ {a} = Ma = fia is the unique C(M)-semispace at a. (d) From Propositions 2.6(e) and 2.5(a) and Corollary 2.2 it follows that B = C(M) above has a smallest base, which is a (possibly proper) subset of M.

(3) Let X be a complete lattice, and let M c 2x be the family of all subsets M = Ma of X of the form M a = IX

E XIX

a),

(2.79)

where a e X (note that, in the terminology of Remark 1.8(e) with Y = E, these are the lower conical subsets of X).


68

Proposition 2.7 (a) We have

(2.80)

C(M) = M. (b) For any subset G of X we have

com G = {x E X ix

(2.81)

sup G).

Proof (a) If G E C(M) and x g G, then, by (2.25), there exists a E X such that a for all g E G (or, equivalently, G C M" , x E X \ Ma, i.e., by (2.79), such that g sup G a) and x a. But then x sup G (since otherwise x ( sup G a, a contradiction). Thus ix E XIx sup G) C G, whence, since the opposite inclusion D holds for any set G c X, we obtain

G = ix E Xlx

sup G) = MG E M.

(2.82)

This proves that C(M) c M, which, together with (1.7), yields (2.80). (b) By (2.19) and (2.79), we have for any set G C X,

ma

co m G = aEX

GCM"

aEX sup Ga

Indeed, to show the last equality, observe that the inclusion c is obvious (taking a = sup G); conversely, if x sup G and a e X, sup G a, then x a. Remark 2.14 (a) For Ma and Ma of (2.79) and (2.71) we have

M a C Ma U {a)

(a E X).

(2.83)

Note also that, by (2.69) and (2.79), a set G C X is an order ideal if and only if Ug eG M g g G. (b) Taking the family M c 2x of all sets of the form Ma = ix E X la

(2.84)

x),

where a c X (note that, by Definition 1.6 with Y = E, these are the upper conical subsets of X), one can prove a corresponding proposition with (2.81) replaced by com G = ix e X I inf G x) (simply byapplying Proposition 2.7 to the complete lattice X - = (X, ,)). Clearly for Ma and Ma of (2.84) and (2.76) we have

fia U {a }

C -

(a E X).

Note also that G is an upward set if and only if U gEG convex if and only if U gi,g2EG,g2Q1 Mg n mg2 c G.

Mg C

(2.85) G and that G is order


69

(4) Let X be a complete lattice, and let M = {Ma I a

E XI U

a E

(2.86)

where for each a E X, Ma and fia are the sets (2.79) and (2.84), respectively.

Proposition 2.8 (a) We have

C(M) =

(Mai

n fia2 l ai, a2 E

(2.87)

(b) For any subset G of X we have

com G = tx E X I inf G

x

sup GI.

(2.88)

Proof The proof is similar to the above proof of Proposition 2.7 (formula (2.82) will be replaced now by G = msup G fiinf G E .A4) .n El

Remark 2.15 (a) If X is a complete lattice, there are some obvious relations between the families C(M) and hence between the sets com G, occurring in parts (1)—(4) above. For example, every set Ma E M = C(M) of Proposition 2.7 is an order ideal, and G is an order convex set containing —oc if and only if it is an order ideal. (b) Applying the results of (1)—(4) above, in particular, to X = (2 F , D) or X = (RF , (for the latter complete lattice see (3.1), (3.2)), where F is a set, one can obtain results on families of subsets of F and families of functions on F. (c) For each a E X, the sets Ma of (2.71) and Ma of (2.84) are complementary in X, and so are the sets fia of (2.76) and AV of (2.79). Hence the sets of the family M of (2.86) are the complements of the sets of the family M of (2.77). These remarks, together with parts (1)—(4) above, suggest to consider, for any family M c 2x , not only the family C(M) but also C (I T), where ./V- c 2x is the family of the complements of the sets in M:

..Ar={X\MIMEM).

(2.89)

Actually this has been done above for some other families M too. Indeed, for M of Section 2.2b, part (1), Ar of (2.89) is the family of all subsets N of the locally convex space X, of the form

N = fx E X l 4:19 (X) >

=

E

X I ( —(1) )(x) < —d},

(2.90)

that is, the family of all open half-spaces in X considered in Section 2.2c, part (1). For M of Section 2.2b, part (3), Ar is the family of all homogeneous open half-spaces considered in Section 2.2c, part (3). For M of Section 2.2b, part (4), IV is the family of all open half-spaces containing 0, of Section 2.2c, part (2). For M of Section 2.2d, Part (1), Al- is the family of all complements of closed hyperplanes, considered in Section 2.2e, and so on.

70


2.2i Parametrizations of Families »1 C 2x , Where X Is a Set Definition 2.8 Let X be a set, and let M C 2. A parametrization of .A4 is any pair (W, 0) where W is a set (called set of parameters) and 0 is a mapping of W onto M (note that 9 need not to be one-to-one). Thus we can write each M E M in the form Mw = 9(w) for some w E W. Remark 2.16 (a) A parametrization (W, 0) of a family M c 2' permits us to replace each M E M by an element w E W such that 0(w) = M, and hence to replace the family M by the set W. This will be useful because of the various posibilities of choosing W (e.g., if X is a locally convex space then often one can choose W = X* x R or W = X*, as we will see below). Parametrizations of families M C 2X will be used, for example, to "parametrize" the "Minkowski type dualities" AM : 2 x —> 2m , that is, to replace them by "equivalent" (but more convenient to work with) dualities A : 2 x —> 2 w (see Chapter 6), and for other applications too. (b) Most of the families M C 2x considered in the preceding sections admit natural parametrizations, which follow from the sentences "Let M be the family of all subsets M of X of the form . . . ," or from the remarks made above, that M E M if and only if it is of the form ... . For example, for the family M of all closed halfspaces of a locally convex space X, considered in Section 2.2b, part (1), one can take the parametrization

W = (X* \ ( 0)) x R, 0(I., d)

(2.91) = Mcp,d =

E

X I (I)(x)

((m, d)

d}

E

W);

note that 0 of (2.91) is not one-to-one, since 9cI, )■.d) = 0(0, d) (A of Section 2.2b, part (2), one can take either

0). For M

W = (X* \ {0}) x (R + \ {0 } ), 0(0, d)

={ X" E where R + =

E RId

X

(2.92)

I 0(X)

d}

( ( ), d)

E

W),

0), or

W -= X* \ {0}, 0(0) = M = {x

E

X I (I)(x)

1)

((1)

E

W);

(2.93)

note that 0 of (2.93) is a one-to-one mapping of W onto M. Thus we have two different useful parametrizations of M. For M of Section 2.2f, one can take ((a, d) E W).

W = X x R + , 0(a, d) = Ma,d

(2.94)

If X is a poset and .A4 is the family of all subsets Ma of X of the form (2.71), where a E X, one can take the parametrization (W, 0) of M, where

W = X, 0(a) = Ma = (x E Xla

x}

(a E X).

(2.95)


For .A4 of Section 2.2h, part (2), one can take any W = (X x {z}) U({z} x

71

ZE

X and (a E X), (2.96)

0(a, z) = M,, 9(z, a) = fia

with M, and M0 of (2.71) and (2.76), respectively. For .A4 of Section 2.2h, part (4), one can take any Z E X and , 0(z, a) = fia

W = (X x tzl)U ({z} x X), 0(a, z) =

(a

E

X), (2.97)

with M" and ki of (2.79) and (2.84), respectively. (c) Given a locally convex space X, for the families M C 2x which are of the form M = [Mq,, d C X HID E X* \ (0), d E P), where P

C

(2.98)

R, one can consider various conditions, for example, E

X*

\

tin

d1, d2 E

P, d1

d

Md c m q,, d2 ,

(2.99)

which is satisfied e.g. by M of Sections 2.2b, part (1), and 2.2c, part (1), but not by M of Sections 2.2d, part (1), or 2.2e, or the condition X E Mc1),(13(x)

(x

E

X, (ID E X * \ {0}),

(2.100)

satisfied by M of Sections 2.2b, part (1), and 2.2d, part (1), but not by M of Sections 2.2c, part (1), or 2.2e. For a study of families (2.98), with P = R, satisfying various conditions, see [2611. Let us mention that for M c 2 )( satisfying (2.100) and for the parametrization W = (X* \{0)) x R, 0(0,d) = X\

M,

E

X*

\

d

e R)

(2.101)

of the family Ar of (2.89), a subset G of X has been called, in [261], 0-convex (or, surrogate convex) if for each x V G there exists (I) E X* \ MI such that G

n

e(o, cD(x))) =

G

n m) (x) = 0.

(2.102)

The term "parametrization" for (W, 0) of (2.101) has not been used in [261], being replaced there by the use of "multifunctions" 0 : (X* \ (0)) x R —> 2 x ; in fact this has been sufficient, since W of (2.101) is fixed. If .A4 of (2.98) satisfies (2.100), then, since (2.102) and (2.100) are equivalent to G ç X \ AlcD, q) (x) = 19(0, (I)(x)), x V X \ 0 -convexity of G implies that G e C(AI), with [266], p. 53, under the additional assumption X E Mq) ,d

Mq),cp(x ) =

0(0, (1) (x)),

(2.103)

Ar of (2.89). As has been noted

> 0, Mx.c1),X(13(x) C Mc13, 1

in

(2.104)


72

the converse implication is also valid, i.e., 0-convexity is then equivalent to Ar-convexity, with Al of (2.89).

2.3 An Equivalent Approach, via Separation by Functions: W-Convexity of Subsets of a Set X, Where W C R x We recall that if X is any set, R x denotes the set of all functions f : X —> R = [—oc, +Do]. For any function f E R X , we will denote by Sd (f ), respectively, Ad (f ) (d E T), the (sub)level sets and, respectively, the strict (sub)level sets of f, defined by d}

(d E R),

(2.105)

Ad(f) = x E Xlf (x) < d}

(d E R).

(2.106)

Sd(f) =

E Xlf

(x)

{

Remark 2.17 (a) We have already encountered such level sets in Section 2.2. For example, (2.42), (2.53), and (2.65) are nothing else than S, (I), A1(0), and Sd(p (a, .)), respectively, where p (a, .)(x) = p (a, x) (a, X E X). (b) In particular, for d = ±co (and any f E R x ) we have, by (2.105) and (2.106), S+00(f) = {-)C E Xif (x)

+00 = X,

(2.107)

}

S(f) = {X E Xif (x) —oc} tx E Xlf (x) = —0,01 —

n

(2.108)

sd(f),

dER

A+00(f) =

fx e

Xlf (x) —oc; or, equivalently,

sup w(x) > —oc Wcui

(x E X).

(2.118)

Proof of (b). By Proposition 2.1 for M = S(W x R), we have 0 E /C(W) = C(S(W x R)) if and only if

o

=n

(w,d)EWxR

{x

sd(w) -

E X I W(X)

d}

wEW.dER

n

E X W(X) = —001,

wEW

that is, if and only if there exists no xo E X such that w(x 0 ) = —oc for all w E W, Which is equivalent to the condition in (b). Similarly to Definition 2.2, it is natural to give:

74


re. For any G C X the W -convex

Definition 2.10 Let X be a set, and let W C hull of G is the set co w G C X defined by

CO W G = fl (H E )C(W) I G C H}. Remark 2.19

(2.119)

(a) By Definitions 2.9, 2.10, and by

Sd( - 00) = { x e XI — oc Sd(-1- 00) ={xE XI

d} =X v 1C(W)

(d E R), (2.120) (d E R),

d} = 0

oo

(2.121)

we have 1C(W U {—cop = 1C(W),

co w 0_} = co w ,

1C(W U (+Do)) = 1C(W) U {0},

(2.122) (2.123)

and if (and only if) 0 E 1C(W), then also cowo+Dol = co w ,

k(W U {+co}) = /C(W),

(2.124)

which suggest to adjoin to W the constant function —oc and, when 0 E ic(W), also the constant function +co. However, this is often inconvenient (e.g., when one wants W to contain only finite functions). (b) By Definitions 2.9, 2.10, and by X Sd(0) = tx E X

10(x)

if d 0,

d) =

(2.125) {

0

if d < 0,

we have, whenever 0 E K(W), K(W U {0}) = K(W),

co w oo} = co w ,

(2.126)

which suggest to adjoin the constant function 0 to W if 0 V W. This will be useful in the sequel, e.g., when X is a locally convex space, W = X* \ {0} and 0 E iC(W). (C) By Definition 2.2 applied to M = S(W x R) of (2.111) and Definitions 2.10 and 2.9, we have CO W G = cos(w x R ) G

(G c 2x ).

(2.127)

Also, by (2.125), we have S((W U {0}) x R) = S(W x R) U (X, 0),

(2.128)

which, together with (2.127) and Remark 2.1 (applied to M = S(W x R)), yields again (2.126) whenever 0 E K(W), indeed, then co woo } G = cos(wxR)u{x,0) G = cos(wxR) G = cow G

(G c X).

2.3 W-Convexity

of Subsets of a Set X, Where W c R x

75

By (2.127), (2.112), and Propositon 1.4(a), co w G is the smallest W -convex set containing G. Theorem 2.1 remains valid for M c 2' and com G replaced by W c and co w G, respectively. Also, by the equivalence (2.114), Theorem 2.2 (applied to M = S(W x R) of (2.111)) yields:

Theorem 2.6 For any G c X we have

n

CO W G =

(w,d)EwxR

n

sd(o=

IX

d).

E X I W(X)

(2.129)

(w.d)EwxR

sup w(G) d, 3g

E

G, w(g) > d)

(G c X), K(W) = (G c X I Vx g G, 3 w

E

W, sup w(G) < w(x)}.

(2.131) (2.132)

Proof By the first equality in (2.23), applied to .A4 = S(W x R), and by (2.114) and (2.105), we have

co w G

EX

I

/9

SAW)

E S(W X E

E

XI

(w, d)

E

XI

/Tti) E W,

EX

I W(X)

sup

R), G Sd(w),

X E

X \ SAW)}

WxR, sup w(G) d, 3 g

E

G, w(g) > d).


76

Remark 2.20 (a) Let us recall that a function is said to separate strictly G l c X from G2 C X if sup w(Gi) < inf w(G2);

(2.133)

when G2 = {x}, a singleton, in (2.133), w is said to separate strictly G 1 from x. Thus the first equality in (2.131) says that cow G is the set of all x E X for which G cannot be separated strictly from x by a function w E W. Also (2.132) means that a set G C X is W -convex if and only if for each x g G there exists a function tV E W that separates strictly G from x. This condition of separation can be taken as a convenient definition of W-convexity (and, in fact, it was historically the first definition of it; see the Notes and Remarks). Thus W-convexity is an "outer" approach to abstract convexity of subsets G of X (see Remark 2.2). (b) Note that, in general, the above concept of separation is nonsymmetric, when — W. For example, if X = R2 , w (R2)+* \ {01, G = ay], y2 ) E R 2 —1 W 01, x = ( - 2, —2), then sup w(G) = 0 and w(x) < 0 for all w E W, so Yi = Y2 there is no w E W separating strictly G from x, but each w E W separates strictly x from G (i.e., satisfies w(x) < inf w(G)). In the particular case where X is a locally convex space and W = X* \ {0} (so W = —W), (2.133) is the usual concept of "strict separation" (e.g., see 10the [148], Schaefer [2441), called also "strong separation" (e.g., see Klee [142], Kelley and Namioka [139]), which is symmetric; indeed, if w separates strictly G1 from G2, then —w separates strictly G2 from G1. We have used other nonsymmetric separation concepts in Remarks 2.7(a) and 2.9 above. Corollary 2.3 Let X be a set, and let W C R x . For a subset G of X the following statements are

equivalent: 1'. 20 .

G E iC(W).

There exists a function h E Te such that G =

3'.

E X

I W(X)

h(w) (w E W)).

(2.134)

sup w(G) (w E W)}.

(2.135)

We have G =

E XIw(x)

Proof The equivalence 1' . 3' follows from (2.130) and (2.131). Furthermore, if 3 0 holds, then for the function h E R iv defined by h(w) = sup w(G)

we have (2.134); thus 3° = 20 .

(w G W)

(2.136)

2.3 W-Convexity of Subsets of a Set X, Where W c

77

Finally, if 2° holds, then G= n

E

wEw

so G

E k(W)

n

X I W(X) h(w)} =

sh(w)(w),

wEw

(by Remarks 2.18(c) and 1.1(c)); thus 2° = 1'.

Remark 2.21 (a) If 2° holds, then (2.136) is the least function satisfying (2.134), since for any he R w satisfying (2.134) we have sup w(G) =

sup (iZEW)

w(x) ( h(w)

sup

w(x)

xEx

xEx

E

W).

w(x) 0100

sup w(G)) (b) For any set G C X and any W E W, the set Ix E XIw(x) is also called [ 143] the w-support of G. Thus G is W-convex if and only if it is the intersection of its w-supports, where W E W. Corollary 2.4 Let X be a set, and let W 1 0.

C

Te, W = —W W. The following statements are equivalent:

Each singleton is W -convex: {xo} E /Coin

2°. W separates the points of X (i.e., for any xl, w E W with w(xi) w (x2)). Proof By W = —W, for each

{x c X I w(x) = w(x0)

x0 E

(2.138)

(Xo E X). X2 E

X, xi 0 x2, there exists

X there holds

E W)) = {X E

XIw(x)

w(x()) (w E W)).

(2.139) Hence, by Corollary 2.3, we have (2.138) if and only if (X0) =

E

X I W(X) = W(x0)

G W))

(Xo E

which is obviously equivalent to 2°.

X),

III

Remark 2.22 Replacing the last equality in (2.139) by the inclusion C, we see that the implication 1° = 2° is valid for any W c Te (without assuming that

W — w). One can also characterize the sets G E 16(W) in terms of the carriers introduced in Definition 1.2(b) (applied to the complete lattice E = (R w , under a certain additional assumption. To this end, let us recall that for any set X and any W C R X , there exists a canonical mapping K = KW of X into R w , defined by K (x)(w) = w(x)

(x EX, wE W);

(2.140)


78

we write K instead of Kw, which will lead to no confusion. Thus K (X) = 1K (X) I x E XI is a subset of R w . We can identify X and K (X) if and only if K is one-to-one (or, equivalently, W separates the points of X). If this is the case, then we can identify each subset G of X with a set of functions on W, namely with K (G) = (K (g) I g E G). and M = K (X), and Corollary Then, applying Proposition 1.3 for E = (R w , 2.3, we arrive at

Corollary 2.5 Let X and W C R x be such that the canonical mapping K : X —> R w , defined by (2.140), is one-to-one (i.e., such that W separates the points of X). Then for a subset G of X the following statements are equivalent: 1 0.

2°.

G E 1C(W). K(G) is a carrier in M = K (X)(c R w ).

Proof By Proposition 1.3, U = K (G) is a carrier in M = K(G) -= {K(X) E

which, since

K

K

(X ) I K(X)

K

sup K(G)b

(X) if and only if

(2.141)

LII

is one-to-one, is equivalent to (2.135).

Remark 2.23 Since (2.135) always implies (2.141), the implication 1' = 2° is valid even without assuming that K is one-to-one. Given two sets X and W C R X , by (2.132) (applied to W and K (X) C R w instead of X and W, respectively, where the canonical mapping K : X -> R W need not be one-to-one), we have that a subset P of W is K(X)-convex if and only if for each w G W, P, there exists x E X such that (2.142)

sup p(x) < w(x). pEP

Similarly to the above, we obtain the following characterizations of such sets:

Corollary 2.6 Let X and W C R x be two sets, and let K be the canonical mapping (2.140) of X into R w . Then for a subset P of W the following statements are equivalent: 1°. P E iC(K(X)). 2°. There exists a function f E R x such that P = {w

WIW

f}

(2.143)

sup p).

(2.144)

(i.e., P is a carrierin W=MCE= (R x , 3 0• We have

P=

E Wlw

pcp

2.3 W-Convexity of Subsets of a Set X, Where W c R x

79

Remark 2.24 (a) If 2° holds, then f = SUP PE P p E -R X is the least function f satisfying (2.143). (b) If K is the canonical mapping (2.140) of X into R w , and îc' is the canonical mapping of W into RK (x) , then (x E X, w E W),

ic- (w)(K(x)) = K(x)(w) = w(x)

(2.145)

and hence 7c.' is always one-to-one (even when K is not so). Given two sets X and W c R x , the canonical mapping K : X —> Te of (2.140) (not necessarily one-to-one) can be extended, in the usual way, to a mapping 2(R') by defining K (G) = {K(g) E K(X) 1g E G} (C RW ) for each K : 2 X —> G E 2. There are also two important mappings a : —> Rw and /3 : 2 x —> 2 w , defined in a natural way, as follows: Definition 2.11 Let X be a set, and let W c -R-x. (a) For any subset G of X the function aG E W W defined by

crG (w) = sup w(G)

(w

E

W)

(2.146)

(i.e., the function (2.136)) is called the W -support function of G. (b) For any subset G of X the subset G/3 of W defined by = (w c WI sup w(G) < -Foo)(= domo-G )

(2.147)

is called the W -barrier set of G. Remark 2.25

(a) By (2.146) (for G a{x} =

= (xi) and (2.140), we have

K(X)

(x E X).

(2.148)

(b) For any sets X and W c Te we have the implication (w

G 1 C G2 (C X) = aG,(w) aG2 (w)

(c) For any sets

E

W).

(2.149)

X and W C R X we have

aG = acow G

(G

C

X).

(2.150)

Indeed, let G C X. Then, by G C co w G, we have the inequality 0, then, since Xgo E supx ,. 0 w(Xgo) = w (go) sup x ,. 0 X = + Do, so G (X > 0), we obtain sup w(G) w G 19 ; this proves the first equality in (2.151). Furthermore, since In- g E G (g c w g) ;t- w(g) —> 0 (w E X # ), G, n = 1, 2, . . .), we have sup w(G) whence sup w(G) 0, which, together with the first part of (2.151), proves the second part of (2.151). (e) In particular, for X and W as in (d), if G is a linear subspace of X, then G )9 =

= 1w

I w(g) = 0 (g

E

E

(2.152)

G)).

We will give some properties of the above mappings o- : 2 x —> R w and : 2w in Chapters 3 and 8, respectively. Here we make only the following observation on a. Remark 2.26

For any sets X and W c X , the restriction a

,C(W)

: 1C(W)

le is one-to-one. Indeed, if G1, G2 E /WV) and aG, = aG2 , then, by Corollary 2.3,G 1 = E X I W(X) CrGI (W) (W E W)} ={x E X I W(X) ( aG2(W) (w e W)) = G2. Since R x has more structures than 2, there appear naturally some additional properties of 1C( W). For example, recalling that, by definition, W R = {w +dlw e W, d E R}, let us prove: Proposition 2.10 Let X be a set, and let W

. Then

C

(2.153)

1C(W) = 1C(W R). Proof We have sup w(G) < w(x) if and only if, for any d sup(w

E

R,

d)(G) < (w d)(x).

Hence, the result follows from (2.132). Remark 2.27 Let us also mention how Proposition 2.10 can be deduced directly from Definition 2.9. Since w = w 0 (w e W), we have W C W R, whence S(W x R) C S((W R) x R). On the other hand, since Sr (W +

d) = Sr-d(W)

we have S((W

R) x R)

E S(W X C S(W X

R)

(W E

W, r, d

E

R),

(2.154)

R), whence

S(W x R) = S((W R) x R). Consequently, by (2.112), there follows (2.153).

(2.155)

2.3 W-Convexity of Subsets of a Set X, Where W c R x

81

For any subset G of a set X, we will denote by x G the indicator function of G, defined by if x E G,

{0

(2.156)

XG(x) =

if x E X \ G.

By Definition 2.9, W-convex sets are a particular case of M-convex sets. In the converse direction, we have the following basic result. Theorem 2.8 Let X be a set, and let M c 2x . Define Jm,

C R x by

JA4 = {—XX\M 1 M E M), /..A4 =

(2.157)

fxm I M E M).

(2.158)

Then we have C(M U

= C (.A4 ) = ( J.A 4 ) = IC ( J.A4

R),

C(M U (0)) = C(M) U (0) = 1C(IM) = JC(/),4 R).

(2.159) (2.160)

Proof The first equalities in (2.159) and (2.160) are part of (2.9). The equality C(M) = iC(Jm) follows from Definition 2.1 and Remark 2.18(b), observing that for any set M c X there holds M

if d < 0, (2.161)

Sd( — XX\M) — 1

X

d < +oo.

if 0

Furthermore, by (2.156) and (1.4), we have (x e X),

sup x m (x) 0> —00

(2.162)

ME.A4

whence, by (2.158) and Proposition 2.9(b), 0 E /C(N) (however, we need not have E C(M), as shown by Proposition 2.1(b)). Hence the equality C(M U {0}) = K(/m ) follows from Definition 2.1 and Remark 2.18(b), observing that {0

if d < 0 (2.163)

Sd(Xm) = M

if 0

d < +oo.

Finally, the last equalities in (2.159) and (2.160) hold by Proposition 2.10.

LII

Remark 2.28 (a) Theorem 2.8 can be also deduced from (2.132) of Theorem 2.7, observing that for any nonempty sets G, M C X and any x E X we have, by (2.156),


82

the equivalences G C M, X E X\M . — Xx\m(g)

= — 00

(g E G), — Xx\m(x) = 0

sup ( — Xx\m)(G) < G c M, x e X\M xm(g) = O (g e G), Xm(x) = -koo sup xm(G) < Xm(x).

(b) By (2.111), (2.157), and (2.161), we have S(Jm x R) = {Sd( — Xx\m)I(M , d) E M

X

R) = M U (X).

(2.166)

Similarly, by (2.111), (2.158), and (2.163), we have

S(Im x R) = {Sd(Xm)I(M, d) e M x R) = .A4 U {0 } .

(2.167)

(c) By (2.112) and (2.159) (with M = S(W x R)), we have

1C(W) = C(S(W x R)) = 1C(.1,s(wxR)),

(2.168)

where, by (2.157), (2.111), and (2.156), -1S(W xR) =

XX\Sd(w)

I (w,

d) E W

X

R) c {—oo, 0) x .

(2.169)

Similarly, by (2.112) and (2.160) (with M = S(W x R)), we have

K(W) U {0 } = C(S(W x R)) U 101 = k(is(wxR)),

(2.170)

where, by (2.158), (2.111), and (2.156), x R) = { Xsd(w)

I (w,

d) E W x R) c 10, -1-oo} x .

(2.171)

(d) By (2.8), (2.159), and (2.119), for any M c 2x we have

com G = coi, G = coim+R G

(G C 2 X ).

(2.172)

(e) From the above we see that the theories of M-convex sets, where M C 2X, and W -convex sets, where W C --R-x , are equivalent; indeed, by Definition 2.9, for each W c R x there exists M c 2x (e.g., M = S(W x R)) such that

K(W) = C(M),

(2.173)

and conversely, by Theorem 2.8, for each M c 2x there exists W c Rx (e.g., W = JA4 of (2.157)) such that we have (2.173). However, each of these theories has its own interest.

2.5 Other Concepts of Convexity of a Set X, with Respect to W c R x

83

(f) The family C(M) of all M-convex subsets of X, where M c 2 )c , can be represented in several ways as the family 1C(W) of all W-convex subsets of X, with Wc — R x . Indeed, if W is such a set, then, by (2.153), so is W R; also, by Theorem 2.8, one can choose the set of {0, —oo}-valued functions Jm of (2.157) and, when 0 E C(M), also the set of {0, +oo}-valued functions 'M of (2.158) to represent C(M).

2.4 A Particular Case: Closed Convex Sets Revisited Let X be a locally convex space, and let W = X* ; or equivalently, one could take W = X* ± R, the set of all finite continuous affine functions on X. Then for w c W, w 0 constant, the level sets Sd (W) occurring in Definition 2.9 are closed halfspaces in X (see (2.42)), and for w = constant they are either X or 0 (by (2.125) and (2.154)). Thus S(X* x R) = S((X* R) x R) = M U

IX, 0),

(2.174)

where M is the family of all closed half-spaces in X. Hence, by (2.112), (2.9), and (2.10), 1C(X*) = 1C(X* R) = C(A4 U {X, 0 } ) = C(M),

(2.175)

and thus by the remarks of Section 2.2b, part (1), a subset G of X is X*-convex (or, equivalently, (X* ± R)-convex) if and only if it is closed and convex. Alternatively, this also follows, for example, from (2.132), (2.153), and the strict separation theorem. Remark 2.29 By (2.175) and (2.159), we have

1C(X*) = /C(JA4 ),

(2.176)

where .A4 is the family of all closed half-spaces in X. Note that W = ./.A4 is a much smaller subset of R x than W = X*, with the property K( W) = C(M); on the other hand, we have X* C R x , while JAA C (0, —00} X (C R X ).

2.5

Other Concepts of Convexity of Subsets of a Set X, with Respect to a Set of Functions W C Rv

(1) By the remarks of Section 2.4, the theory of W-convex subsets of a set X, that is, of the sets G e 1C(W) = C(S( W x R)) where W c R x , may be regarded as an extension of the theory of closed convex subsets of a locally convex space X, i.e., of the sets G e 1C(X*) = C(S(X* x R)) where S(X* x R) is the family of subsets of X, consisting of X, 0 and all closed half-spaces (2.42) in X. One can use a similar procedure for the other families M C 2 x (instead of M = S((X* \ {0}) X R)) of the form (2.98), occurring in Section 2.2. Namely for any family (2.98) (where X is

84


a locally convex space), to which we adjoin Mod I d e R) = {X, 0 } , proceed as follows: Let X be an arbitrary set, and let W c T?x . Then, replacing (1) E X* \ {0} by_ w E W in the definition of the respective family (2.98), one obtains a family M w c 2 x of the form (

ffvt— w = {Mw,d C X w E W, dEPU { 0 }},

(2.177)

with P of (2.98), and one can regard the sets G e C(M) as being "convex" (with respect to the set of functions W). For example, in order to extend in this way the theory of evenly convex subsets of a locally convex space X, let X be an arbitrary set, W c Rx and

Mw = A(W x R) = {Ad(w) I (w, d) E W x R),

(2.178)

with Ad (w) of (2.106) (this extends (2.53), from (1). c X* \ {0} to w E W); then one can call the sets G E C(./qw), "evenly convex sets with respect to the set W," and prove corresponding results for them. For example, by (2.24), we have G E C(IV/i w ) if and only if for each x V G there exist w E W and d E R such that w(g) < d(g E G), w(x) d, or, equivalently, C(Mw) = {G c X I Vx V G, ]w

E

W, w(g) < w(x) (w E W)). (2.179)

Clearly, by (2.132) and (2.179), we have 1C(W) c C(M w ), where the inclusion may be strict. For example, if X is a locally convex space, then Mx* = A(X* x R) consists of X, 0 and all open half-spaces in X, and 1C(X*) and C(Mx) are the families of all closed convex and, respectively, all evenly convex subsets of X. (2) Let us also mention the following concept of convexity of subsets G of a set X, with respect to a nonempty set of functions W c Te, obtained by replacing the inequality by strict inequality < in the characterization (2.135) of the sets G E k(W). Definition 2.12 Let X be a set and 0 of X is W -convexlike if G = {x

Remark 2.30

W CTe. We will say that a subset G

e X I w(x) < sup w(G) (w

E

W)).

(2.180)

W);

(2.181)

(a) The inclusion c in (2.180) means that w(g) < sup w(G)

on the other hand, the inclusion exists w E W such that

D

(g

E

G, w

E

in (2.180) holds if and only if for each x g G there

sup w(G)

w(x).

(2.182)

By (2.132), every W-convex set satisfies the latter condition, but the converse is not true, as shown by W = {0) (or, by any open convex set G in a locally convex space X, and W = X*).

2.6 (W, cp)-Convexity of Subsets of a Set X, Where W Is a Set and cp : X x W

R

85

(b) If there exists a W-convexlike subset G of X, then, by (2.181), W contains no W). constant function (in particular, 0 W c R x ). (c) The empty set 0 is not W -convexlike (by (2.181), (1.1), and 0 Lemma 2.1

If G is a nonempty open convex set in a locally convex space X and W = X* \ {0}, then we have (2.181). sup w(G). If w(g) = sup w(G), Proof Let g E G and w E X * \ {Oh SO w(g) 0, take x E X such that w (x) < sup w(G), and let then, since w xx

1—X

1 —A

x

(0 < X < 1).

(2.183)

Then, since G is open and g e G, for sufficiently small X we have xx e G. But from w E X * \ {0), w(g) = sup w(G) and w(x) < sup w(G), we obtain w(xx)

1 = 1— A

w(g)

1—

w(x) > sup w(G),

in contradiction with xx E G. This proves (2.181).

El

Remark 2.31 (a) If G and W are as in Lemma 2.1, then, by the usual separation theorem, for each x V G there exists w E W satisfying (2.182). Hence, by Remark 2.30(a) and Lemma 2.1, every nonempty open convex subset G of X is (X* \ {0})convexlike. In the converse direction, by (2.180) for W = X* \ {0}, every (X* \ {0})-convexlike subset G of X is convex. However, an (X* \ {0})-convexlike set need not he open. Indeed, there exists (Klee {141}) a bounded closed convex (and hence (X* \ {0})-convex) subset G of a dense linear subspace X of a Hilbert space, with no support points (we recall that g o E G is called a support point of G, if w(go) = sup w(G) for some w E X* \ {0}), and hence satisfying (2.181) with W = X* \ {0}, so G is (X* \ {0})-convexfike. (b) For topological spaces X there are also other concepts of convexity of subsets of X defined with the aid of functions on X or with the aid of support functions (see

the Notes and Remarks).

2.6 (W, (p)-Convexity of Subsets of a Set X, Where W Is a Set and ço :X x 14/ —>R Is a Coupling Function Definition 2.13 (a) Let X and W be two sets. Then any function yo : X x W

R is called a coupling function. (b) Let X be a set and W ç Rx. Then the function yonat :X x W —> R defined by gOnat(X, w) = w (x)

(x e X, w e W)

(2.184)


86

is called the natural coupling function. Remark 2.32 (a) Coupling functions are a useful tool for the study of a pair of sets X and W, both in the general case and in many particular cases, as we will see in the sequel. (b) Often the form of the elements of a set X (or, of 2 x , etc.) suggests the choice of a set W, and of a coupling function q) : X x W —> R, suitable for the study of certain problems (e.g., see Section 8.4 below; or for some examples in combinatorial optimization, see [275] and [276]). Definition 2.14 Let X and W be two sets, and let go : X x W —> R be a coupling function. A set G C X is said to be convex with respect to the pair (W, 0, or, briefly (W, go)-convex, if

fl

G =

sd((p(.,

fix

w = ))

e X I yo(x , w)

d} .

(2.185)

WEW,c/eR

WEW,dER sup yo(é; ,w)c1

sup çp(g,w)cl

gcCi

g EG

We will denote by 1C(W, y)) the family of all (W, cp)-convex subsets of X. Remark 2.33 (a) In the particular case where W c Te and yo = ÇOnat, the natural coupling function (defined by (2.184)), condition (2.185) becomes (2.115). Hence (2.186)

k(W, (Pnat) = K(W),

with 1C(W) of Definition 2.9. Thus, W-convexity of subsets of X, where W c is a particular case of (W, 0-convexity. (b) The general case of (W, 0-convexity can be reduced to the above particular case by a suitable choice of a subset V = Vw, v, of R x . Namely, for any sets X and W and any coupling function go : X x W R, choosing (w

vw, ço = (10 ( . , w) Vw, 9, =

I w E W} =

E W),

w) I w c W} c R x ,

(2.187) (2.188)

a subset G of X is (W, cp)-convex if and only if it is V w , ço -convex; that is, we have /C(W, go) = /C(Vw, 92 ).

(2.189)

Indeed, this follows from Definition 2.14 and Remark 2.18(b). Note also that, by (2.112), we have 1C(Vw, v,) = C(S(Vw

,

x R)),

(2.190)

2.6 (W, cp)-Convexity of Subsets of a Set X, Where W Is a Set and ça : X x W —4

R

87

where, by (2.111) and (2.188), S(Vw, ç,, x R) = {Sd(vw, 99) I vw,99

E

Vw

d

E

R} (2.191)

= {Sd(40(•, w)) I w E W, d

E

R).

Thus the theories of (W, 0-convexity and V -convexity (where V C R x ), of subsets of X, are equivalent. Although the latter is simpler, (W, yo)-convexity is also useful, as we will see in the sequel. (c) Given any X, W and yo :XxW —> R, for the natural coupling function çonat : X x Vw4, —> T? (with Vw , of (2.188)) we have, by (2.184) (mutatis mutandis) and (2.187), Pnat(X , Vw,ço) —

(

Vw,y9(X)

=

(p(x , w)

(x

E

X, w E W).

(2.192)

Note also that in the particular case where W c R x , for yanat of (2.184) we have (2.193)

VW , (Pnat = W, whence, by (2.189) (or, alternatively, by (2.186)), we obtain k(141, (Pnat) = 1C(Vw, 99n„t ) = 1C(W).

(2.194)

(d) For two sets X and W and a coupling function yo : X x W —> R, the mapping w —> v,„, ço of W onto V = Vw. 99 (of (2.188)) need not be one-to-one, that is, the relations WI, w2 E W, (p(., w 1 ) = cp(., w 2 ) need not imply w 1 = w2. For example, W —> M is if (W, 0) is a parametrization of a family M Ç 2 x such that 9 not one-to-one (e.g., see (2.91)) and if yoo : X x W —> T? is the coupling function (2.203) below, then for any w 1 , w2 G W with w1 0 w2, 9(w1) = 9(w2), we have (P0( . , w 1 ) = (Po (. w2). (e) In the particular case where W is a linear subspace of R x , the mapping w —> (Pnat (., w) = w(.) (i.e., the identical embedding of W into R x ) is linear, i.e., Pnat X x W —> R is linear in the second variable (but it need not be bilinear when X is also a linear space). More particularly, when X is a linear space and W is a linear subspace of X # , (Pnat : X x W —> R is bilinear. Often the (equivalent) converse way is followed; namely, given two linear spaces X and W, one "sets them into duality" with the aid of a bilinear coupling function (•, •) : X x W —> R, requiring that the relations (x, w) = 0 (x e X) should imply w = 0 (and, often, also that the relations (x, w) = 0 (w E W) should imply x = 0). Thus, the general coupling functions yo permit to extend the usual (bilinear) duality. (

By Remark 2.33(b), (c) above, one can obtain results on general (W, 0-convexity of subsets of X from the results on W-convexity, where W C R x , simply by replacing w(x) by yo(x, w). For example, Proposition 2.9(b) extends as follows: We have 0 e k(W, ço) if and only if sup yo(x, w) > —oc /Dew

(x

E

X).

(2.195)


88

Furthermore, defining the (W, cp)-convex hull co vv4 G of a set G c X in the obvious way, Theorem 2.7 extends to co vv, 9, G = {x e X 1 w E W, sup cp(g, w) < cp(x , w)} gEG

=

E

XIV (w,d)

E

W x R, go(x, w) > d, 3g

E

G, cp(g,w) > d}

(G c X), 1C(W , cp) = IG c X IV x

G, 3w E W, sup cp(g, w) < cp(x , w)I.

(2.196) (2.197)

gEG

Also one can extend Definition 2.11 as follows: Definition 2.15 Let X and W be two sets and cp:XxW—>T2. (a) For any subset G of X the function o-G4 e T?w defined by aGn ,p (w) = sup cp(g, w)

(w E W)

(2.198)

gEG

is called the (W, 0-support function of G. (b) For any subset G of X, the subset G" of W defined by Gfi' 9' = (w c W I sup cp(g, w) < +o0 )

(2.199)

gEG

is called the (W, cp)-barrier set of G. In the sequel we will often use the following important class of coupling functions:

Definition 2.16 Let X and W be two sets. A coupling function yo : X x W —> T? is said to be of type {0, —Doh if ço (X x W) c (0, —co ) , that is, if yo can assume only the values 0 or —oc. Now we will give a result that shows that the theories of (W , cp)-convex subsets of X, where cp : X x W —> R is a coupling function of type (0, —oc) and M-convex subsets of X for M c 2 x parametrized by (W, 0) (with the same W), are equivalent. Theorem 2.9 Let X and W be two sets. (a) For any mapping 0 : W —> 2 x there exists a coupling function cp 9 : X x W ---> R of type (0, —oc) such that 1C(W , cp 0 ) =

(W)),

(2.200)

where 0(W) = (9(w) 11D

E

WI.

(2.201)

2.6 (W, (p)-Convexity of Subsets of a Set X, Where W Is a Set and ça : X x W —> k

89

(b) For any coupling function (p :X x W —> I? of type {0, —co), there exists a mapping Oço : W —> 2 x such that (2.202)

1C(W , (p) = C(0(9 (W)).

Proof (a) Given 9 : W —> 2x , define (po : X x W --- > R by (x E X, W E W).

(P0(x, w) = — Xx\o(w)(x)

(2.203)

Then (po is of type {0, —oc), and, by (2.161) (with M = 0(w)), we have 0(w)

if wE W,d< 0,

X

if w E W, 0

SA(p9(•, w)) = d < +Do,

whence, by Definitions 2.14 and 2.1 (for M = 0(W)), we obtain (2.200). Alternatively, one can observe that for (po of (2.203)) the set Vw,,p, of (2.188) becomes V W , Soo —

{ — XX\H(w)

I WE

W) —

(2.205)

JO(W),

whence, by (2.189) and (2.159) (for M = 0(W)), we obtain k(W , (Po) = k(Vw ) ) = k(J0(w)) = C(O(W)).

(b) Given (p : X x W —> R of type (0, —co), define 9,p : W --> 2 x by 0,p (w) = fx E

X

I (p(x, w) (

—1} = Sli ((P ( . , w))

(w E W).

(2.206)

Then, since ço is of type (0, —oc), for 999 of (2.206) and (po :XxW-->1? of (2.203) we have, by (2.156), (P(x, w) = — X{yEx 199(y,w) , _1}(x)

(2.207) (x E X, W E W),

= — Xx\o(w)(x) = (Po w (x , V))

and hence, by part (a), we obtain (2.202).

0

Remark 2.34 (a) For yoo of (2.203) we have (P0(x , 10 — (Pnat(X, — XX\O(w))

(x E X, w E W),

(2.208)

where (p flat : X x ./9(w) —> R is the natural coupling function (with Jo(w) c R x of (2.205)). (b) Given 9 : W --> 2 x , for (po : X x W --> T? and 9, : W —> 2x of (2.203) and (2.206) we have, by (2.156), 0(w) = (x E X I — Xx\o(w)(x) ,-. yoo (of (2.203)) and yo --> Oço (of (2.206)) give a one-to-one correspondence between multifunctions 0 : W ----> 2' and coupling functions yo :X x W—> R of type {0, —oc). (c) If X is a set and W c R x , then the family .A4 = S(W x R) c 2x of (2.111) admits the natural parametrization (14-7 , 0), where = W x R, 0(w, d) = Sd(w)

((w, d)

(2.210)

E

Then the coupling function (p o : X x W --> T? of (2.203) becomes 409(x, (w, d)) = — Xx\o(w,d)(x)

(2.211) —

X{yEx I w( y )>d}(x)

(x E X, w E W,d E R),

and for this çoe we have, by (2.200), 94) = S(W x R) and (2.112), /C(W x R, (p9) = k( 171" (po) = C(9( 2 )) = C(S(W x R)) = )C(W)•

(2.212)

Let us also note that, in particular, if X is a locally convex space and W = X* \ (0), then )14 = S(W x R) is the family of all closed half-spaces in X, and the parametrization (2.210) of .A4 reduces to (2.91). (d) If X and W are two sets and yo : X x W —> R is a coupling function, one can extend (c) above by applying it to V w4, c —/ix of (2.188). Namely, taking A/1 = S(Vw4, x R) c 2x of (2.191) and parametrizing it by = W x R,

(w, d) = Sd(v„, 4 ) = Sd((P(. , w))

((w, d)

E 1/17/), (2.213)

the (0, —00)-type coupling function (po of (2.211) will be replaced by *0 (x, (w, d)) = — X{yEx I ça(y,w)>d}(x)

(x

E

X,

W E

W,d

E

R),

(2.214)

and (2.212) (for Vw4 ) yields k(W x R, *ow) = C(0q) (W x R)) -= C(S(V w , q, x R)) = k(Vw,O.

(2.215)

(e) One can define : W --> 2 x by (2.206) not only for yo : X x W —> of type R. Then the mapping ço ---> (0, —oc) but for any coupling function ço : X x W 2x of is no longer one-to-one, and for any yo : X x W --> R and Ow : W x R (2.213), we have 60 (w , —1) = 0,p (w)

Note also that, in particular, if ço : X x W —> (0, —oc), then 0 9' of (2.213) becomes E

{

X I (p(x, w)

E

T?

(2.216)

W).

is a coupling function of type

—1)

if d < 0, (2.217)

6 (w, d) = ScA(P( . , w)) = X

if d

0,

2.6 (W, (p)-Convexity of Subsets of a Set X, Where W Is a Set and (,c) : X x W --> R

91

0. so Oç °(w, d) = O(w) of (2.206) if d < 0, and 6w (w , d) = X if d (f) If X and W are two sets and yo : X x W —> ii is a coupling function, one can extend the family (2.178) to

A(Vw, ço x R) = {Ad(4 0 (*, w)) I (w, d) E W

X

R},

(2.218)

and then one can use (2.218) to extend C(Mw) of part (1) of Section 2.5 (from W C R X to any (W, (p)).

Chapter Three

Abstract Convexity of Functions on a Set For an arbitrary set X we will consider now the complete lattice E = (Rx , functions f : X —> R, with the usual pointwise order defined by

f2 fi (x)

fi

E X),

f2(x)

whence the lattice operations sup and inf in E = (Te , well; that is, for any index set I, (sup fi )(x) = sup fi (x), (inf fi )(x) = inf fi (x) iEi

iEi

of all

(3.1)

act pointwise on X as

(x e X).

(3.2)

LEI

lEi

Also the greatest (resp. the least) element of E is the constant function f (resp. f —co).

+cc

3.1 W-Convexity of Functions on a Set X, Where W C In this section we will restate some of the concepts and results of Chapter 1 for the complete lattice E = (Rx , using (3.1) and (3.2), and for suitable infimal generators of E. To this end, as in Sections 2.3-2.5, it will be convenient to denote an arbitrary subset of R x by W (instead of the notation M used in Chapter 1). For E = (Rx , (and M = W), Definition 1.1 yields:

Definition 3.1 Let X be a set and W c R x , i.e., a set of R -valued functions on X. A function f : X —> R is said to be convex with respect to the set W, or, briefly, W -convex, if

f = sup {w

G WIW 92

f),

(3.3)

3.1 W-Convexity of Functions on a Set X, Where W c kx

where sup and

93

are in the sense (3.2), (3.1), i.e., if f (x) =

sup w(x) wEw w(y)f(y ) (YE X)

(x

E

X).

(3.4)

We will denote by C(W) the set of all W-convex functions on X: C(W) = if E R x If = sup{weWlw

f)).

(3.5)

By Proposition 1.2, there holds: Proposition 3.1 (a) C(W) is a convexity system in (R x , - R, the W -convex hull off is the function fco(w) : X —> R defined by Definition 3.2

fco(w) = max th

E

C(W) I h .._ f),

(3.7)

that is, the greatest W-convex minorant of f (observe that we use the notation fco(w) instead of the notation co w f of (1.15)). Applying Definition 1.4 to E = (iix , ., ) , we arrive at Definition 3.3 Let X be a set. A hull operator y on (Ttx ,

Te ---> 17?x satisfying for any f, 7

) is a mapping v :

Rx,

E -

(3.8)

It)

(3.9)

f,

(3.10)

(fv)v = f , ,

where we use the notation fi, instead of v( f). Theorems 1.1 and 1.2 yield that f --->- f,„(w) (of (3.7)) is a hull operator on (--k x , ) and that for any f E r?)( we have fco(w) = sup (w E W lw

f).

(3.11)

94

Abstract Convexity of Functions on a Set

Also, by Corollary 1.2, C(W) is the smallest subset of 1-i x , closed for sup and containing W.

Remark 3.1 It may happen that for each xo E X there exists wx,, tvxo ( f and wx„(xo) = f (x0), so

f(x 0 ) = max (w(x0 ) e T? I w

E

W,w , f)

(x 0

E

E

X),

W such that

(3.12)

but at the same time fco(w) W, i.e., the sup in (3.11) (which is taken in W )o ) is not attained for any wo E W (e.g., when X is a locally convex space and W = X* + R, as shown by Theorem 3.7 below). Let us recall that the operations ± and ± (of upper addition and, respectively, lower addition) on R are defined by (a, b

a -± b=a±b=a+b a --i- (±oo) = (H-oo) ± a = 4-oo

E

R),

(3.13)

(a e R),

a ± (—oo) = (—oo) H.-- a = —oo

(a

E

R);

(3.14) (3.15)

as usual, we will keep the notation a ± b also for a, b E R with R n {a, b) 0 0. These operations are extended pointwise to R x (where X is any set): by definition, (f

-.

F . h)(x) = f (x) 4- h(x), (f H.- h)(x) = .f (x) H.- h(x)

(x c X). (3.16)

In order to give some basic examples of infimal (supremal) generators in E = (— R x , ,), we need the following lemma, which "corresponds" to formula (2.22):

Lemma 3.1 For any set X and any function f : X --->- R, we have

f = inf {x {x) 4- f (x)). xEx

(3.17)

Proof By (2.156) and (3.13), (3.14), there holds

X{x}(Y) j - f (x) =

0 ± f (y) = f (y)

if x = y,

+Do --i- f (x) = +Do

if x 0 y,

whence (y c X).

f(y) = inf {X{x}(Y) + i c (x)) xeX

El

Remark 3.2 (a) If X is any set, then, by Lemma 3.1, the set Y = {X{x} + d i (x, d)

E

X

X

R} c R x

(3.18)

3.1 W-Convexity of Functions on a Set X, Where W c R x

95

Hence, by Proposition 1.6(a), so is the set is an infimal generator of E = (p , Y. Y \ { 4- oc}. Note also that —co R we have (b) Similarly to Lemma 3.1, for any f : X '

f = sup {—x {x} ± f (x)), xEx

(3.19)

and hence the set

= t—x (x ) ± dI (x, d) e X x

c

(3.20)

Hence, by Proposition 1.6(b), so is the set is a supremal generator of E = (Rx , \ {—oc). Note also that +oo (c) For any set X and any function f E R X we will denote by Epi f, respectively, Hypo f, the epigraph and, respectively, the hypograph of f; we recall that these are the subsets of X x R defined by

Epif = {(x, d) eXxR1 f(x)

(3.21)

d),

Hypo f = ((x, d) E X x Rlf(x) ?, d).

(3.22)

Then one can replace (3.17) and (3.19), respectively, by f=

f=

inf

(x.d)E Epi f

sup (x,d)EH ypo

f

(x{x ) + d),

(3.23)

( — x{x} +

(3.24)

d);

indeed, (3.23) has been shown in the proof of Theorem 0.1, implication 3 0 and the proof of (3.24) is similar. Hence the sets

1°,

Yo = {X{x} + d 1(x, d) E X x R) c (R U 1+000 x ,

(3.25)

} = 1—x(x) + d I (x, d) E X x R) C (R u {-00 ) x ,

(3.26)

To

are also an infimal and, respectively, a supremal generator of E = (Rx , Note that Yo eliminates from Y \ {-Hoc} of (a) the functions xfz) + (—Do) E where x E X (and a corresponding remark holds for Yo and i; 1—oc) of (b)). Both infimal generators Y and Yo will be useful in the sequel. (d) Since x (x) d —> (x, d) is a one-to-one mapping of Yo (of (3.25)) onto X x R, we can identify Yo with X x R. Under this identification, (1.45) is nothing else (by the equivalence (3.32) below) than the mapping f —> Epi f of R x onto the family E(R x )=1McXxRi3fER x ,M=Epifl = {Epi f If E R x ) (3.27)

of all epigraphic subsets of X x R (in the sense of Definition 3.4 below). Note also that for the family of all lower conical subsets with respect to 170 , that is, (see Remark 1.8(e)), for lz_1 (4) = {{ — x{x} +dix

E

X, d

E

R, —x{x} + d

)9 I f e

Tel,

(3.28)

96


under the above identification the corresponding mapping w is f —> Hypo f, and it satisfies

w(inf fi ) = n co(fi )

R x ).

(3.29)

tEl

Theorem 1.3 and Remark 3.2 yield now:

Theorem 3.1 Let X be a set and W feo(w) = inf {x{,-) +

C

1-2 ) . Then 1(x, d)

E X X

= inf {x{x} +did e R, X C(W) = ff

E RX

I V (x, d)

R, WE W, f

n

E

E X X

w, w(x) > d) (f

Sd(w)}

R, f (x) > d, ]w

E

E R X ),

(3.30)

W,f w, w(x) > d).

(3.31) Proof Observe that for any w c Rx and x(x ) + d E Y0 (of (3.25)) we have, by

(2.156), the obvious equivalence W

(3.32)

X{x) ± d w(x) > d.

Hence, by (1.51) (with ,A4 = W and Y = Y0) and (3.32), we obtain the first equality in (3.30); the second equality in (3.30) is obvious. The proof of (3.31) is similar, using (1.52).

Remark 3.3 (a) According to Remark 1.9(a) and formula (3.32), the conditions (3.33)

w, w(x) > d

f

can be expressed by saying that w separates f from x{ x } ± d. (b) Formula (3.31) can be deduced directly from (3.11). It can be also expressed in terms of level sets of functions, namely as C(W) = if

E RX

I Vd e R,V x g Sd(f ), Dw

Sc(f) g Sc.(w)

(C E

R), x

E

W,

(3.34)

Sd(w)i•

(c) In terms of level sets, we also have C(W) = ff c R x I Vd

E

R,Vx Sd(f), 3w

Ac(f) g A(w) (c E R), x

E W,

(3.35)

g Ad(tv)).

Indeed, by (3.31), we have the inclusion C in (3.35). Observe now that if A(f) g A c (w) (c E R), then w f. Hence, if f belongs to the right-hand side of (3.35)

3.1 W-Convexity of Functions on a Set X, Where W c Rx

97

and X V Sd(f), then, choosing s > 0 with d d; thus, by (3.31), f E C(W).

re,

W-convexity of functions on X has been defined For a set X and for W c above as convexity with respect to the subset W of the complete lattice of functions (— Rx, Let us show now that W-convexity of functions on X is equivalent to (and hence it can be defined as) convexity of their epigraphs with respect to the family of sets E (W) in the complete lattice of sets (2X xR , D), defined by

e(w) =

{M c X x RI3w E W, M = Epi w} = (Epi wiwE W};

(3.36)

in other words, E(W) is the family of the epigraphs of the functions belonging to W we have used it in Remark 3.2(d)). (for W = Proposition 3.2 Let X be a set, and let W c

. Then

f E C(W) R defined by (3.40) satisfies ktm c C(W) and (3.41). Indeed, the "only if" part has been shown in the above proof of Theorem 3.2. Conversely, if ,u m of (3.40) satisfies Am E C(W) and (3.41), then, by Proposition 3.2, we obtain M = Epi ,u, m E C(E(W)). Theorem 3.3 For any f : X —> R we have Epi (fc0(w)) = cos(w)(EPi f ),

(3.42)


99

and hence

inf

fc0(w)(x) =

(x,d)Ecoe ( w ) (Epi f)

d

(x

E

(3.43)

X).

Proof By (3.11), (3.38), (3.36), and (2.19), we have

Epi (fc0(w) ) = Epi ( sup w) = WE W

fl

Epi w

WE W

wf

w R is different. Indeed, by (1.9) (in E = (Rx , ,)), we have C(W)

C

C(W

(3.44)

R),

but the inclusion may be strict (e.g., see Section 3.2). We now give a characterization of (W R)-convex functions. To this end, we will use the well-known identification of each pair (f, r) E R x x R with the function (f, r) E R x x R defined by (f c R x , x

(f, r)(x, d) = f (x) + rd

E

X, d, r

E

R).

(3.45)

Theorem 3.4 Let X be a set, and let W

C — Rx .

Then

S(W R) = S((W x {-1}) x R), C(E(W

R)) = 1C(W x —1}),

(3.46) (3.47)


100

f E C(W Proof For any w E

R) R are the (W x {-1})-convex subsets of X x R. (b) For further results on (W + R)-convex functions, see section 8.3 below. Let us now prove the following fact, which has been used in the above proofs of Theorems 3.2 and 3.3 and in Remarks 3.4 and 3.6(c): Proposition 3.3 Let X be a set and f : X —> W. Then f is uniquely determined by its epigraph Epi f, namely

f (x) = inf

(x,d)EEpi f

d (= PEpi

E X).

(X

f (X))

Hence f —> Epi f is a one-to-one mapping of R x into 2x x R

(3.51)

.

Proof For any x E X we have, by (3.21),

Id

E

R I (x, d)

E

0

if f (x) = +oc,

Epi f} = /R

if f (x) = —oc,

[f (x), +oc)

(3.52)

if f (x) E R,

whence (3.51) follows (using (1.1) when f (x) = +oc).

0

Remark 3.8 (a) The above one-to-one mapping f —> Epi f has also some nice algebraic properties, such as (3.38) above, and

Epi (inf (fi, .f2)) = (Epi fi ) U (Epi f 2 ).

(3.53)

3.1 W-Convexity of Functions on a Set X, Where W c k's'

101

(b) Proposition 3.3 permits to characterize (or to define) various classes of functions by properties of their epigraphs (e.g., see (3.37), (3.48), Remark 3.11, and Lemmas 3.2, 3.3 below). In connection with Proposition 3.3, it is natural to give: Definition 3.4 Let X be a set. A subset M of X x R is said to be epigraphic, if there exists a (unique) function f : X —> T? such that M = Epi f. The next proposition gives the range of the above mapping f —> Epi f by characterizing the epigraphic subsets of X x R: Proposition 3.4 A subset M of X

xR

is epigraphic if and only if it has the following two properties:

(x, d) (x, dn )

E

E

M = {x} x (d ± R + ) c M,

M (n , 1,2, . . .), lim d„ = d n ---> - oo

E

R

(3.54) (x, d)

E

M,

(3.55)

where R + = [0, -koo). Namely in this case we have

M = Epi A m ,

(3.56)

with A m : X —> R of (3.40). Proof If M = Epi f for some (unique) f

c R x , then, by (3.21), M satisfies

(3.54) and (3.55). Conversely, we will show now that if M is a subset of X x R satisfying (3.54), (3.55), then we have (3.56), with ,u, A4 : X —> R of (3.40). Indeed, by (3.40), we have M c Epi ,u m . On the other hand, let (x, d) E Epi ,u m (i.e., d E R and d), and let Idd c R be any sequence such that (x, dn ) E itm(x) = inf(X,d)Em d' M (n = 1, 2, ...), limn , dn = Am (x)• If Am (x) = —oc, then, by d E R, we have p. m (x) < d, so there exists d' < d with (x, d') E M, whence, by (3.54), we obtain (x, d) G M. If ,u m (x) > —oc, then, by (3.55), we have (x, p, m (x)) E M, and hence, by ,u m (x) d and (3.54), we obtain (x, d) E M. Thus Epi p, m c M, which, together with the opposite inclusion observed above, yields (3.56), so M is an epigraphic subset of X x R. 0 Remark 3.9 Geometrically (3.54) and (3.55) mean that for each (x, d) E M the set M contains the whole "vertical" half-line "above" (x, d) and that the intersection of M with each "vertical line" {x) x R is closed in X x R. Let us return now to abstract convexity. One can use fco(W) to localize the concept of W-convexity of functions, as follows:


102

Definition 3.5 Let X be a set, W c Te and x0 E X. A function f : X said to be W-convex at xo if

f (x0) = fc0(w) (xo)•

T? is

(3.57)

We will denote by C(W; x o ) the set of all such functions:

C(W; xo) = if

I

E R X f(X0) = fco(W)(X0)}.

(3.58)

Since C(W) = If E -k x I f = fco(w) } (by (1.23) in E = (Te, ,)), we have C(W) =

n c(w;

(3.59)

x0);

X0 EX

that is, a function f : X —> R is W-convex if and only if it is W-convex at each xo c X. One can give localized versions of some of the above results. For example, there holds the following localized version of (3.31):

Theorem 3.5 We have

c(w ; x0) = If

E R X IVdE

R,

f(x0) > d, Rw E W, f w, w(x0) > dl.

(3.60) Proof If f c C(W; x0) and d E R, f(x 0 ) > d, then, by (3.11) and (3.58),

sup w(x0) = fc0(w)(xo) = f(x 0) > d,

wew if

so there exists w E W with f w, w(x0) > d. Conversely, if f (x0) fc0(w)(4), then, by (1.31) (in E = (Rx , ()) and (3.11), there exists d c R such that

f(x 0 ) > d > Jeco(w)(x0) = sup

W(X0),

wcW W f

and for this d there exists no w E W satisfying f not belong to the right-hand side of (3.60).

w, w(x0) > d, that is, f does

In Section 2.3 we defined, for any sets X and W c Te and any subset G of X, the W-support function ŒG E Tel of G (by (2.146)), and we showed that the restriction of the mapping a- : G E 2X —> E Te to the family /C(W) of all W-convex subsets of X is one-to-one (see Remark 2.26). Now we will determine the functions h E Rw which are support functions of some G C X or of some (unique) G E /C(W), and the inverse mapping (a I 1C(14) ) -1'


103

Theorem 3.6 Let X be a set, let W C Rx , and let K =- Kw be the canonical mapping (2.140) of X into R w . Then for a function h G R W the following statements are equivalent: 10. 2'.

There exists a set G C X such that h = o-G . There exists a (unique) set G E k(W) such that h = o-G . h E C(K(X)).

3°.

The inverse of the one-to-one mapping G E 1C(W) —> aG E C(C(X)) is the mapping h E C(K(X)) —> G = fx G X I w(x) Proof 1'

h(w) (w E 147 )} E /C(W).

(3.61)

3 0 . If 1 0 holds, then, by (2.146) and (2.140), we have

h(w) = o-G (w) = sup w(G) = sup K(g)(w)

(w E W);

EG

that is, h = supg , G K (g), where K(g) E K (X) c C(K (X)). Thus, by Proposition 3.1(a) (mutatis mutandis), h G C(K (X)). 30 = 2°. If h e C(K (X)), define G as in (3.61). Then, by Corollary 2.3, we have G E /C(W). Also, by h E C(K(X)), (2.140), (3.61), and (2.146) we obtain for any W E W,

h(w) =

sup C (X)EK

K(x)(w) =

(X)

sup X EX

w(x) = sup w(G) = aG(w)• (ivEW)

K (X) R w of (2.140) is one-to-one, SO K(x) may be identified with x for each x E X. First, note that in this case the Xsupport function op of any subset P of W is, by Definition 2.11 (mutatis mutandis), the function p(X) = Crp(K (X)) = sup K(x)(p) = sup p(x) PEP

pE P

(x E X),

(3.62)


104

and hence, by Definition 3.1, a function f : X —> R is W -convex (i.e., f E C(W)) if and only if f = au (f) with U (f) of (3.6). Moreover, by Remark 1.1(c), we have f E C(W) if and only if there exists a (possibly empty) set P C W such that f = Cfp; in other words, the W-convex functions f : X —> R are nothing else than the support functions of the subsets P of W. Now Theorem 3.6 yields the following sharpening of these observations: Corollary 3.1 Let X be a set, and let W C R x be such that the canonical mapping lc = KW : X —> R w of (2.140) is one-to-one. Then for a function f E R x the following statements are equivalent:

10. 20 . 3°.

There exists a set P C W such that f = ap. There exists a (unique) set P e 1C(K (X)) = IC(X) such that f = o- p.

f E C(W).

The inverse of the one-to-one mapping P E 1C(X) —> o - p E C(W) is the mapping

f E C(W) —> P . (w E Wlw “) G /C(X).

(3.63)

Proof Apply Theorem 3.6 with X and W c T?x replaced by W and K (X) =

X

C

R w , respectively.

El

In Chapter 2, for any family of subsets M of X we have considered the family of functions J.A4 ± R c .7R x with JA4 of (2.157). It is now natural to ask: What is the family C(Jm ± R) of all (Jm ± R)-convex functions on X? We will give the answer in Section 4.4 (see Theorem 4.9).

3.2

Some Particular Cases

3.2a C(X* ± R), Where X Is a Locally Convex Space If X is a locally convex space, then, by Definition 3.1 (for W = X* ± R) and Remark 1.1(c), a function f : X —> R is (X* ± R)-convex if and only if there exists a (possibly empty) set Wf c X* ± R such that

f= sup (4) — r ),

(3.64)

(1)-rEW1

that is, if and only if f is the (pointwise) supremum of a (possibly empty) set of continuous affine functions.


105

Remark 3.11 For any (13. — r E X* ± R we have, by (3.49), Epi (4) — r) =

, d) EX x RI (1)(x) —

d) (3.65)

= {(x, d) EXxRI ((I), —1)(x, d) ( r), which is a closed half-space in X x R, bounded by the "nonvertical" closed hyperplane

H = {(x, d) E X x RI (ti, —1)(x, d) = r)

(3.66)

(the "vertical" closed hyperplanes in X x R are, by definition, the sets of the form {(x, d) E X x RI ((I), 0)(x, d) = r} = ((x, d) E X x RI (13.(x) = r), where ci) E X* , 0, r E R). Moreover (3.65) is "the upper closed half-space" bounded by H, in the sense that if (x, d) E H, then (x} x (d ± R+) C Epi (CD — r). Conversely, every nonvertical closed hyperplane H = {(x, d) E X xRI (4; , s)(x , d) = 79 in and hence X x R can be written in the form (3.66) (by taking (1). = — C1, r = — every upper closed half-space in X x R bounded by a nonvertical closed hyperplane is the epigraph of some continuous affine function (I) — r E X* ± R. Thus, by (3.38), condition (3.64) admits the following equivalent geometric formulation: we have f c C(X* R) if and only if Epi f is the intersection of a family of upper closed half-spaces bounded by nonvertical closed hyperplanes. R is called: (a) convex, if We recall that a function f : X

f (Xx ± (1 — À).,7)

Àf

(X)

(1 — À)f (Y)

(b) proper, if f # +Do and f E (R U {±

(x,

E X, 0

X

1); (3.67)

}) x (i.e., f (x) > —oc for all X E X).

Lemma 3.2 Let X be a locally convex space and f E R X . (a) f is convex if and only if Epi f is convex (in X x R).

(b) f is lower semicontinuous if and only if Epi f is closed. (c)f is proper if and only if Epi f 0 and Epi f contains no vertical line (x) x R, where x E X. Proof (a) If f is convex, (x i , d1 ), (x2, d2) c Epi f (i.e., f(x i ) 1, then À d2 E R) and 0 R, f (x2) f (Xxi + (1 — x)x2) X.Rx i) + (1 — x) Px2) Ad

di E

1 + —

whence X(x , (11) + (1 — À)(x2 , d2) = (Xxi + ( 1 — X)x2, Ad 1 ± (1 — X)C12) E Epi f , so Epi f is convex. Conversely, assume now that Epi f is convex, and let x, E X, 0 À 1. If f (x) = +x or f() = +co, then (3.67) holds. If both f (x), f ('e') < +oo, then (x, d), a, (7) E Epi f for all d > f (x), d > f() (including also the cases when f (x) = —oc or f = —oc). Therefore, since Epi f is convex, for all


106

d > f (x) , 21 > f() we obtain (Ax ± (1— X)3 , Xd ± (1— Â.);1) = X(x , d)

(1 — A)(,)

E

Epi f,

that is, f (Xx ± (1 — X)5 ) Ad ± (1 — A)cl, whence we infer (3.67). (b) Assume that f is lower semicontinuous. That is to say for each xo e X and do E R with do < f (xo) there exists a neighborhood Vd„ (x0) of xo such that • < (x) for all x E Vd„(X0). Hence, if do < f (x0), then, choosing E > 0 such that do ± E < f (x0), we have d < f (x) for all (x, d) E Vd0 +8 (4) X (do — E, do ± 8), which proves that (X x R) \ Epi f is open, so Epi f is closed. Conversely, assume now that Epi f is closed, so (X x R) \ Epi f is open. Then for any (x0, d()) with do < f (x 0 ) there exist a neighborhood V(x 0 ) of xo and £0 > such that d < f (x) for all (x, d) G V(X0) X (do — E, do ± e), whence do < (x) for all x G V (x0); that is, f is lower semicontinuous. Finally, (c) is immediate from the definitions.

Theorem 3.7 Let X be a locally convex space. For a function f : X are equivalent:

1'. 20 .

R the following statements

f E C(X * R). Either f +oo, or f is a proper lower semicontinuous convex function.

Proof 1 = 2'. If f E C(X * R), f # —oo, that is, if (3.64) holds with 0 W f C X* ± R, then, by (3.38), we have Epi f =RD-rEw EPi ( 4) — r), where for each IT, — r E Wf, Epi (14) — r) is the upper closed half-space (3.65) in X x R. Also, if f ±oo, then Epi f 0. Hence, if f +oo, then Epi f is a nonempty closed convex subset of X x R containing no line {x} x R, where x E X. Consequently, by Lemma 3.2, f is a proper lower semicontinuous convex function. 2' V. If f —cc, then for Wf = 0 we have (3.64). If f _ -= +00, then for WI = R we have again (3.64). Hence in both cases f E C(X * R). Assume now that f +oo is as in 2'. Then, by Lemma 3.2, Epi f is a nonempty closed convex subset of X x R containing no vertical line {x} x R where X E X. Let (x, d) E (X x R) \ Epi f. Then, by the strict separation theorem, there exists (11), r) G X* x R (X x R)* such that sup(, r)(Epi f) < (4), r)(x, d), that is (by (3.45)), 0

sup (c1). (.7)

(3.68)

rcl) < (I)(x) ± rd.

(.7,2-1)eEpi f CASE

(x) < -F oc. Then (x, f(x)) c Epi f, whence, by (3.68), (I)(x)

rf (x)

sup ( (7) ± rd) < )(x)

rd,

(3.69)

(i,j)EEpi f

and hence r < 0 (since if r 0, then, by the assumption d < (x), we get rd r f (x), in contradiction with (3.69)). Therefore, denoting the left-hand side of (3.68)


107

by p and multiplying both sides of (3.68) by — (> 0), we obtain that the relation (.7C, "ci) E Epi f implies that — (1) (V) — d — p, i.e., — cl)(i) ± p d so ,

Epi f c Epi (—

(I) ± TI /3), and —

i6 , so (— (13 + /3) (x ) >

(x) — d > —

d. f (x) = +oo and r O. By (3.68) and since the relation (V, d) E +oo) implies (Z, t) E Epi f for ali t 0, we must have 0, by f Epi f r < 0. Then we arrive at the same conclusion as in case 1. CASE 2:

CASE 3: f (x) = +00 and r = 0. Then, taking any X0 E X with —cc < f (xo) < -1-00 (which exists, by our assumptions on f) and any do < f (x 0 ), from case 1 0 above it follows that there exists a continuous affine function, say, (Do ro E X* + R, such f (and (1) 0 (x0) ro > do). Also, by r = 0, we have for (I) and fi of that (Do ro case 1, (I) (i) — /3 -.“) (.3c- E dom f). Hence for any 0 < c < +00 the continuous affine function we = (Do + ro + c(cD — fi) e X*

R

(3.70)

satisfies w r (.5) E dom f), and clearly wc (V) d-oo = f(') (.5cf(') X \ dom f), so we ( f. On the other hand, by p < .4)(x) + rd = 0(x), for any sufficiently large c we have we (x) > d. Consequently, by the conclusions of cases l-3 and by Theorem 3.1, formula (3.31) (with W = X* + R), it follows that f e C(X* R). Remark 3.12 (a) By Section 2.2b, part (1), Epi f is a closed convex subset of X x R if and only if it is the intersection of a family of closed half-spaces of X x R. By Theorem 3.7, Lemma 3.2, and Remark 3.11, we have that Epi f is a closed convex subset of X x R containing no vertical lines (x) x R if and only if it is either empty or the intersection of a family of upper closed half-spaces bounded by nonvertical closed hyperplanes. (b) One can obtain results on C(X* R) either from those on C(W), applied to W = X* ± R, or from those on C( W R), applied to W = X*. For another strong tool (namely, biconjugates), see Chapter 8. Since the (X* + R)-convex functions on a locally convex space X (which are closely related to the usual convex functions f : X —> R, as shown by Theorem 3.7) are those of the form (3.64) where (I) — r are continuous affine functions, it is natural to introduce the following terminology: Definition 3.6 Let X be any set and let W c R x . Then the functions in W that is, the functions of the form w+r are called W-affine functions.

(W E

W, r

E

R),

R,

(3.71)

108


Remark 3.13 (a) By Definitions 3.1 and 3.6, a function f : X —> R is (W R)convex if and only if it is a supremum of a set of W-affine functions. Some authors call "W-convex functions" what we call here (W R)-convex functions (see the Notes and Remarks). (b) The functions w E W are sometimes called W-linear functions. (c) The functions w±r E W±R are called W -elementary functions [205], [206], and for any f : X —> R the function fc„(w+R) is also called the (W R)-regularization (or -regularization [205], [206]) of f.

3.2b C(X*), Where X Is a Locally Convex Space If X is a locally convex space, then, by Definition 3.1 (for W = X*) and Remark 1.1(c), a function f : X —> R is X*-convex if and only if there exists a (possibly empty) set Wj c X* such that f= sup (1),

(3.72)

1) E WI

(

that is, if and only if either f —oc or f is the (pointwise) supremum of a nonempty set of continuous linear functions.

Remark 3.14 (a) Note that we have +00 E C(X * + R), but +cc g C(X*). Indeed, by (3.72) and (1)(0) = 0 ((1) G W./ C X* we have f (0) = 0

(3.73)

(f E C(X) \ {—oo}).

(b) By the particular case r = 0 of Remark 3.11, we have f E C(X) if and only if Epi f is the intersection of a family of (homogeneous) upper closed half-spaces bounded by homogeneous nonvertical closed hyperplanes. We recall that a function f : X —> R is called positively homogeneous if (x E X, 0 < A < +Do).

f(Ax) = Xf (x)

(3.74)

It is well-known and easy to check (e.g., see [235], thm. 4.7) that a positively homogeneous function f : X —> R U {±oo} is convex if and only if it is subadditive, i.e., f +

(x)

f

(x,

E

X).

(3.75)

Lemma 3.3 Let X be a locally convex space. A function f E R X is positively homogeneous if and only if Epi f is a cone with vertex (0, 0). Proof If f is positively homogeneous and (x, d) E Epi f, 0 < À < ±oo, then, by (3.74), f(Ax) = Xf (x) Ad, so X(x , d) = (Ax, Ad) E Epi f. Conversely, assume now that Epi f is a cone with vertex (0, 0), and let X E X, 0 < À < +c.o. If f (x) = -Foo, then f(Ax) Xf (x). If f (x) < -Foo, then


109

(x, d) E Epi f for all d > f (x), whence (Ax, Ad) = X(x, d) E AEpi f c Epi f;

Ad for all d > f (x). Thus

that is, f(Ax)

f(Ax)

(x E X, 0 < A < d-oo),

Af(x)

(3.76)

f (x) for all xEX,O R w of (2.140) is one-to-one, and we have IC(X) E X **

(x E X).

(3.82)

Thus K (X) = (X*, weak*)*, where (X*, weak*) denotes X* endowed with the weak* topology. Hence, by Theorem 3.8 (applied to W = (X*, weak*) and W* = (X*, weak*)* = (X), instead of X and X*), a function h E R w belongs to C(K (X)) if and only if either h —oo or h is a proper weak* lower semicontinuous positively homogeneous convex function. Consequently, by Theorem 3.6 (with W = X*) and Section 2.4 (on 1C(X*)), we obtain: Theorem 3.9 Let X be a locally convex space. A function h : X* —> R is the support function of a (unique) closed convex subset of X if and only if h is a proper weak* lower semicontinuous positively homogeneous convex function. Remark 3.16 (a) Theorem 3.9 implies (and thus, by the above, it is equivalent to) Theorem 3.8. Indeed, by Theorem 3.9 applied for X and h : X* —> R replaced by (X*, weak*) and f : X = K (X) -> R, respectively, and using that a convex function f : X —> T? is weakly lower semicontinuous if and only if it is lower semicontinuous, it follows that we have 2° of Theorem 3.8 if and only if f is the support function of a

111


(unique) weak* closed convex subset of X*, which, by Corollary 3.1 (with W = X*) and and Section 2.4 (mutatis mutandis), is equivalent to f E C(X * ). aG E C(K(X)) is one-to-one, with (b) Since the mapping a : G E k(X * ) mapping (3.61), it is natural to ask (see Remark 3.10(b)) what special propinverse erties of the sets G c 1C(X*) correspond to certain special properties of the support functions a-G E C(K (X)), and conversely. For example, a closed convex subset G of a locally convex space X is said to be continuous (Gale and Klee [94]) if is a-G is finite and weak* continuous on X*; for some recent results on such sets, see Auslender and Coutat [15], and Coutat, Volle, and Martfnez-Legaz [44].

The Case Where X = {0,1 } " and W C (R")* ix

3.2c

Since all norms on R" are equivalent (e.g., see [148], p. 154, thm. 1), we will write (R n ) * instead of (R", Il• ID*, where I lis any norm on R". Now let

= tX =

X = 10,

R" 1

= 0 or 1 (i = 1, . . . , n)),

(3.83)

that is, the set of all vertices of the "unit cube"

c R" 10

[0, 1 ]" = {x = where 1

1 (i = 1, .

, n)1,

(3.84)

n < +Do, and let W C (Rn)* 1X = ( w E R X 13 (Do E (R")* , (1)1x

(3.85)

This case is important for applications to combinatorial optimization. Indeed, if e, = {0 . . . 0 1 0 . . . 0) is the ith unit vector in R" (i = 1, . . . , n) and 9

EiE0

7 7 9 9

7

ei = 0, then, using the natural one-to-one mapping S

xs

= E ei iEs

(S

c 11,

n1)

(3.86)

of the family 2 0 —n } (of all subsets of the .set {1, nl) onto {0, 11' (thus, x s is the "incidence vector", or "characteristic vector" of the set S), one can regard each function f : 2 11 —n } R as a function f : (0, 1)" T? (i.e., as an W-valued "pseudo-Boolean" function). In this way, each problem of minimizing or maximizing a function f : 2 11 --" 1 —> -k can be identified with a "0-1 optimization problem." Let us note that, by (3.85) and (I)(0) = 0 (:1) E (R n ) * ), we have f (0) = 0 for all f E C(W) {-00}; hence d-co t C(W). We recall (see (2.109)) that for any set X and any f E — R X , the domain (or, the effective domain) of f is the subset dom f of X defined by dom f = fx E Xlf (x) < +00).

(3.87)

Theorem 3.10 Let X = {0, 1)" and W C (Rn)*Ix. For a function f E the following statements are equivalent:

, denoting D = dom f,


112

1 0. f

2°.

E

C(W).

—oc or f has the following properties:

Either f

(R U {-Foo }) x ; f(0), (iii)for any A. x 0 (x E D) such that (i) f (ii) 0

E

ExED X x X E X, we have ExED A. x x

E

D

and

xx f (x).

f (Ek,x) xED

(3.88)

xED

30 .

Either f —oo or f can be extended to a proper lower semicontinuous positively homogeneous convex function T: R" —› R. Proof 1'

3°. If 1° holds and f * —oo, define 7: R n

7(x) = sup {4)(x) I (I) c (Rn)*,

(DI x

R by

(x e

f}

R 0 ).

(3.89)

Then, by Remark 1.1(c), we have Î e C((Rn)*), and hence, by Theorem 3.8, either f _Do or f is a proper lower semicontinuous positively homogeneous —oc and 1°, there exists w E W such that w convex function. But, by f f, and hence, by (3.85), there exists I E (R")* with (1)1 x = w f, whence, by (3.89), —00. Finally, by (3.89), (3.85), (3.11), and 1 0 , we have ffl x =- fco(w) = f. The implication 3° = 2° is obvious. 2° = 1°. If f —oc, then, by Proposition 3.1(a), f E C(W). Assume now that f E R X has the properties (i)—(iii) (then, by property (i), f —oc). We claim that {W E WIW

Indeed, by X = (0, Wk E W defined by

On and

f}

(3.90)

0.

property (i), for any sufficiently large k the function

(X = V ni E X)

wk ( x) = —

(3.91)

i= 1

satisfies wk (x) f (x) (x E X \ {0}), which, together with (3.91) for x = 0 and property (ii), implies that wk f, proving the claim (3.90). Furthermore we claim that for any .7C0 E X we have, by (3.11), (3.90), and linear programming duality theory, fw( w )(xo)

=

sup (w(xo) I w

E

W, w(x)

f (x) (x

E xxf(x) E Ax = xo, A...x xED

xED

E

D)}

0 (x

E

D) I .

(3.92)


113

Indeed, the first program in (3.92) is feasible by (3.90). If xo e D, then the dual program, that is, the second one in (3.92), is also feasible, since for Ax() = 1, A.x = 0 (x E D \ {x0 } ), we have E xED X x X = x0; hence, both programs admit optimal solutions and have the same value (e.g., see Schrijver [250], p. 92). On the other hand, if xo g D, then, by property (iii) and D , dom f (of (3.87)), we have Ax E xED

0 (X E

= xo, )1/4.x

D)}

= 0;

(3.93)

that is, the second program in (3.92) is not feasible. Hence we have again (3.92), with all terms being = +oo (e.g., see 12501, p. 92). This proves the claim (3.92). Now, if -)C0 E D, then, by (3.92) and property (iii), we obtain

fc0(w)(xo)

inf f f (

E

),xx)

= f (x())

E Ax , x0,

0(x G D) }

A,-

xeD

xED

fco(w)(xo),

whence f (xo) = fc0(w)(4). Finally, if x0

g D, then, by (3.92) and (3.93), we obtain

.f (xo) ?, Lo ( w ) (xo) =- inf 0 =-- +00,

whence f(x0) = fc0(w)(4). Thus f E

o

C(W).

Remark 3.17 (a) IffEC(W), f# —oo, then 0 E D 0 0 and f (0) = 0. Indeed, this is obvious from W C (Rn)* x and the definition of C(W). Also it can be deduced from 2', as follows: If D = 0, then, by the usual convention, 0 = ExE0 ).„,-x E X, but 0 ig D, in contradiction with property (iii); thus D 0 0. Then, by property (iii) for ),.x = 0 (x E D), we have 0 E D and f(0) ( 0, which, together with property (ii), yields f (0) = 0. (b) For W = (R")*1 x , one can also give the following proof of the implication 3' 1°, which does not use linear programming duality theory: Assume that f G — R x \ —oc) and r : Rn —> R are as in 3°. Then, by Theorem 3.8, we have f E C((R")*), that is, (

TOC) whence, by W I 11)

sup { (1) (X) I 14) E (R n

f1 x

= f, (3.85) and f }, we obtain

f (x) , ,T. (x)

) *

,

CI)

14)1 x E

sup {w(x) I w E W,

TI) ',.

11

WO)

f) =

(X E

E (Rn)* ,

f (w) (x)

Ra),

{w E

(x E [0, 1}"),

and hence f e C(W). Thus 3' = 1'. We will now give for the R-valued functions belonging to C(W), a stronger extension property. To this end, we recall that a function T : R n --> R is said to

114


be polyhedral convex if there exist a finite number of continuous affine functions 41, = + ri : Rn —> R (where (D i E (Rn)*, ri E R for i = 1, . . . , k; k < -f-oc), such that f(x) = max i 41, (x) (x E Rn). Clearly every polyhedral convex function is convex and continuous.

Theorem 3.11 Let X = {0, l } n and W C (R")*I x . Then every function f : X —> R belonging to C(W) can be extended to a positively homogeneous polyhedral convex function : R" —> R. Proof If f : X —> R belongs to C(W), then, by the above proof of Theorem 3.10, implication 2° = 1°, for each .X0 E D = X the sup in (3.92) is attained for some wx„ E W, and we have

w 0 (x)

f (x)

(x0 , x e D = X),

wx 0 (xo) = feo(w)(xo) = f(x 0 ) By (3.85), let (I)x„ e (Rn)* be such that (1)x„

(.1C0 E

(3.94) X).

= wx„, and define

(3.95)

7: Rn

R

by

T(x) = max (1)4 (x) X() EX -

(x

E

R").

(3.96)

Then fispolyhedral convex and positively homogeneous, and, by (3.96) and (3.94), we have f x = maxx„E x (Px() I x = maxx 0 Ex wx () f. On the other hand, by (3.96) and (3.95), we obtain f(x) = max x„ E x w 0 (x) wx (x) = .f (x) (x e X), whence, finally, = f.

Tx

3.2d The Case Where X = 10,11" and W = Now we consider the case where W

( Rn ) * Ix

R

{w

E

X = {0, 1}" (with 1

RX I

3 (i) E

(Rn)* , 3 r

(Rn)*

R

n < +oo) and E

R, (1)1 x +r = w). (3.97)

Theorem 3.12 Let X = {0, 1}n and W = (Rn)* x ± R. For a function f E statements are equivalent:

the following

1°. f E C(W). 2'. f E (R U I+ U {—oo}. —oc, or f can be extended to a proper lower semicontinuous 3°. Either f convex function f : Proof 1° = 3°. Assume 1°. If f

=

+oo, then one can take, for example,

X Ix=(,17€1?"

(3.98)


If f

±oo, define

f

1(x) = sup (1)(x)

115

: Rn —> R by

r

E (R")* , r

e R, (1)1 x +r

(x

E Ra). (3.99)

Then f G C((Rn)* R), and hence, by Theorem 3.7, f is a proper lower semicontinuous convex function on Rn . Also, by (3.99), (3.97), (3.11), and 1°, we have

TOO = suP {w(x) I w e W, w

(x E (0, 1}").

f} = feo(w)(x) = f (x)

The implication 3' 2° is obvious. —oo, then, by Propostion 3.1(a), f e C(W). 2° = 1°. If f C R U {+oo}, and let x o = Assume now that f e (R U {-Foop x , so f (x0). Then, since X is finite, there exist E X = 10, 11" and XER, X < , • • • , Tl X E- R such that

(X)

if 4.10

min

Tl i

0,

x=1,17Ex

(3.100)

x00, f (x) R the following statements are equivalent: 10 . f E C(V,,N + R) (i.e., f is the supremum of a set of functions of the form

(3.124)). 2°. — f E C(Va,N + R). 3°. Either f (X) C R and f is a-Holder continuous with constant N or f ±oo.

Proof 1 0 3°. Assume 1' and f # ±oo, and let X 1 , (3.123) and (3.122), we obtain f (xi) =

sup

{—NP(xl, Y

) a

E X. Then, by 1°,

r}

yEX,rER —Np(•,yr

sup

{ —Np (x 2 , y)a

(3.125)

r} — Np(x l , x2)

yEX,rER —Np(.,y)a+rf

= f (x2) - NP(xi, x2 )' Now, if there existed x l EX such that f(x 1 ) = —oo, then, by (3.125), we would obtain f (x2) = —oo for all x2 E X, in contradiction with our assumption; similarly, the existence of X2 E X such that f (x2) = +Do would imply that f +oo, contradicting our assumption. Thus f (X) C R. Finally, from (3.125) (applied also to x2 interchanged with x 1 ), we obtain (3.120). 3' 1°. We have +00 E C(Va. N + R) (since sup y EX.rER{ —N P(X, Y } ) gm-) r} X), and —oo = = +oo for all X E sup 0 suPrERI—Np(x, xr E = --r rER r C(Va,N + R). Assume now that f (X) C R and that f is a-Holder continuous with constant N. Then, by (3.120), we have )

—Np(x, x) a f (x) = f (x)

—N p(x, y )a f (y)

r

(x, y E X), (3.126)

whence f (x) = max —N p(x , y)a f (y)}

(x E X)

(3.127)

yEX

(the max being attained for y = x), so f E C(Va,N + R). 1° 2°. This equivalence follows from the equivalence 1° 3° (since f satisfies 3' if and only if so does — f ). Remark 3.21 (a) The equivalence 1° 2°, which can be also written in the form

C(Va,N

R) = —C(V c„N

R),

(3.128)

is rather surprising if we compare it, for example, with Theorem 3.7. Indeed, applying Theorem 3.7 to — f , we obtain that if X is a locally convex space, then — f E C(X*+


121

±oo or f is an upper semicontinuous concave function R) if and only if either f + R) if such that —f is proper; hence, by Theorem 3.7, f E C(X * R) n either f ±oo or f E f (X) C R only if X* ± R (i.e., is continuous and f and affine). (b) Denoting by Ha ,N the set of all a-Holder continuous functions f : X —> R with constant N, the equivalence 1' 3' says that

-c(x*

C(17,,N

R) = H,,,N U {-00, +Do } .

(3.129)

Corollary 3.3 We have C(11,,N) = C(V,,N fc0(14,,,,) — fc0(v„ N +R) =

R) = Ha,N U {-00, +oo},

max h hEH,y.N o-cc,+001

(f E R x ).

(3.130) (3.131)

Proof By (1.9), (3.129), and Proposition 1.5, we have C(I-1„,N) c

U {—cc, +00}) = C(C(Va.N R))

= C(Va,N R) = Ha,N U {-00, +00}.

Conversely, by (1.7) and (1.11), we have Ha,N U {-00} C C(Ha,N). Also, -Foo = suprER r E C(Ha,N), since the constant functions r E R belong to Ha,N (moreover they belong to Va ,N + R). This proves (3.130). Finally, formula (3.131) follows from (3.130), (3.7), and (3.129).

3.2f Suprema of a-illilder Continuous Functions, Where 0 < a

1

Let (X, p) be a metric space, and let 0 < a 1 (fixed). We recall that a function f : X —> R is said to be a-Holder continuous if there exists a constant 0 < N < +oo such that f is a-Holder continuous with constant N, i.e., such that we have (3.120). Thus, denoting by Ha the set of al a-Holder continuous functions and using the notation Ha,N of Remark 3.21(b), we can write

H.

(3.132)

O 0),

(3.134)

that is, those of (3.124) but with N variable (for simplicity we often write N > 0 instead of 0 < N < ±oo). We will also need the following concept: Definition 3.7 Let (X, p) be a metric space and 0 < a 1. We say that a function f : X —> R satisfies the a-growth condition, if there exist a constant 0 < N < ±oo and an element y E X such that the function f ± Np • , yr is bounded from below. Lemma 3.4 For a function f E R x the following statements are equivalent:

1°. f satisfies the a-growth condition. 2'. There exists a constant 0 < N < +Do such that all functions f+ N pe, -y-r (57 E X) are bounded from below 3'. There exists a Va -affine minorant off (with V, of (3.133)). Proof 1°

2'. Assume that there exist 0 < N < +Do, y E X, and r E R such

that f (x) ± N p(x, y)a ,- r

(x c X),

(3.135)

and let '5)- E X. Then, by (3.122) and (3.135), we obtain f (x) + Nio(x,T'r

f (x) + N(P(x, Y)

r — N p(y , -y-)a (>

a

— P(Y,'Y -))

—

oc)

(x E X).

The implication 2' 1' is obvious. 1' 3'. Clearly (3.135) holds for some y E X,N >0 and r e R if and only if f (x)

— NP(x,

y)" + r

(X

E X),

where —Np(., yr ± r is Va -affine.

(3.136) 0

Remark 3.22 (a) If f satisfies the a-growth condition, then f (x 0 ) > —oc (Xo E X), and every majorant f f satisfies the a-growth condition. (b) Every function f bounded from below, and (by (3.126)) every a-Holder continuous function, satisfy the a-growth condition. (c) In particular, if (X, p) is the metric space associated to a normed linear space (X, II • 11), that is, P(x, Y)

= I Ix — Yll

(x, y E X),

(3.137)


123

then, by (3.135) with y = 0 and Lemma 3.4, equivalence 1' 2°, and by (3.122) with x2 = 0, f E R x satisfies the a-growth condition if and only if the function is bounded from below. f Nil •

r

Theorem 3.15

For a function f E R x the following statements are equivalent: R) (i.e., f is a supremum of Va -affine functions (3.134)). 1'. f E C(V, 2'. f E C(1-1,) (i.e., f is a supremum of a-Holder continuous functions). 3°. Either f is lower semicontinuous and satisfies the a-growth condition or f —oc. Proof 1' •#`,. 2°. By (3.132), (1.9), (3.130), (1.35), (3.133), and (1.7) (in E = (— Rx , we obtain

C(1-1,) = C (N>0

C

Ha.N = C U(Ha,N U { -00 , +00 1) N>0

C(V,,N R))

C

C(C(Va R)) = C(V, R)

N>0

C

(L.)I (Va.N R)) N>0

cc

u C(Va,N (

R)) , (3.138)

N>0

whence C(Ha ) = C(V, R). 1° 3'. We will prove a stronger result, namely the following local version: For f E R x and xo E X the following statements are equivalent: f E C(07, R); x0 ). 3'. Either f is lower semicontinuous at xo and satisfies the a-growth condition or f (x 0 ) = —oc.

1'.

1'

3'. Observe, first, that for any f E R X , fco(Voi +R) is a minorant of f which is lower semicontinuous (by (3.11) and since the Va -affine functions (3.134) are continuous). Thus we have f

feo(v,y +R)

f

(f E R x ),

(3.139)

where J denotes the lower semicontinuous hull of f, that is, the greatest lower semicontinuous minorant of f. Hence, if 1' holds, i.e., if fc0(7,y +R)(xo) = then 7(4) = f (x 0 ), and thus f is lower semicontinuous at x0 . Assume now and that f(x 0) > d > —oo. Then, by Theorem 3.5 (with W = Va R), there exist y E X, N > 0, and r E R such that

Yr

r

f, — NP(xo,

+ r > d.

(3.140)

124


Hence, by the first inequality in (3.140) and by Lemma 3.4, implication 3 0 = 1 0 , f satisfies the a-growth condition. 3' 1'. If f(x0) = —oo, then f c C(17„ ± R; x0) (by fco(w) f and (3.58)). Assume now 3' and that f(x 0 ) > d > —oo. We will show that there exist 0 < N < +oo, y E X, and r E R satisfying (3.140), which, by Theorem 3.5, will complete the proof. Let

e = -,( f (x0) — d) (> 0).

(3.141)

Then, since f(x0) > d, we have f(x0) > -. ( f (x 0 ) + d) = d + e. Hence, since f is lower semicontinuous at xo, there exists 61 > 0 such that f (x) > d + e for all X E X with p(x, xo) < 6 1 , and therefore also

f (x) > d + e — Np(x, y)a

(x E X, p(x, xo) < 61, N > 0). (3.142)

On the other hand, since f satisfies the a-growth condition, there exist 0 < No < +Do, yo e X and ro e R such that

(x E X).

f (x) —NoP(x, Yof + ro We claim that there exists N > 0 such that for all N

(3.143)

N,

N p(x, y)a — N o p(x, y o )a ?- -Np(x, y)a > -N a01-., 2 (x, y E X, p(x, x o )

6 1 , p(x o , y)
Sr/2", whence, by (3.122),

P (x , Yo) P (x, y) Œ

1+

P (Y , Yo) a P (x , Yr

(x, y E X, p(x, x0)

1+

(3.144)

2

P (x , x0) — P (x0, y) > 2a p(y , Yor 6a1

C

61 2

Si, P(xo, Y) < — ) ,

(3.145)

where C = supil + (2a P(Y, Yo)a)/6Ply C X, P(xo, Y) < 61121 < +oo. Hence

a NP(x, y) — N0P(x, yo)a = NP(x, y) a {i N p(x, y)a (1

N0 C) N

No 10(x, Yo) a 1

N p (x , y)a j

(x, y E X, P(X, x()) i

Si SI, P(X0, Y)
0, for any 0 < (5 < ( ) a we have

NP(xo, y)a < NS'
—E + r = d, that is, the second inequality in (3.140).

Corollary 3.4 For any f E R x we have

fco(Ha

Proof By

)=

I

if f satisfies the a-growth condition,

f

(3.148)

otherwise.

(3.138) and (3.7), we have for any f E RA', (3.149)

fco(Ha) — fc0(vce +R)•

Assume now that feo(va +R) * —oo. Then, since fcc,(voi +R) E C(Va ± R), from Theorem 3.15, implication 1° 3 0 , it follows that fco(va+R) satisfies the a-growth condition, and hence, by (3.139) and Remark 3.22(a), both y and f satisfy the agrowth condition (the latter, together with (3.149), proves the second part of (3.148)). Consequently, by Theorem 3.15, implication 3' = 1' (applied to f), we have f E C(V, + R) , whence, by f f, (1.30), and (3.139), we obtain

7

—

7co(v„+R) -- R with constant N are still defined by condition (3.120), but properties (3.121), (3.122) no longer hold for such a. In fact for 1 < a < -koo we only have (e.g., see [148], p. 161) la + bl a

2a-1 (lal a ± Ibl a )

(a, b E R).

(3.150)


126

Therefore, defining again Va,N C R X by (3.123), the above proof of Theorem 3.14, implication 1' = 3', no longer works for a > 1. However, for a = 2 we have: Theorem 3.16 If (X, p) is the metric space associated to a real Hilbert space (X, (• , .)), that is, if

= Ilx — Y11 2 = (x — y, x

P(x,

y)

= 11x 1I 2 + ( -2 y, x) + 11Y11 2 then for a function f 1°. f

E

2°. Either f function.

(3.151)

R x the following statements are equivalent:

R). ±oo, or f

E C(172,N

(x, y e X),

Nil • 11 2 is a proper lower semicontinuous convex

Proof By (3.5), (3.123) (with a = 2), and (3.151), we have 1' if and only if f (x) =

sup

( — Mx YII 2 + r

(y,r)EXx R

—NH--y11 2 +r

-N 'ix 1 2

+

sup

.

{( 2 NY, x) — Nilyli 2 + r)

(y,r)EXx R

—IV

11. — Y11 2 +r“

(3.152) 1' 2'. If (3.152) holds, then, since the functions x ---> (2Ny, x) — NII Y11 2 + r are affine, from Theorem 3.7 (applied to f N11 • II 2 ) there follows 2'. 2' 1'. If 2' holds, then, by Theorem 3.7,

f (x)

+ N Ilx 11 2 =

sup (z,b)ExxR

1(z, x)

E

X),

(3.153)

N 11 . 11 2

f

whence, taking y = (1/2N)z, r = b r — (1/ 4 N)IIz11 2 = r N II Y I1 2 ), we obtain

f (x) + N Ilx 11 2 =

(x

b)

sup

( 1 /4 N)IIz11 2 (so z = 2Ny, b

(( 2 NY, x) r — N be )

(x c X).

(y,r)EX x R

(2N y,.)-I-r — NI1Y111/4f +N11 . 1 2

But, by (3.151), we have (2N y , •) — yll 2

r = —Nlix11 2

r — N ilY11 2

(2Ny,x) —

whence, by (3.154), we obtain (3.152).

(3.154) f + Nil • 11 2 if and only if r

f (x)

(X E

X), LII

Remark 3.23 A function f on a normed linear space (X, • I1) is called weakly convex (Vial [296]; Martfnez-Legaz [175]) if there exists N > 0 such that f N •I12

3.3 (W, (p)-Convexity of Functions on a Set X

127

is proper, lower semicontinuous and convex. Therefore the functions f * ±oo in 2' above may be called weakly convex with constant N (or N -weakly convex). For a > 1, defining again V, c RA' by (3.133) and the a-growth condition by Definition 3.7, the equivalence 1° 3° of Lemma 3.4 remains still valid, with the same proof. Moreover we have: Theorem 3.17 In any metric space (X, p) the equivalence 1' .(#. 3° of Theorem 3.15 remains also valid for any a > 1. Proof Again it is sufficient to prove the local version, that is, the equivalence l' R is said to be convex with respect to the pair (W, (p), or, briefly, (W, (p)-convex, if f = sup Iço(., w) I w c W, (P( . , w)

(3.156)

fl.

We will denote by C(W, ço) the set of all (W, (p)-convex function on X. Remark 3.25 (a) In the particular case where W c R x and ("a = condition (3.156) becomes (3.3). Hence C(W, (Nat) = C(W),

with C(W) of (3.5).

çOnat

(of (2.18 4)),

(3.157)

128


(b) The general case of (W, (p)-convexity can be reduced to the above particular case using the subset V = V1.47,,,, of R x defined by (2.188); namely a function f : X ---> R is (W, (p)-convex if and only if it is V w4 -convex: C(W, go) = C(Vw 4 ).

(3.158)

Thus the theories of (W , yo)-convexity and V -convexity (where V C -R-x ) offunctions on X are equivalent. Again, although the latter is simpler, (W, 0-convexity has the advantage of more versatility. (c) Similarly to Remark 2.33(c), if W c R x , then for (pnat :XxW--->R of (2.184) we have, by (3.158) and (2.193) (or, alternatively, by (3.158) and (3.157)), C(W, (Nat) = C(Vw,, p„,„) = C(W).

(3.159)

By (3.159) and (3.158), one can obtain results on general (W, 0-convexity of functions f : X ---> W from the results on W-convexity, where W c W x , simply by replacing w(x) by ço(x, w). For example, defining the (W, (p)-convex hull fcc,(w,„,) of a function f : X —> W in the obvious way (replacing C(W) by C(W, (p) in (3.7)), formula (3.11) extends to feo(w,) = sup ko( . , w)lw c W, (p•, w)

f).

(3.160)

(d) In Chapter 8 we will see that given X, W and (p : X x W ---> R, the (17w,,p R)-convex functions f : X —> -k, i.e., those satisfying

f=

sup

{go(., w)

r},

(3.161)

wEW,rER yo(-,w)-Frf

are of interest due to the flexibility in the possible choice of W and go : X x W ---> The functions go(., w) + r (w E W,rE R) occurring in (3.161) (i.e., the V w4 -affine functions in the sense of Definition 3.6) are called (W, (p)-affine functions. Extending the terminology of Remark 3.13(c), the functions go(., w) ± r E Vw4,± R (w E W, r G R) are called (W, (p)-elementary functions. Let us also mention that the functions (toe, w) E Vw, (p are sometimes called (W, go)-linear functions. A useful tool for the study of the functions satisfying (3.161) is the machinery of conjugations (see Section 8.3).

Chapter Four

Abstract Quasi-Convexity of Functions on a Set Whenever there is defined a family of abstract "convex subsets" of a set X, in some sense, one can define a corresponding concept of abstract "quasi-convex functions" f on X by requiring that all level sets Sd(f) (d E R) be "convex", in the given sense. In the present chapter we will consider the quasi-convex functions on a set X defined in this way, for various concepts of abstract convex subsets of X introduced in Chapter 2.

4.1 .A4-Quasi-Convexity of Functions on a Set X, Where .A4 C 2 x In Section 2.1 we singled out the family C(M) of all M-convex subsets of a set X, where M c 2x . Using this family, one can introduce M-quasi-convex functions on X, as follows: Definition 4.1 Let X be a set, and let M c 2 x . We say that a function f : X —> R is quasi-convex with respect to the family M, or, briefly, M-quasi-convex, if

(4.1)

(d E R).

Sd(f) E C(M)

We will denote by Q (M) the set of all M-quasi-convex functions on X:

Q(M) = If

E RX

I Sd(f) E C(M) (d

E

R)).

(4.2)

Definition 4.1 and the preceding results on C(M) yield results on Q (M). Thus, first, by Definitions 4.1 and 2.1, we have f E Q (M) if and only if SAP =

DM

E M

I SAP

g

All

(d E R),

(4.3)

or, equivalently (by (2.19) or (2.12)),

Q(M) =

If

E R X I Sd(f) = COM S1(f) 129

(d

E

R)).

(4.4)

Abstract Quasi-Convexity of Functions on a Set

130

Also, by (4.2) and (2.24),

Q(M) = If E R x I V (x, d) E X x R,x S d (f), 3M

E

M, Sd(f)

M, X

C

X\

E

(4.5)

Remark 4.1 We have

Q(M) = if

I V (x, d)

E

]M

E M, Ad(

E

f)

X

C

X

R, x Sd(f),

M,x

X \ M}.

E

(4.6)

Indeed, the proof is similar to that of Remark 3.3(c) (i.e., use (4.5) and Ad(f) c S1(f) C Ad+F(f) for all d E R and E > 0). In the sequel, for simplicity, in many similar situations (e.g., see Proposition 4.4, Corollaries 8.14, 8.15), we will state the results only with Sd(f) C M (omitting their equivalent formulations with Ad(f) C 11/). By Definition 4.1 and by (1.9) (in E = (2x , D)), we have C M2 =

(4.7)

Q(A41) g Q(M2)•

Proposition 4.1

(a) Q(./14) is a convexity system in (Tix , () (i.e., closed for sup). In particular, —00 E Q(M). (b) Given r ERU {±oo), we have f r E Q(M) if and only if 0 E C(M). This holds if and only if there exists f E Q (M) such that inf f (X) > —oc. and fi E Q(M) Proof (a) If / R), then, by Proposition 2.1(a) and Sd (sup

E

(4.8)

I), that is, Sd(fi) E C(M) (i

fi ) =

E

I, d

G

(4.9)

(d

E

R),

we obtain Sd(suPicr fi) E C(M) (d E R), so suPia f, and (2.6), —00 E C(M). (b) The first statement follows from (2.6) and

E

Q(M). Also, by (2.120)

iEr

icI

X Sd(r)

if r d < +Do,

(4.10)

= 0

if d < r

+a).

Assume now that there exists f E Q(M) satisfying (4.8), and let d E R, inf f(X) > d. Then 0 = Sd(f) E C(M). Conversely, if 0 E C(M), then, +00 E Q(./v), and by (4.10), we have Sd(+00) E C(M) for all d E R, so f clearly this f satisfies (4.8).

4.1 M-Quasi-Convexity of Functions on a Set X, Where M c 2x

Definition 4.2 Let X be a set, and let M C 2x . For any f : X M-quasi-convex hull off is the function from) : X —> R defined by

fq(m) = sup th

E

Q(M ) I h

131

R the

(4.11)

f 1-

Remark 4.2 By Definition 4.2 and Remark 1.11 applied to the convexity system (see Proposition 4.1(a)), fq(m) is nothing else than Q(M) in E = (Rx , fc00,,, (of (1.59)), the Q (M) -convex hull of f. Also, by the last part of Proposition 4.1(a), we have — OC E th E Q (.A4) 1 h

(f

0

f

E

R x ).

(4.12)

By Remark 4.2, from Proposition 1.9, Corollary 1.3, and Remark 1.10(b), applied we obtain, respectively: to the convexity system B = Q(.114) in E = (Rx , Proposition 4.2 (a) For any f

E

R x we have (4.13)

fq (m) E Q(M), and fq (m) is the greatest M-quasi-convex minorant of f:

fq(m ) = max {h

(4.14)

f).

E Q(M)lh

(b) We have

Q(M)

= If

E

= f 1.

R X I fom)

(4.15)

Corollary 4.1 (a) We have

—ooq(m ) = —oo, (+oo) q(m ) = max Q(M).

(4.16)

(b) There holds

Q(M)

= tfq(m)1

f

E

(4.17)

R X )-

Theorem 4.1 The mapping f —> fq(m) is a hull operator on (R x ,

Remark 4.3 (a) By (4.15), we have +ooq(m) = +oo if and only if +00 E Q(M), that is, Sd(+00) E C(M) (d E R), i.e., 0 E C(M) (by (2.121)). By Proposition 2.1(b), this holds if and only if nmEm M= 0. (b) By Remark 2.1 and Definitions 4.1, 4.2, we have

Q(A/1 U (X)) = Q(M), fq(mu{x}) = fq(m)

(f

E

RA'),

(4.18)


132

and if (and only if) 0 E C(M), then also Q04 U {O}) = Q(M), fq(mu(0)) = f q (m)

(f E R X ).

(4.19)

Hence we can replace .A4 by M i = M U (X) and, if and only if n mEm M = 0, then also by M2 = M U {X, 0) without modifying Q(M) and q(M). (c) Since Definition 4.1 and formula (4.9) are "similar" to Proposition 3.2 (which could be also used as a definition of W-convex functions) and formula (3.38), respectively, it is natural that one can obtain results on level sets and Q(M) parallel to some "corresponding" ones on epigraphs and C(W). We will give now some further results in this direction. We will need the following result "corresponding to" formula (3.41): Lemma 4.1 Let W 1 1, ER be a family of subsets of a set X, satisfying

r R ("corresponding to" the function Am of (3.40)), defined by p lut }(x) = inf r XEU,

(x E X),

(4.21)

we have sd(p{u,})

= t>d n ut

(d E R).

(4.22)

Proof If x E Sd(p{u r )) and to > d, then infx , u, r = pfud (x) d < to , so there exists r < to such that x e Ur C Uto (by (4.20)), whence, since to > d has been arbitrary, x E Ut.

n

t>d

Conversely, if x E Uto , then p{u,}(x) = infxEu, r to . Thus, if x E then piu,)(x) d. to for all to > d, whence ptui )(x)

nto>d U10 ,

Corollary 4.2 Let fUritER be a family of subsets of a set X satisfying (4.20). If M C 2 X and

Ut E C(M)

(t E R),

(4.23)

then for the function m il defined by (4.21) we have P{u,} e Q(M).

(4.24)

Proof By (4.23), (4.22), and Proposition 2.1(a), we have Sd(pfu,}) E C(M) (d E

R).

133

4.1 »1 Quasi Convexity of Functions on a Set X, Where M c 2x -

-

There holds the following result, corresponding to formula (3.43) of Theorem 3.3: Theorem 4.2 For any f : X —> R we have fq(M ) (x) =

inf d = inf d xEcom Sd(f) x Ecom Ad(f)

(x E X).

(4.25)

Sd(f ) (d E R) there holds

Proof By (4.21) and com Sd (f)

inf d P{co», s,(f)}(x) = xEco.A,, st,(f)

(x E X).

inf d = f (x)

xEsd (f)

Furthermore, by (4.22) with Ut = com St (f) (t E R) and by (1.21) and (2.5), we , d com st(f) E C(M) for all d E R, so Ptcom Si(f)} E have Sd(Pfcom st(f))) =

nr

Q ( M) •

Finally, if h E Q (.A4) , h f, then Sd(h) c C(M) and Sd(h ) D Sd(f) (d whence, by (2.12) and (2.16), Sd (h) D COm Sd (f) (d E R), and hence h(x) =

inf d

xEsd (h)

inf

xEc0 m Su(f)

E

R),

(x E X).

d = P { c°, st (f)}(x)

Thus P{com st(f)} is the greatest M-quasi-convex minorant of f, which proves the first equality in (4.25). The second equality in (4.25) is obvious, by (2.16) and Sd(

(d

Ad+E(f) Ç S + (f)

E

R,

8>

El (4.26)

0).

Corollary 4.3 (a) For any f : X —> R we have sd(fq(

,

) )

n

co.A, St (f) =

>d

n

com /(f)

(d

E

R),

(4.28)

P(S,(f, ( m ) )) = P{com S t (f)) = P(A,(km ) )) = P{com A t (f)} •

(b) If 0

E

(4.27)

r>d

C(M), then for any f : X ---> R,

inf fq(m ) (X) = inf f (X).

(4.29)

Proof (a) By Theorem 4.2 and Lemma 4.1 with Ut = com St (f) (t E R), we have the first equality in (4.27). The second equality in (4.27) is obvious, by (2.16) and (4.26). Furthermore, by (4.21) and (4.25), we obtain P(s,(f, t 01 (x) =

inf

d = fq(m ) (x)

inf

d = P{com St (f)}(x)

xEsd(f i ( .m)) x Ecom Sd( f)

(x c X),


134

and similar relations for St replaced by At (b) By fq(m) f, we have

inf A uk/0 (X)

inf f (X).

Now, if we had inf fq(M) (X) < inf f (X), then there would exist X0 E X such that AGA/0(x°) < inf f (X), whence, by (4.25), there would exist d < inf f (X) such that x0 G COM Sd (f ), SO COM Sd (f ) 0. Therefore, if 0 E C(M), then Sd (f ) 0 (since otherwise Sd (f) = 0 E C(M), whence, by (2.12), corn Sd(f) = Sd(f) = 0), so inf f (X) d, a contradiction. This proves (4.29). Remark 4.4 (a) Formula (4.28) implies again Theorem 4.2. Indeed, if (4.28) holds, then, by (4.21) and (4.28), fq( M)(x) =

inf

d = pls,(fi(m) ))(x) — P{co m st (f))(x)

inf

d

xesd(fq(m)) xEcom

SI( f)

X).

E

(b) Despite (4.28) and (4.27) we need not have Sd(fq(M)) = COM Sd(f ) (d E R) (see e.g. Remark 4.8 below), so a result corresponding to formula (3.42) of Theorem 3.3 does not hold. (c) By (4.27), com (f) G C(M) and (2.5), we have Sd( fq(M)) E C(M) (d E R), which proves again (4.13). There holds the following basic result on f ( rn ) : Theorem 4.3 For any f : X

R we have

A K A/0 (x) =

sup inf f (X \ M) mE.A4 xEx\m

(x E X).

(4.30)

Proof Let X E X. If x E COm Sd(f ), where d E R, then, by (2.23) of Theorem 2.3, for each M E »I with x E X \M there exists yd,m E Sd(f) n (x• M), whence

inf f (X \ M)

sup mEm xEx\ni

f(Yd,m)

sup

d.

MEJvl

x EX \M

Consequently, by Theorem 4.2, we obtain sup mEm xEx\m

inf f (X \ M)

inf

xEco m sd (f)

By (1.1), this inequality remains also valid if x

d = fq (m ) (x).

corn Sd (f ) for all d e R.

4.1 A/I-Quasi-Convexity of Functions on a Set X, Where .A4 c 2x

135

Assume now, a contrario, that this inequality is strict, so there exists d E R such that sup inf f (X \ M) < d < f(04 ) (x). Em xcx\ni

(4.31)

Sd (fq(m ) ), where Sd(fq (m ) ) E C(M) (by fq (m) E Q(M)), and hence, Then x by (2.24) of Theorem 2.3, there exists Mo E M such that Sd (fq(m ) ) c Mo (so fq(m) (y) > d for all y e X \ Mo) and x E X \ Mo. Consequently, by fq (m) f, we obtain

d

inf fq (m)(X \ Mo)

inf f (X \ Mo)

inf f (X \ M),

sup MEA4 xcx\m

which contradicts (4.31). This proves (4.30).

11]

Proposition 4.3 Let X be a set and M C 2x . Then Q(M) R = Q(M).

(4.32)

R D Q(M) ± O = Q(M). Conversely, Proof Since 0 c R, we have Q(A/1) LII if f E Q(M) and d E R, then, by (2.154), we have f + d E Q(M).

The following theorem shows some connections between .A4-convexity of sets G C X and .A4-quasi-convexity of certain related indicator functions: Theorem 4.4 Let X be a set, and let M C 2x . Then for any set G C X the following statements are equivalent: V. 2".

G E C(M). —

Y n.X\G

(

AA

3° For each d E R, — xx\G + d E Q(M).

These statements are implied by—and if and only if 0 E C (M), they are equivalent to—the following statements (equivalent to each other): 4° .

5°.

XG E Q(M). For each d E R, xG +d E Q(M).

Proof I' - 2°. By (2.161) and (2.6), we have the equivalence G E C(M) Sd( — Xx\G) E C(M)

(d E R).

The equivalences 2° .#>- 3' and 4° .- 5' follow by Proposition 4.3.

(4.33)


136

The implication 4° = 1° and, if 0 E C(M), the equivalence 1° - 4° follow from (2.163) and Definition 4.1 (with f = XG). Finally, if the implication 1° = 4° holds, then, by (2.6), we obtain 0 = Xx c Q(.114), whence, by Proposition 4.1(b), 0 E C(M). E Corollary 4.4 Let X be a set, and let M c 2 x . Then S(Q(M) x R) = C(M).

(4.34)

Proof By (2.111) (for W = Q(M)) and (4.2), we have S(Q(M) x R) = {Sd(f)I f e Q(M), d e I?) g C(M).

(4.35)

Conversely, if G E C(M), then, by Theorem 4.4, —xx\G E Q(M), whence, by (2.161), G = S-1( — Xx\G) E S(Q(.A4) x R). El The next result shows some relations between M-convex hulls of sets and Mquasi-convex hulls of certain related indicator functions. Theorem 4.5 Let X be a set, and let M C 2x . Then (a) For any set G C X we have (— XX\G)q(M) — —

(b) If 0 E C(M), then for any set G

C

(4.36)

XX \com G-

X we have

(XG)q(.10) — Xcom G = (Xcom

(4.37)

Oci(A/1)-

Conversely, the first equality in (4.37) implies that 0 E C(M). Proof (a) By (2.161) and com X = X (i.e., (1.25) in E = (2x , o)), we have { com G COm

S d( — X X\G)

if d < 0, (4.38)

—

X

if 0

d < +00.

Thus, if x E X \ com G, then x V com Sd(—Xx\G) for d < 0 and x E d < +Do, while if x E COm G, then X E E Com Sd( — XX\G) for 0 COm Sd( — XX\G) for all d E R. Hence, by Theorem 4.2, we obtain (— XX\G)q(.114)(X) =

inf xEcom S / ( --x x\G), d 7--- — X X\co

.A4 G (X)

(X

E X).

4.1 M-Quasi-Convexity of Functions on a Set X, Where ,A4 c 2x

137

(b) The proof of the first equality in (4.37) is similar, observing that if 0 e C(M), so com 0 = 0, then, by (2.163), if d < 0,

(4.39)

com Sd(XG) — 1 0 com G

if 0

d < +Do.

Furthermore, by Theorem 4.1, we have (fq(m ) ), ( 4 ) = fq(m ) (f e W X ), whence, by the first equality in (4.37), we obtain (XG)q(JvI) — ((XG)ci(M))q(M) — (Xcom G)q(M);

or, alternatively, by 0, com G E C(M) and Theorem 4.4, we have Xco m G E Q(M), whence the second equality in (4.37). Finally, if the first equality in (4.37) holds, then, by (2.6), we obtain (Xx)q(m) = LI XCOm X = Xx, so O = Xx E Q(M), whence, by Proposition 4.1(b), 0 E C(M). Remark 4.5 (a) Theorem 4.5 implies again Theorem 4.4. Indeed, if G E C(M), so G = com G, then, by (4.36), — Xx\G = XX\co m G = ( XX\G)q(M) E Q(M), which proves the implication 1' = 2'. Conversely, if - XX\G E Q(M), so then, by (4.36), — XX\co m G XX\G, whence G = (— XX\G)q(M) = — X X\G, 1'. The proof of the com G E C(M), which proves the implication 2') equivalence 1° - 4° is similar, using (4.37). (b) For any subset G of X, the representation function TI G of G is defined by slightly modifying the indicator function x G , namely by —

—oo

if X E G,

(4.40)

11G(x)

if x E X \ G; this function will be used later (especially in Section 8.10 and Chapter 9). Note that, by (4.40), we have for any G C X,

(4.41)

(d e R),

SAG) = G

this corresponds to (2.163). Hence, by Theorem 4.2, we obtain if x E COm G, ( 77G) q (m)(x) =

inf

xEeom Sd (11G)

d = {

if x

co.A4 G;

that is, by (4.40), ( 1 1 G)cl(M) = licom G •

(4.42)

This formula is similar to the first part of (4.37), but now we do not need the assumption 0 E C(M). Note also that, by (4.41), for any set G c X we have the


138

equivalence G E C(M)

E

(4.43)

Q(M).

Using Jm of (2.157), we are now able to give a subset of Q(M) that is a supremal generator of Q(M) in E = (Rx , and hence (by Propositions 4.1(a) and 1.10) a base of the convexity system Q(M) in (R x , Theorem 4.6 Let X be a set, and let M

C

2x . Then Jm R c Q(M),

(4.44)

sup (w ± d) (w,d)EhA xR

fq(m) =

(f

E

R x ),

(4.45)

d)),

(4.46)

w+d, R is M-quasi-convex at xo if saidtobe f (xo) = fq(M) (X0)-

(4.50)

We will denote by Q (A4; xo ) the set of all such functions: Q(.1\4; xo) =

If

E R x I f (xo) = fq (m)(x()))•

(4.51)

By (4.15), we have

Q(M) =

n

Q(M; xo).

(4.52)

X0 E X

One can give localized versions of some of the above results. For example, there hold the following localized versions of (4.4) and (4.5):

140


Proposition 4.4 Let X be a set, .11/1

Q(M;

x0)

C

2x and xo

= If c X I = If

G

{x0)

G

X. Then

n

S(f) = {x01

R X IV d e R, xo

n com

sd(f) (c1

G

R)}

(4.53)

g Sd(f),

3M e .A4, Sd(f) M, xo c X \ M}. Proof By (2.17), we have for any f c

{x0 } n sd(f) g

{x0 }

Te,

n com

S 1 (f)

(c1

G

R).

On the other hand, by Theorem 4.2 and fq (m) f, we have f g Q(A/1; x0) if and only if A(M)(4) = infx,Gcom S„(f) d < f (x0 ), which happens if and only if there exists d e R with xo E cox/ Sd(f) \ Sd(f), that is, with {x0 n com Sd(f) g (x0 ) n & ( f. ). This proves the first part of (4.53). Finally, by (2.23) of Theorem 2.3, we have xo E com S, (f) \ Sd ( ) if and only if xo g Sd(f) and there exists no M c M with Sd(f) c M, xo E X \ M. This proves the second equality in (4.53). }

4.2 4.2a


Quasi-Convex Functions on a Linear Space X

We recall that if X is a linear space, a function f : X R is said to be quasi-convex if all level sets Sd (f) (d c R) are convex (in the usual sense), or, equivalently (by (0.1), (0.2) and (4.26)), all level sets Ad ( f ) (d c R) are convex. It is well-known (and easy to see) that this happens if and only if

1). (4.54) Now let X be a linear space, and let M be the family of all semispaces in X. Then, by the main statement of Section 2.2a, part (1), a function f : X R is M quasiconvex if and only if it is quasi-convex. Hence, by Corollary 4.2, if U, c X (t E R) are convex and satisfy (4.20), then plud of (4.21) is quasi-convex. For any f : X R we will denote by fy the quasi-convex hull off, that is, the greatest quasi-convex minorant of f. Then for the family M of all semispaces in X we have fci(m ) = f4 (f R x ). Theorems 4.2, 4.3, and formukt (2.36) yield f (kxi + (1 — )0x2)

(x i , x2 c X, 0

max (f (xi), f (x2))

X

-

fq (x) = inf

xEco Sd(f)

d =

inf

x Eco A d

f4 (x) = sup inf f (A) = AA(x)

(

f)

d

(f

sup inf f (A) A€:4(x)

G

Te , x c X),

(4.55)

(f G R X , x e X), (4.56)


141

where A(x) denotes the family of all complements X\M of semispaces M in X, with x E X\M, and :4-(x) is the family of all complements of semispaces at x (the last equality can be seen by a simple translation). It is well-known (e.g., see [148], p. 188, thm. 2) that the complement X \ S of a semispace S = x + So at x (where So is a semispace at 0) is the union of {x} and the "diametral" semispace x — So at {x}; thus 71(x) is the family of all sets of the form {x} U Sx , where Sx is a semispace at x. Remark 4.7 In particular, let X = R". Then, since the lexicographical order y or x ?-z, Y), the is a total order on R" (i.e., for any x, y E R" we have either x complement of the set M of (2.39) is the set X \ M = { y E R n I u(Y) L z . }

(4.57)

Therefore, denoting by 11(R") the set of all linear isomorphisms u : R" —> R", from (4.56) we obtain fq (x) =

sup

inf

uEll(R")

YER n u 1 u(x)

f (y)

(f E R R " , X E R").

(4.58)

One can also show (see [171], prop. 2.4 and cor. 2.5) that

(x) =

sup

inf

uEL(Rn)

f (y)

" t1(y) L u(x)

(f

R R " , x E X),

(4.59)

VER

R" (in [1711, of all where ,C(R") denotes the set of all linear mappings u : R" n x n matrices); since 11(R") c r(R"), (4.58) implies the inequality t in (4.59). Let us return to an arbitrary linear space X. Corollary 4.3(a) and formula (2.36) yield Sd (fq ) =

fl co Si(f) = n co At(f)

t>d

(f E R x , d G R).

(4.60)

t>d

Now we will obtain a simpler expression than (4.60) for the level sets Ad( fq ), which makes them more convenient for the study of fq than the level sets Sd(f) (d E R). To this end, we will need the following counterpart to Lemma 4.1: Lemma 4.2

Let {Ut ), E R be a family of subsets of a set X. Then, for the function p fut) : X —> defined by (4.21), we have Ad

(p{u,})

=

U Ut

(d E R).

(4.61)

«41

Proof We have p {ui }(x) = infxEu, r < d if and only if there exists r < d such that x E Ur , that is, if and only if X G U t< d tit •


142

Now we can prove the following counterpart to (4.60): Corollary 4.5 Let X be a linear space. Then

Ad(f q )

= co

(f e X , d E R).

A d(f)

(4.62)

Proof By (4.55), (4.21), and Lemma 4.2, we have for any f c 7?-x and d E R,

(4.63)

Ad(f q ) = Ad(P{cok(i)}) = U co A, (f), t T? we will denote by fc-f the lower semicontinuous quasi-convex hull off, that is, the greatest lower semicontinuous quasi-convex minorant of f. Then fq (m) = f(T (f E Te). Since the complement of the closed half-space (2.42) (where (I) c X* \ {0}, d E R) is the open half-space

D = y {

GX

I

> d} = ly

G X I( -1 (y) < —d}, )

(4.66)


143

Theorems 4.2, 4.3, and formula (2.44) (for the family M of all closed half-spaces in X) yield

fq(x) = inf d = inf d xEé--6,5,1 (f) xEE6Ad(i) f(x) =

sup inf f (D)

(f e R x , x e X), (f e T?x , x e X),

(4.67) (4.68)

DE00C)

where 0(x) denotes the family of all open half-spaces in X containing x. Since D of (4.66) contains x if and only if (ID (x) > d, formula (4.68) is equivalent to f(x) = sup sup dER d) EX* (1)(x)>d

inf

f (y)

yeX c13(y)>d

(f

E RX , X E

X).

(4.69)

Remark 4.9 (a) Every open half-space in X containing x can be also written in the form

D = ly

GX

I (i)(y) > 4;(x) — l 1,

(4.70)

with suitable ci; e X* \ In indeed, if D of (4.66) contains x, it is enough to take Cl; = (1/()(x) — d))(1) (note also that, conversely, it is obvious that every D of the form (4.70) contains x). Hence formula (4.68) is equivalent to

f(x) = sup

inf

f (y)

(f

inf

f (y)

(f E R x , x

vEX

G RX , X E

X).

(4.71)

X).

(4.72)

(b) We have also sup

f(x) =

(13 EX* e>0

yEX

E

Indeed, taking d = (I) (x) — E in (4.69), we, obtain the inequality ,.. in (4.72); on the other hand, from (4.71) we obtain the inequality in (4.72). (c) Corollary 4.3(a) and formula (2.44) yield sd(fz,-)

=n t>d

z)S (f) =

n

(f E R x , d E R).

(4.73)

t>d

4.2c Evenly Quasi-Convex Functions on a Locally Convex Space X We recall that if X is a locally convex space, a function f : X —> R is said to be evenly quasi-convex [213] if all level sets Sd(f) (d E R) are evenly convex. Now let X be a locally convex space, and let .A4 be the family of all open halfspaces in X. By the main statement of Section 2.2c, part (1), a function f : X ---> R is »I-quasi-convex if and only if it is evenly quasi-convex. Hence, by Corollary 4.2, if U, c X (t E R) are evenly convex and satisfy (4.20), then plud of (4.21) is evenly quasi-convex.


144

For any f : X —> R we will denote by feq the evenly quasi-convex hull off, that is, the greatest evenly quasi-convex minorant of f. Then fq (m) = feq (f e R x ). Since the complement of the open half-space (2.53) is the closed half-space D = ly G X

I (I) ( Y )

dI = ty

G

X I ( -0 )(y)

—d},

(4.74)

Theorems 4.2, 4.3, and formula (2.54) (for the family M of all open half-spaces in X) yield fey (x) =

inf

xeeco Sd(f)

d =

inf

xEeco Ad(i)

d

feq (x) = sup inf f (D)

(f E R ' ,

X

E X),

(f G R x , x G X),

(4.75) (4.76)

DED(x)

where D(x) denotes the family of all closed half-spaces in X containing x. Since D of (4.74) contains x if and only if ct(x) d, formula (4.76) is equivalent to fey (x) = sup sup

inf

VEX

dER cr.Ex.

rt. (x)d

f(y)

(f E — R X , x e X).

(4.77)

(I) (Y) ?- d

Remark 4.10 (a) One can replace D(x) of (4.76) by its subfamily D(x) consisting of all closed half-spaces that contain x on their boundary:

feq(x) = sup ci)Ex.

inf yEx

f (y)

(f G R X , X G X).

(4.78)

Indeed, taking 4 = cp. (x) in (4.77), we obtain the inequality in (4.78); on the other hand, since (I)(x) d implies ty e X I (D(Y) d} D fy e X I (NY) from (4.77) we obtain the inequality in (4.78). (b) Corollary 4.3(a) and formula (2.54) yield Sd(feq ) =

n t >d

4.2d

eco St (f) =

n

eco A f (f)

(f E -k x , d e R).

(4.79)

1>d

Evenly Quasi-Coaffine Functions on a Locally Convex Space X

Definition 4.4 Let X be a locally convex space. We will say that a function f : X ---> R is evenly quasi-coaffine if all level sets Sd(f) (d E R) are evenly coaffine (in the sense of Section 2.2e). Now let X be a locally convex space, and let M be the family of all complements of closed hyperplanes (2.63) in X. Then, by the main statement of Section 2.2e, a function f : X —> R is M-quasi-convex if and only if f is evenly quasi-coaffine. Also, by Proposition 2.2, every evenly quasi-convex function belongs to Q(M), and conversely, every function f E Q(M) with connected level sets is evenly quasiconvex.


145

For any f : X —> T? we will denote by fqc„ the evenly quasi-coaffine hull of f, that is, the greatest evenly quasi-coaffine minorant of f. Then fq(m ) = fq,a ( f e Te). Since the complement of the set M of (2.63) (where (ID e X* \ (0), d E R) is the hyperplane H = (y e X I (I)(y) = d},

(4.80)

Theorem 4.3 (for the family M of all complements of closed hyperplanes in X) yields sup inf f (H)

fq,,,(x) =

(f

E

RX ,

XE

X),

(4.81)

HEH(x)

where H(x) denotes the family of all closed hyperplanes containing x. Since H of (4.80) contains x if and only if cl)(x) = d, formula (4.81) is equivalent to fqca (x) = sup sup dER q), EX*

inf

f (y)

yEX

q ( Y)=d

cl)(x)=d

(4.82) = sup

inf

ci3 EX*

4.2e

(f

f(v)

,x

E

E

X).

YEX

Lower Semicontinuous Functions on a Topological Space X

Let X be a topological space, and let M be an intersectional base for the closed subsets of X. Then, by the main statement of Section 2.2g, a function f : X —> R is M-quasi-convex if and only if all level sets Sd( f) (d E R) are closed, that is, if and only if f is lower semicontinuous. Hence, by Corollary 4.2, if U, C X (t e R) are closed and satisfy (4.20), then plui l of (4.21) is lower semicontinuous. As in (3.139), for any f : X R let us denote by f the lower semicontinuous hull off, that is, the greatest lower semicontinuous minorant of f. Then for the above family M we have f(M) = j(f E Te). Theorems 4.2, 4.3, and formula (2.66) yield 7(x) =

inf

d =

inf

(f E T?x , x e X),

d

(4.83)

xeSd(f)

f (x) =

sup inf f (V)

(f

E

RX,

x e X),

(4.84)

VeV(x)

where V (x) denotes a fundamental system of open neighborhoods of x. Also Corollary 4.3(a) and formula (2.66) yield Sd (f) = n S,(f) = t>d

n

A r (f)

(f c R x , d c R).

(4.85)

t>d

4.2f Nondecreasing Functions on a Poset X Let X be a poset, and let M be the family of all subsets Ma of X of the form (2.71) where a e X. Then, by Proposition 2.3(a), a function f : X — R is M-quasi-convex

146


if and only if all level sets Sd (f) (d

E

R) are order ideals in X: (x E Sd(f), d E R).

1,Y e X I “, x } g Sd( f )

(4.86)

But this happens if ond only if f is nondecreasing on X: V, x E X, y

x = f(v)

(4.87)

f (x).

Indeed, (4.86) means that the relations x, y E X, y ( x, d E R, and f(x) d imply that f (y) R we will denote by f the nondecreasing hull off, that is, the greatest nondecreasing minorant of f. Then for the above family M, Theorem 4.3 yields f< (x) = sup inf f (y) = inf f (y)

(x G X);

VEX

aEX VEX a<xav

(4.88)

x R is quasi-convex with respect to the set W, or, briefly, W -quasi-convex, if

Sd(f)

E k(W)

(d e R).

(4.89)

We will denote by Q(W) the set of all W-quasi-convex functions on X: Q(W) = If E

I S1(f) e

1C(W) (d E R)).

(4.90)

Remark 4.11 One can also introduce other concepts of M -quasi-convexity (where M c 2x ) and W-quasi-convexity (where W c R x ) of functions f E R X , replacing the level sets S1(f) by the level sets A d (f) (d E R) in Definitions 4.1 and 4.5, respectively, and/or the family 1C(W) in (4.89) by a family of "W-convex sets" in a different sense, such as, by C(Mw) of (2.179). However, here we will not consider these concepts. There are two natural ways of obtaining results on Q(W) as particular cases of the preceding results. Indeed, on the one hand, Definition 4.5 and the results of Section

4.3 W-Quasi-Convexity of Functions on a Set X, Where W c R x

147

2.3 on 1C(W) yield results on Q(W). On the other hand, by Definitions 4.5 and 2.9, we have Q(w)

= If e

R x I sd(f)

e

C(S(w

x

R)) (d e R)} = Q(S(W x R)), (4.91)

with S(W x R) of (2.111), and hence the results of Section 4.1 on Q(M), applied, in particular, to M = S(W x R) of (2.111), yield results on Q(W), which can be formulated simply by replacing M E M, C(M) and Q(M) by Sd(w) E S(W x R), 1C(W) and Q(W), respectively. Let us give now some results obtained with these methods. Proposition 4.5

Let X be a set, and let W C Te . Then (a) Q(W) is a convexity system in (lix , () (i.e., closed for sup). In particular, —00 E Q(W). (b) Given any rERU f±oo), we have f r E Q(W) if and only if (2.118) holds.

Proof This follows from Definition 4.5, formulas (4.9), (4.10), and Proposition 2.9. Alternatively, this also follows from Proposition 4.1 applied to M = S(W x R) El and Proposition 2.9(b). Proposition 4.6

Let X be a set and W C WX . Then W C Q(W),

(4.92)

W1 C W2 = Q(W1) g

(4.93)

W ± R C Q(W) ± R = Q(W) = Q(W ± R).

(4.94)

Proof By (2.116) and (4.90), we have (4.92). Also, by (2.117) and (4.90), we have (4.93). Furthermore, by (4.92), we have W ± R C Q(W) ± R, and by (4.32) of Proposition 4.3 applied to M = S(W x 10, we have Q(W) ± R = Q(W). Finally, by (2.153) of Proposition 2.10 and (4.90), we obtain Q(W) = Q(W + R). 0 Remark 4.12 (a) There is no relation for M and Q(M) similar to (4.92), since

.M C 2 X and Q(./1/1) c R x (while both W and Q(W) are in R x ). (b) By (4.91) and Theorem 4.6 (applied to M = S(W x R)), the set Js(w x R) ± R (with Js(w x R) of (2.169)) is a base (and hence a supremal generator) of the convexity system Q(W) in (R x , -.). Proposition 4.7 Let X be a set and W C Tx . Then

Q(W) =

ff

G

R x I Sd(f) =

n

St (w) (d E R)}

(w,t)EWxR

sup w (Sd ( f

Nt


148

----

(f e R x IV (x, d) ]w

E W,

E

X

X

R, f (x) > d, (4.95)

sup w(Sd(f)) < w(x)}.

Proof By Definition 4.5 and Remark 2.18(b), we have the first equality in (4.95). Also, by Definition 4.5 and (2.132) of Theorem 2.7, we have the second equality in (4.95). El By Definition 4.2 and Proposition 4.2(a) for .A/1 = S(W x R), we arrive at: Definition 4.6 Let X be a set, and let W c R x . For any f : X -± R the W -quasi-convex hull of f is the function fq(w) : X —> R defined by

fl,

fq(w) = max (11 G Q(W)1h

(4.96)

i.e., the greatest W-quasi-convex minorant of f. Note that, by (4.96), (4.91), and (4.14) (for .A4 = S(W x R) of (2.111)), we have fy(W) — fq(S(W

(f

x R))

E

(4.97)

R X ).

Using (4.97), from Theorems 4.2, 4.3 and formula (2.127) we obtain Theorem 4.7 Let X be a set, W C le and f : X --÷ T. Then

fq(w) (x) = fq(W)(X) =

inf d = inf d xeco w S, (f) xecow Ad(f) sup sup dER weW w(x)>d

inf

(x

(x

f(y)

yEX w(y)>d

E

E

X),

X).

(4.98) (4.99)

Also, by (4.97), Corollary 4.3(a) and (2.127), we have

sd(fq(w)) , ii cow St(f) = t>d

fl cow Ar(f)

(f e lix , d e R). (4.100)

t>d

By (4.91), W-quasi-convex functions are particular cases of M-quasi-convex functions. In the converse direction, there holds: Theorem 4.8 Let X be a set, and let .A4 c 2x- . Then for Jm c Te defined by (2.157), we have

Q(M) = QUM = Q(JA4 + R).

(4.101)

Proof By (4.2), (4.90) (with W = Jm and W = Jm ± R), and Theorem 2.8, we 0 have (4.101).

4.3 W-Quasi-Convexity of Functions on a Set X, Where W c R x

149

Remark 4.13 (a) By (4.14), (4.101), and (4.96), for any M c 2x we have

fq(M)

—

fq(Jm)

—

(f

fq(J.m+R)

E

(4.102)

R ic )•

Hence, applying Theorem 4.7 and formula (4.100) to W = Jm C R X of (2.157) and using (4.102), (2.172) and (2.156), we obtain again Theorems 4.2 and 4.3 and formula (4.27). (b) From the above we see that the theories of W -quasi-convex and M-quasiconvex functions are equivalent; indeed, by (4.91), for each W c T?x there exists M c 2 x (e.g., M = S(W x R)), such that

(4.103)

Q(W) = and, conversely, by Theorem 4.8, for each M c 2 x there exists W c W = J./1/44 of (2.157)) such that we have (4.103).

R -

x- (e.g.,

By (4.91) and Theorem 4.8 (applied to M = S(W x R)), we obtain: Corollary 4.6 Let X be a set, and let W c R x . Then

(4.104)

Q(W) = Q(.1s(wxR)) = Q(Js(wxR) + R). By (1.35), for any set X and any M this line, let us also prove:

C

2, we have C(C(M)) = C(M). Along

Proposition 4.8

Let X be a set and .11/1

C

2x . Then

(4.105)

k(Q(M)) = C(M),

(4.106)

Q(Q(M)) = Q(M) = Q(C(M)). Proof By (2.112) for W = Q(M), (4.34), and (1.35), we have

1C(Q(M)) = C(S(Q(M) x R)) = C(C(M)) = C(M). Furthermore, by (4.90) for W = Q(M), (4.105), and (4.2), we have the equivalences f

E

Q(Q(M)) .:=> Sd(f)

E

k(Q(M)) = C(M) (d

E

R) f

E

Q (M). Finaly,b

(4.2) and (1.35), we have f

E

Q(M) .:=> sd(f)

E C(M) =

C(C(A4)) (d E R) . f

One can obtain corresponding results for W .A4 = S(W x R) and using (2.112) and (4.91):

C

E

Q(C(M)).

El

R X by applying the above to


150

Proposition 4.9 Let X be a set and W c le. Then

S(Q(W) x R) = 1C(W).

(4.107)

Proof By (4.91), (4.34) for M = S(W x R) and (2.112), we obtain S(Q(W) x R) = S(Q(S(W x R)) x R) = C(S(W x R)) = 1C(W).

El

Proposition 4.10 Let X be a set W c T? x . Then

C(K(W)) = 1C(W) = Me(W)) = k(Q(W)),

(4.108)

Q(Q(W)) = Q(W) = Q(K(W)).

(4.109)

Proof By (2.112) and (1.35) (for M = S(W x R) in E = (2X, 2)), C(1C(W)) = C(C(S(W x R))) = C(S(W x R)) = 1C(W). Furthermore, by (4.91), (4.105) (for M = S(W x R)), and (2.112), we have

1C(Q(W)) = 1C(Q(S(W x R))) = C(S(W x R)) = which, together with W c C(W) c Q(W) (by (1.7) and (4.115) below) and (2.117), proves (4.108). Also, by (4.91) and (4.106) (for M = S(W x R)), Q(Q(W)) = Q(Q(S(W x R))) = Q(S(W x R)) = Q(W). Finally, by (2.112), (4.106) (for M = S(W x R)), and (4.91), we obtain

Q(1C(W)) = Q(C(S(W x R))) = Q(S(W x R)) = Q(W).

El

One can use fy(w) to localize the concept of W-quasi-convexity of functions. Indeed, applying Definition 4.3 to M = S(W x R) and using (4.97), we arrive at:

Definition 4.7 Let X be a set, W c R x and .7c0 G X. A function f : X ---> R is said to be W -quasi-convex at xo if f(x0) = fq(w)(xo)• We will denote by Q(W; x0) the set of all such functions: Q(W; xo) =

If

G

R X I f (xo) = fq (w)(xo)) •

(4.110)

4.4 Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X

151

One can prove localized versions of some of the results on Q(W). For example, (4.95) admits the following localization (in the same way as (3.60) is a localized version of (3.31)): Q(W; x0) = If E R X I \yid E R, f(x0) > d, ]tV G

(4.112)

W, sup w(Sd(f )) < w(xo)}.

4.4 Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X, Where W C R x Proposition 4.11 Let X be a set and W C Te . Then

C(W) C C(W ± R) C Q(W + R) = Q(W).

(4.113)

Proof By (1.9) for »11 = W, M2 = W ± R, in E = (Rx , (), we have C(W) c C(W ± R). Also, by (4.94), Q(W ± R) = Q(W). Finally, if f c C(W), then, by (3.3), (4.9), (2.116), and Proposition 2.9(a), we

have sd(f) = sd(sup{w E W I w

= Wolf n

sd(w) E 1C(W)

f})

(d E R),

(4.114)

11),,f

so f E Q(W) (by Definition 4.5). Thus for any W C R X , C(W) Ç Q(14), whence, replacing W by W ± R, we obtain C(W + R) C Q(W + R). Remark 4.14 4.5a below).

(4.115) 0

The inclusions c in (4.113) and (4.115) may be strict (see Section

From Proposition 4.11 and Definitions 3.2 and 4.6, we obtain: Corollary 4.7 Let X be a set, and let W c TV ( . Then for any f : X ---> -.R7- we have

.fco(w) '-- W. Thus, for example, (3.37) of Proposition 3.2 (which could be also used to define the concept of W-convex functions f : X —> R), is "parallel to" (4.90) of Definition 4.5, and formulas (3.31) of Theorem 3.1 and (3.43) of Theorem 3.3 are "parallel to" Proposition 4.7 and formula (4.98) of Theorem 4.7, respectively. For W = JM, where »1 c 2 ic and ././vf is the set (2.157), one can improve part of (4.113) as follows: Theorem 4.9 Let X be a set, and let M c 2 x . Then

C(Jm R) = Q(Jm R).

(4.117)

Proof By Theorem 4.6, JM R is a base of the convexity system B = Q(M) in E = (— Rx, and hence, by Definition 1.8, we have Q(M) = C(J.A4

R).

(4.118) CI

Hence, by (4.118) and Theorem 4.8, we obtain (4.117).

Remark 4.16 Since JM R is a base of Q(M) = C(Jm R), the results on yield, in particular, results on the base »1 of C(M) in a complete lattice E = (E, the base Jm R of Q(M) in E = (le , Thus, for example, replacing in (1.5) R and Q(M) respectively, we obtain .A4 and C(M) by J m

f} o

{ — Xx\m ± d G JM ± RI — Xx\m d

(4.119)

(f G Q(M) \ {-00}), or, equivalently (by (4.47)), {(M,d) E .A4 x Rid (f

E

inff(X \M)}

0 (4.120)

Q(M) \ {—oo}),

which is equivalent, obviously, to {M c

MI inf f (X \ M) > —00}

0

(f

E

Q(.A4)\ {—oo}).

(4.121)

4.4 Relations Between W-Convexity and W-Quasi-Convexity of Functions on a Set X

153

One can also verify (4.120) by taking any f e Q(M) \ {—oo}, d c R and x E X with d < f (x), and any M E »1 satisfying Sd(f)cM,xEX\M (which exists, by Sd(f) E C(M) and (2.24) of Theorem 2.3); indeed, then X \ M d. inf f (X \ Sd(f)) X \ Sd(f), whence inf f (X \ M)

C

Corollary 4.8 Let X be a set, and let W

C — Rx.

C(Js(w.R)

Then R) = Q(Js(w xR)

R).

(4.122)

Proof By (4.117) for .A4 = S(W x R), we have (4.122). Remark 4.17 Formula (4.118) shows thatfor each M c 2 x there exists W c R x bit R, with JM of (2.157)) such that (e.g,W= (4.123)

C(W) = Q(M),

but we have no converse result (saying that for each W c Rx there exists »1 C 2 x in fact, we only know (by (4.115) and (4.91)) that for anysuchta(4.123)old; W C — R X , C(W) C Q(W) = Q(S(W x R)). For W = Jm, where .A4

2 x , one can improve part of (4.116) as follows:

C

Corollary 4.9 Let X be a set, and let M c 2 x . Then for any f : X fco(Jm+R)

—

R we have

fq(Jm+R)•

Proof By Theorem 4.9 and Definitions 3.2 and 4.6, we have (4.124).

(4.124) LI

Remark 4.18 (a) Similarly to Remark 4.15(a), Corollary 4.9 implies again Theorem 4.9. (b) From (4.124), (4.102), and (3.11) there follows again formula (4.45) of Theorem 4.6. Since the Definitions 3.5, 4.3, and 4.7 of the localized concepts of convexity and quasi-convexity have used only the corresponding functional hulls, from (4.124) and (4.102) there follows: Corollary 4.10 Let X be a set, M

C

2x and X0 E X. Then

C(Jm R; xo) = Q(J.A4

xo) = Q(J.A4; xo) = QCM; xo).

(4.125)

154


Remark 4.19 (a) Corollary 4.10 shows that Remark 4.17 can be carried over to the localized concepts. (b) Applying Corollary 4.9 and formula (4.102) for M = S(W x R) and using (4.97), we obtain fco(is(vxR)+R) C(JS(WxR)

=

fq(Js ( wR ) +R = fq(Js(wxR)) = fy(W),

R; x0) — Q(Js(wxR)

(4.126)

R; x0) = Q(Js(wxR); x0)

= Q(W; x0).

(4.127)

If X is a set and W c R x , then, by Proposition 1.5 (applied to M = W in the complete lattice E = (Rx , ,)), we have C(C(W)) = C(W).

(4.128)

Along this line let us prove: Proposition 4.12 Let X be a set and W c X . Then Q(C(W)) = Q(W) = C(Q(W)).

(4.129)

Proof By (1.7) and (4.115), we have W c C(W) c Q(W), whence, by (4.93) and (4.109), we obtain Q(W) g Q(C(W)) g Q(Q(W)) = Q(W), which yields Q(C(W)) = Q(W). Alternatively, this also follows from (4.108) and (4.90). Finally, by Theorem 1.4 applied to the convexity system B = Q(W) in E = (R x , we have C(Q(W)) = Q(W). 0 Remark 4.20 Similarly, if X is a set and MC 2x , then, by Theorem 1.4 applied to the convexity system B = Q(M) in E = (Rx , we have C(Q(M)) = Q(M)-

4.5

(4.130)


We now show that for some particular families M c 2x considered in Section 4.2, the equality (4.118) (with Jm of (2.157)) yields further characterizations of M-quasi-convex functions.

4.5a

Lower Semicontinuous Quasi-Convex Functions Revisited

Let X be a locally convex space, and let M be the family of all closed half-spaces (2.42) in X; then (see Section 4.2b) a function f : X --> R is M-quasi-convex

4.5 Some Particular

Cases

155

if and only if it is lower semicontinuous and quasi-convex. Also, if W = X* (or, W = X* ± R), then, by the main statement of Section 2.4, a function f : X —> is W-quasi-convex if and only if it is lower semicontinuous and quasi-convex; this fact, together with Theorems 3.8 and 3.7, shows that for W = X* the inclusions in R) are strict. (4.113) (and hence in (4.115) for W = X* and W = X* On the other hand, let us observe that for the closed half-space (2.42) (where cl) E X* \ {0}, d G R) we have, by (2.156), XX\M

— — X{yEX I c(v)>d} —

X(d,H-co) °

3,

(4.131)

and hence for im of (2.157) we obtain k1/44

(4.132)

R = K o X* ,

where (4.133)

K = { — x(d,+,0) ±tId,t c R} c

Thus, using (4.118), there follows: Theorem 4.10 Let X be a locally convex space. A function f : X and quasi-convex if and only if

f=

sup

( — X(d,+00)

— R is lower semicontinuous

t) o

(4.134)

d,tE R,c1)E X* (—X( 1 .+-,00±t)oc13 f

We recall that a function f : X --> R is said to be quasi-affine if both f and — f are quasi-convex (this term is used due to the analogy with the well-known fact that if f (X) C R and both f and — f are convex, then f is affine). From Theorem 4.10 we obtain the following result, corresponding to Theorem 3.7: Corollary 4.11 Let X be a locally convex space. A function f : X ----> R is lower semicontinuous and quasi-convex if and only if it is a supremum of a set of lower semicontinuous quasi-affine functions. Proof If f is lower semicontinuous and quasi-convex, then, by Theorem 4.10, we have (4.134), where, by Proposition 4.13 below, implication 2' = 1', each ( — X(d,+00) -1- t) o c13 is lower semicontinuous and quasi-affine (since each — x(d,±00) t is nondecreasing and lower semicontinuous on R; indeed, Sr( — x(d,+00) t) = (—oo, d] if r < t and R if r t, so each Sr (—x (d, +) t) is closed in R). Conversely, assume now that f = supwcw w, where W C R X is a set of lower semicontinuous quasi-affine functions w. Then Sd f = nwE w Sd (W) (see (4.9)), where each Sd(w) (w E W, d c R), and hence each Sd (f) (d c R), is a closed convex set, so f is lower semicontinuous and quasi-convex.

156


Thus, in order to complete the proof of Corollary 4.11, it will be enough to prove (the implication 2° = 1° of) the following proposition, which gives useful characterizations of lower semicontinuous quasi-affine functions: Proposition 4.13 Let X be a locally convex space. For a function w : X —> R the following statements are equivalent:

I°. w is lower semicontinuous and quasi-affine. 2°. There exist c13 E X* and a lower semicontinuous nondecreasing function k : R --> R such that w = k o 1. 3°. For each d E R, Sd (W) is 0, or X, or a closed half-space in X. Proof 1° = 30 . If 1' holds, then, for each d E R, Sd (W) is closed and convex, and X \ Sd (w) = ly G X I W(Y) > d) = {y G X I w(y) < —d} = A_d(—w) is open and convex. Hence, if S d (W) 0, then, by the separation theorem 0 and X\ Sd (W) (e.g., see [461, p. 24, thm. 4), there exists (I) E X* \ {0 } such that sup cl) (Sd(w)) —

inf (I)(X \

Sd (W)),

and thus

Sd (w) g_ ly E X I (1). (y)

inf (I)(X \ Sd(w)))•

(4.135)

But, since X \ Sd (W) is open and convex and (I) c X* \ 10), we have X \ Sd(W) C G X I cl(y) > inf (I)(X \ Sd (D))) (by Lemma 2.1 for w = —(1)), whence the opposite inclusion to (4.135), and hence the equality Sd (w) = {y

G

X

I (NY)

inf (1)(X \ Sd(w))}-

(4.136)

Thus Sd (W) is a closed half-space in X. 3 0 = 2°. Assume first that for each d E R, Sd (w) is either 0 or X. Then for d E R and any x E X we have X E Sd (W) if and only if S d (W) = X, whence w(x) = inf

dER XESd(W)

d = inf

del?

d=d

(x E X),

Sd(W)=X

that is, w = c7 (a constant function). Hence, taking k = cl and an arbitrary (I) E X*, there holds k((I)(x)) = d = w(x) (x E X), i.e., w = k o (I) where k is a constant function. Assume now that 3° holds and that there exists do E R such that Sdo (w) is a closed half-space, say, Sdo (W) = ty G X I

'D(y)

(4.137)

do . Then Sd (w) C 5d0(w) 0 X, so where (I) E X* \ (0), td( c R. Let d Sd (W) X. If Sd (W) is a closed half-space, say,

Sd(w) = ( y G X 1 11/ (Y)

t),

(4.138)


157

where kIl E X * \ (0), t G R, then by Sd(w) C Sdo (W), (4.137), (4.138) and an infinite dimensional version of the Farkas lemma (e.g., see [81], Theorem 4), there exists A c (0, ±Do) such that cl) = AW tdo At; thus, by (4.138), we obtain ,

Sd(w) = ly

E X I (1) (Y)

(4.139)

td},

tdo . Finally, if Sd(w) = 0, then we can still write (4.139), with with td = Ar td = —00 < tdo• On the other hand, assume now that d > do. Then Sd(w) D Sdo (W) 0 0, SO Sd(W) 0 0. If Sd(W) is a closed half-space, then, similarly to the above, we obtain (4.139) for some td G R with td tdo . Finally, if Sd(W) X, then we can still write (4.139), with td = -{-oo > tdo • I? by Define k : R k(t) = inf d

(t

dER t-td

c R),

(4.140)

7- implies {d E R with ltd I d G R) as above. Then, since t ( td) C {d E RIIt td } , whence k(t) k(7:), the function k is nondecreasing. Furthermore for the family Wd IdER of closed convex subsets of R, defined by

if td Ud —

/0(-0C), td] R

= C°'

if td

G

if td

= ±°°'

(4.141)

R,

we have (4.20) and k = ', { t'd } (of (4.21), with X = R), and hence k is lower semicontinuous (by the consequence of Corollary 4.2, mentioned in Section 4.2b). Finally, by (4.139) and (4.140), we have w(x) = inf

dER xESd(w)

d=

inf

dER c13(x)rd

d = k(cD(x))

(x E X),

that is, w = k o 1. 20 = 1 0 . If 2' holds, then, since k is lower semicontinuous and cl) is continuous, w = k o cl) is lower semicontinuous. Now let x,:x- G X, say, cl(x) (t(), and let 1. Then (1) (x) = (1)(Ax

± (1 —

(I) (A x

A)x)

(1 — ))

(A ' ± (1 — )) =

whence, since k is nondecreasing and w(x) = k(cI)(x)) (x e X), we obtain min (w(x), w(Y)) = w(x)

w(Ax + (1 — A.)5C')

w( x--) = max (w(x), w(.77));

thus both w and —w are quasi-convex, that is, w is quasi-affine.

El


158

Remark 4.21 Theorem 4.10 gives a representation of each lower semicontinuous quasi-convex function f as a supremum of a much smaller set (in fact, parametrized by RxRx X*) of lower semicontinuous quasi-affine functions than that of all lower semicontinuous quasi-affine functions majorized by f (asserted by Corollary 4.11).

4.5b

Evenly Quasi-Convex Functions Revisited

Let X be a locally convex space, and let M be the family of all open half-spaces (2.53) in X; then (see Section 4.2c) a function f : X —> R belongs to Q (M) if and only if it is evenly quasi-convex. On the other hand, let us observe that for the open half-space M of (2.53) (where .43 E X* \ {0), d E R) we have, by (2.156), XX\M =

X{yEX

1:13 (Y)?-(1 } =

Xid,±00) 0

(D,

(4.142)

and hence for Jm of (2.157) we obtain (4.132), where K = {— xid.+00) ±tid,t E /2 } c R R .

(4.143)

Thus, using (4.118), there follows: Theorem 4.11 Let X be a locally convex space. A function f : X —> R is evenly quasi-convex if and only if sup

(—xid.+00)

+ t) 0 (D.

(4.144)

d,tER,c1)Ex*

We recall that a function f : X —> R is said to be evenly quasi-affine, if it is quasi-affine and evenly quasi-convex. From Theorem 4.11 we obtain: Corollary 4.12 A function f : X --> R is evenly quasi-convex if and only if it is a supremum of a set of evenly quasi-affine functions. Proof The proof is similar to the above proof of Corollary 4.11, using now that each —x Ed , ±0„) is nondecreasing on R and using, instead of Proposition 4.13, the (implication 2' 1° of the) following proposition, which gives useful characterizations of evenly quasi-affine functions: Proposition 4.14 Let X be a locally convex space. For a function w : X -- R the following statements are equivalent: 1°. w is evenly quasi -affine


159

2 0 . There exist (1) E X* and a nondecreasing function k : R -÷ T? such that w=k0(1). 3'. For each d E R, Sd (W) is 0, or X, or a (closed or open) half-space in X. Proof 1' = 3 0 . If 1' holds, then, for each d E R, Sd(w) is evenly convex and X \ S, (w) = A —d (— tV) is convex. Therefore, if Sd (W) 0 0 and X \ Sd (W) 0 0, then

the interior Int Sd (W) 0 0, and hence, as in the proof of Proposition 4.13, there exists (I) c X* \ [0) satisfying (4.135). Since (I(y) ?. inf cI)(X \ S(/(w)) (y E X \ Sd(w)), we have also

fy

C

XI (NY) < inf (I)(X \ SAO)} g SAO.

(4.145)

We will show that Sd (W) coincides with one of the two half-spaces determined by (I?, occurring in (4.135) or (4.145). Indeed, if not, then both inclusions (4.135) and (4.145) are strict. In other words, there exist X1 E X\ Sd (W) and x2 E Sd(W) such that cI)(x i )..< inf cI)(X \ Sd (0), whence, by Xi E X \ S d (w), x2 E

Sd(W)

(1) (x2)

inf (1)(X \ SAO),

and (4.135), we obtain

(I)(x i ) -= inf (1)(X \ SAO) = (1) (x2)•

(4.146)

Now, since Sd (W) is evenly convex and x l g Sd (W), there exists an open half-space M = ( y E X I (Do(y) < do) (where CD0 E X* \ (0), d E R) such that Sd(W) g

M = ( y E X I cDo(Y) < do), xi g M.

(4.147)

But, by (4.145) and (4.147), we have the inclusion of open half-spaces ly c X I (I)(y) < inf (1)(X \ Sd (w))) ç (y E X I (D0(y) < d0), whence also the inclusion of their closures, that is, ly E X I (I)(y)

inf (I)(X \ S, (w)) C (y E X I 4:1310(y)

de ),

and hence (e.g., by [81], thm. 4) there exists A E (0, +00) such that (Do = X 44), do ,X inf (I)(X \ Sd (W)). Thus, by M = ly E X 1 (I)0(y) < do ), we obtain M = (Y E X

I CD(Y)
R is evenly quasi-coaffine and only if

sup

(—xi } + t) o (D.

if

(4.155)

daER.(1)EX* ( — X1di+t) °(1) f

From Theorem 4.12 we obtain the following result corresponding to Corollaries 4.11 and 4.12: Corollary 4.13 A function f : X —> R is evenly quasi-coaffine if and only if it is a supremum of a set offunctions of the form k i o cl) i , where k i E R R and cDi E X* for each i. Proof The "only if" part is obvious by Theorem 4.12. Conversely, assume now that f = supi "(k i o cl)i ), where k E RR and c1) i E X* (i E I). Then, by Proposition 4.1(a), it is enough to show that each ki o 443, i is evenly quasi-coaffine. But this follows from the fact that for any x E X, dc R,kc and (I) G X* we have X G

0 (D) - k (cD (x)) X

E

d (1).(x)

A t(t E R, k(t) > d)

fl ly E X 10(y)

tER,k(t)>d

A t).

CI

162


Remark 4.23 (a) Corollaries 4.13 and 4.12, combined with Proposition 4.14, implication 1° = 2', imply again the fact, observed in Section 4.2d, that every evenly quasi-convex function is evenly quasi-coaffine. (b) Theorem 4.12 gives a representation of each evenly quasi-coaffine function f as a supremum of a much smaller set of functions (in fact, parametrized by RxRx X*) than that occurring in Corollary 4.13. (c) By Corollary 4.11 (resp. Corollary 4.12) and Proposition 4.13 (resp. Proposition 4.14), equivalence 1' . 2°, a function f : X —> R is lower semicontinuous (resp. evenly) quasi-convex if and only if it is a supremum of a set of functions of the form k, o (I)„ where each (P i E X* and each k, : R ---> R is nondecreasing and lower semicontinuous (resp. nondecreasing); in the "only if" part, such a representation as a supremum is given by (4.134) (resp. (4.144)). Next, Corollary 4.13 gives a similar 1? characterization of evenly quasi-coaffine functions f : X —> R, with k, : R ---> — being arbitrary. Also, by Theorem 3.7, f is a proper lower semicontinuous convex function, or f ±oo, if and only if f is a supremum of a set of functions of the form ki o cD i , where each cD i E X* and each k, : R —> R is of the form k,(t) = t b, (t e R) for some b, G R. Generalizing these facts, Martinez-Legaz [168], [170] has introduced and studied the following concept: Let K C R R be a set of functions k : R ---> R, closed for supremum (i.e., such that k C K implies supk a k E K), and let X be a locally convex space. A function f : X -÷ R is said to be K-convex if it is a supremum of a set of functions of the form k, o CI, where each CD'i E X* and each k, E K. Also Martfnez-Legaz [168], [170] has observed that obviously f : X —> R is K -convex if and only if it is W -convex, where W = K 0 X* =- {k o(Dik K, (I) G X*);

(4.156)

recently Penot and Volle [219], using this latter observation, have obtained some further results on K -convexity.

4.6 (W, (p)-Quasi-Convexity of Functions on a Set X, Where W Is a Set and cp : X >CHI R Is a Coupling Function Definition 4.8 Let X and W be two sets and ço : X x W —> R a coupling function. A function f : X —> R is said to be quasi-convex with respect to the pair (W, (p), or, briefly, (W, 0-quasi-convex, if Sd(f)

C

1C(W, (p )

(d E R)

(with 1C(W , (p) of Definition 2.14). We will denote by Q(W, (W, 0-quasi-convex functions on X.

(4.157) the family of all

Remark 4.24 (a) By Definitions 4.8 and 4.5 and Remark 2.33(b), (c), the theories of (W, 0-quasi-convexity and V -quasi-convexity (where V C R x ) of functions on X, are equivalent.

4.6 (W, (p)-Quasi-Convexity of Functions on a Set X

163

(b) Similarly to the observations of Sections 2.6 and 3.3, one can obtain results on general (W, 0-quasi-convexity of functions on X from the results of W-quasiconvexity, where W C RA', simply by replacing w(x) by ça(x, w). For example, defining the (W, ço)-quasi-convex hull fy(w4) of a function f : X —> R in the obvious way (replacing Q(W) by Q(W , ço) in (4.96)), Theorem 4.7 extends to inf

fq (w,p ) (x)

xEco w ., S' (f)

fq(w.co(x)

inf

d=

(f E R A' , x E X),

d

XECOw .w Ad(f)

sup sup

inf

dER w EW

y EX

f (y)

(f E R X , x E X).

(4.158) (4.159)

yo(y,w)> d

d

Also formula (4.100) extends to sd(fq(w,42)) _ n cow,, st(f) = n cow,, At(f)

(f E R x , d E R).

t>d

t>d

(4.160) (c) One can use fq( w.,p) to localize the concept of (W, 0-quasi-convexity of functions, defining for any xo c X, Q(( 1 §0 );

X()) =

If E

I

(xo) = fq(w,o(xo)},

(4.161)

and one can prove localized versions of some of the results on Q(W , ço). For example, extending (4.95) and (4.112) as in (b) above, we have Q(W , ço) = (f E R X I V(X, d) cXxR, f (x) > d, ]/.1) E W,

sup ça(y, w) < ço(x, w)},

(4.162)

YESd(f)

and its localized version: Q((W, (p); xo) =

If

E RX

1Vd E R, f(x 0 ) > d,

3w E W,

sup (P(Y , ID)

< OXO, 0).

(4.163)

YESd(f)

Now we will give a result that shows that the theories of (W, ço)-quasi-convex functions on X, where ço :Xx W—> R is a coupling function of type {0, —00}, and M -quasi-convex functions on X, for M C 2 X parametrized by (W, 0) (with the same W), are equivalent. Theorem 4.13 Let X and W be two sets. (a) For any mapping :W ---> 2X, there exists a coupling function ço o : X x W ---> R of type {0, —oo} such that Q(W (P) = Q(9 (W)).

(4.164)


164

(b) For any coupling function (p : X x W —> R of type mapping 0,p : W ---> 2x such that

{0, —00), there exists a

(4.165)

Q(W, go) = Q(9 2 x , there exists a coupling function (po : X x W — R of type ( 0, —00) such that C(Vw,p,

R) = Q(9(W)) = Q(W, (Po) = Q(vw,)•

(b) For any coupling function (p : X x W —> R of type mapping Ov : W —> 2' such that C(V w , cp

(4.166)

{0, —00), there exists a

R) = Q(0,p (W)) = Q(W, (p) = Q(V w ,ço ).

(4.167)

Proof: (a) Given O : W —> 2 x , define (po : X x W —> R by (2.203). Then, by (2.205), (4.118) (for M = 0(W)), (4.164), and (2.189), we obtain C(Vw

R) = C(.19(w) R) = Q(0(W)) = Q(W (Po) = Q(Vw4„)•

(b) The proof is similar, defining Oço : W —> 2 x by (2.206).

Remark 4.25 (a) If X is a set, W c R x and M = S(W x R), parametrized as in Remark 2.34(c), then for the ( 0, —00)-type coupling function (A) of (2.211), we obtain from (2.212) and Definitions 4.1 and 4.5, Q(W x R, (Po) = Q(f4-I , goo) = Q(9C14-7 )) = Q(S(W x R)) = Q(W). (4.168) (b) If X and W are two sets and ça : X x W —> R is a coupling function, and if M = S(V w4 x R) c 2x of (2.191) is parametrized as in Remark 2.34(d), then for the ( 0, —00)-type coupling function *ow of (2.214), we obtain from (2.215) and Definitions 4.1 and 4.5,

Q(W x R,

= Q(099 (W x R)) = Q(S(V w ,,,, x R)) = Q(V w4).

(4.169)

(c) Theorem 4.14(b) shows, in particular, thatfor any sets X, W and any coupling function yo : X x W —> T? of type ( 0, —00), we have the equality R) = Q(W , (p),

(4.170)

4.7 Other Equivalent Approaches to Abstract Quasi-Convexity of Functions

165

in which 0 does not occur explicitly. Note that (4.170) need not hold when yo is not of type 10, —oo) (e.g., when X is a locally convex space, W . X*, and yo =

4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on a Set X, with Respect to Convexity Systems B C 2x and Hull Operators u : For the two approaches to abstract convexity of subsets of X mentioned in Definitions 2.4 and 2.7, we arrive at the following two concepts of quasi-convex functions f : X —> R, respectively:

Definition 4.9 Let X be a set and B c 2 x a convexity system. We will say that a function f : X —> R is quasi-convex with respect to B, or, briefly, B-quasi-convex, if Sd(f) e B

(d

(4.171)

R).

E

We will denote by Q(B) the set of all B-quasi-convex functions on X: Q(B) =

If

E

R x I Sd(f)

E

B (d

R)).

E

(4.172)

Definition 4.10 Let X be a set and u : 2x ---> 2x a hull operator on (2 x , D). We will say that a function f : X —> R is quasi-convex with respect to u, or, briefly, u-quasi-convex, if Sd ( f )

E

(d

C (u )

E

(4.173)

R)

(with C(u) of (2.35)). We will denote by Q(u) the set of all u-quasi-convex functions on X: Q(u) =

if

E

R X I Sd(f)

E

C(u) (d

E

R)).

(4.174)

Remark 4.26 (a) By (4.174) and (2.35), we have Q(u) =

If

E

R X I Sd(f) = u(Sd(f)) (d

E

R)).

(4.175)

(b) By Remarks 2.3(b), 2.5, and 2.28(e), the approaches to quasi-convexity of functions f : X —> T?, via Q(B) (where B c 2x is a convexity system) and Q(u) (where u : 2x —> 2 x is a hull operator) are equivalent to each other, and to the approaches via Q (M) (where M c 2x ) and Q(W) (where W c R x ). Each of these approaches has its own interest, and in certain applications one may be preferable to other equivalent approaches. (c) Since the convexity system B of Definition 4.9 is a subfamily of 2, one can obtain results on Q(B) by applying the theory of Section 4.1, in particular, for

166


M = B. For example, by (4.14) (with M = B), we have for any f E RA', fq(B) =

max th E Q(B) I h

f

(4.176)

where fq(B) is the B-quasi-convex hull of f, in the sense of Definition 4.2 (with M = B). Corresponding to Definitions 4.2, 4.6, and to (4.176), it is natural to give: Definition 4.11 Let X be a set and u : 2x ---> 2x a hull operator on (2 x , D). _ For any f : X —> T? the u-quasi-convex hull off is the function fq(u ) : X ---> R defined by

fq (u) = max th E

Q(U)

Ih

(4.177)

f

with Q(u) of Definition 4.10. Remark 4.27 (a) Similarly to Remark 4.2(a), considering the convexity system Q(u) in (R x , fq(u) is nothing else than fc„ou , (of (1.59)), the Q(u)-convex hull of f. (b) One can use fq(B) and fq(u) to localize the concepts of B-quasi-convexity and u-quasi-convexity of functions, defining, for any ,C0 E X, Q(B; x0) = (f E R X I f(x 0 ) = fq (5)(x0)}

(4.178)

Q(u; xo) = ff . E R X I f (x0) = fq (u)(xo)) •

(4.179)

4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators on i-?x Definition 4.12 Let X be a set. A hull operator y : R x —> R x (in the sense of Definition 3.3) will be called a quasi-convex hull operator if there exists a hull operator u : 2x —> 2x (in the sense of Definition 2.3) such that y is the quasi-convex hull operator with respect to u, i.e., such that fv = fq (u)

(4.180)

(f E R X ),

where f, is that of Definition 3.3. Note that, by (1.70), for any hull operator y :

f(x) = max h(x) h EC (v) 1 f

R x we have

(f E R x , x E X).

(4.181)

4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators 167

In the present section we will give some characterizations of quasi convex hull operators among hull operators 1) : R x —> R x . We will denote by C the set of all functions k : R —> R satisfying k(sup di ) = sup k(di )

(I 0 0, {c1,}, Ei C R),

(4.182)

(I 0 0, {(l i } jel C R),

(4.183)

iEi

k(inf di ) = inf k(di ) iEr

and we will denote by K the set of all functions k : R —> R satisfying (4.182). Furthermore we will use the notations Coe = (k E C I k(—oo) = —00),

(4.184)

= fk E K I k(—oc) = —oo);

(4.185)

note that C c K, and hence Coo c K. Theorem 4.15

Let X be a set. For a hull operator y : R x ---> R x the following statements are equivalent: P.

There exists a hull operator u : 2x —> 2 x such that

fv 20 .

fq(u)

4°.

(4.186)

We have (k o f)„ = k o f,

3'.

(f E Rx).

(f E R x , k E Cco ).

(4.187)

We have kof E C(v)

(f E C(v), k E Coe ).

(4.188)

kof E C(v)

(f E C(v), k E K oe ).

(4.189)

We have

Proof 1° = 2°. Assume 1°, and let .A4 = C(u) c 2 x (of (2.35)). Then, by Theorem 1.4 (for B = C(u)), we have C(u) = C(C(u)) = C(M), whence, by (4.186), iv = fq(u) = fq(M) ( f E R X ). Let x E X.

CASE A: {M E MIX E X\ M) 0 0 (or, equivalently, x g nme./vt M com 0). Then, by Theorem 4.3, (4.183), and (4.182) we obtain for any f E R X and

168

k e


C, (k o f),(x) = (k o f) q(M ) (x) =

sup

inf(k o f)(X \ M)

sup mEm xEx\m

k(inf f (X \ M)) = k(

MEM

sup

inf f (X \ M))

MEM xEX\M

x EX \M

= k(fq(m ) (x)) = (k o f,)(x).

CASE B: M I x E X \ M) =- 0 (or, equivalently, x E com 0). Then, by Theorem 4.3 and (4.184), we obtain, for any f e R x and k E Coo , (k o f),(x) = sup inf(k o f)(X \ M) = —oc = k(—oo)

mo

= k(sup inf f (X \ M)) = k(f q(m ) (x)) = (k o f,)(x). M Eø

2° = 3°. If 2° holds and f E C(v), k E Coo , then f = f,„ whence, by (4.187), we obtain (k o f), = k o f, = k 0 f,

that is, kof E C(v). 2' be as in the above proof of the 1° = 4°. Assume 1°, and let .A/1 = C(u) implication 1° = 2°, so f = fq(m) (f E Te), whence C(v) = Q(M). Then, by Theorem 4.6, (4.182), (4.47), and (4.120), we have sup

(k o f)(x) = k(f (x)) = k(f q(m ) (x)) = k(

(—x x\m t)(x))

(mmEm xR

_x x\m +t1

=

sup

(k o (—x x\A4

t))(x)

(MMEMxR —

Xx\to+t_çf

(f G C(v) \ (—co) = Q(M) \ (—oc), k E K„, x E X)

and, by (4.185) and Proposition 3.1(a) we have k o (—oo) = k(— oc) = — 00 E C(V). Thus, to prove (4.189), it will be sufficient (by Proposition 4.1(a)) to show that k o ( — Xx\Aft t)

(k E K oo , M E M, t

E Q(M)

E

R).

(4.190)

But, by (2.156) and (4.185), for any k E K oo , M c M, and t e R we have

k(—oo) = —oc

if x E M,

k(t)

if x E X M,

k( — xx\m(x) t)

4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators 169

whence, by (1.7) and (2.6) we obtain for any d

E

R,

Sd(k 0 (—x x\A4 t)) = {x E X I k(— x x\m (x) t)

M

if d < k(t)

X

if d

E

d}

C(M),

(4.191)

k(t)

which proves (4.190). The implication 4° 3' is obvious (since C oo c K 00 ). 3° 1'. We will show that if 3" holds, then the mapping u : 2x —> 2 x defined by u(G) = tx E X I (XG)v(x)

0}

(G

C

X)

(4.192)

is a hull operator satisfying (4.186). Let G1, G2 C X, G1 C G2. Then xG, xG 1 , whence, by (3.8), (XG 2 )v (XG1)v, and hence, by (4.192), u(G 1 ) C U(G2)• x G (g) = 0 (g E G); Let G C X. Then, by (3.9) and (2.156), we have (x G ) v (g) that is, G C u(G). Again, let G C X. Then, by G C u(G), we have Xu(G) XG, whence, by (3.8), 0 = X u (G)(X) (XG)v • On the other hand, by (4.192), we have (X(;))(x) (X,,,(G))v u(G), whence (XG)v for all x E X(G) (since xu(G)(x) = +00 for x E X \u(G)). Therefore, by (3.10) and (3.8), (XG)v = (XG)vv (X.(G))v, which, together with proved above, inequality yields (Xu(G))v = (XG)v, whence, by (4.192), the oposite satisfies (2.13)—(2.15); that is, u is a hull operator. u(u (G)) = u(G). Thus u Now, let us prove (4.186), or, equivalently, that (4.193)

C(v) = Q(u).

In order to show the inclusion C in (4.193), let f E C(V), and let us show (see (4.175)) that S 1 (f ) = u (Sd(f )) (d E R) , or, equivalently (by (2.14) and (4.192)), that for any d E R we have the implication X E X, f (x) > d = (Xsd (f))v(x) > 0.

(4.194)

Let x E X, f (x) > d, and take any nondecreasing continuous function k : R 7i (where R is endowed with the usual extension of the topology of R) such that if p

d,

(4.195)

k(p,) =

if p,

f (x).

Then k E Coo and, by (4.195) (with p, = f (y)), we have k(f (y)) = —oo < 0 (y E Sd(f)), whence k o f xs (,(f). Also, by f E C()), k E Coo , and 3", there holds kof E C(v). Finally, by (4.195) (with p, = f (x)), we have (k o f)(x) =


170

k(f (x)) = +oo. Hence, by (4.181), we obtain (Xs d (f)),(x) =

max

h(x)

(k o f)(x) = -Foo,

h€C(v) lix.sd(1)

which proves (4.194) and the incluson c in (4.193). In order to show the inclusion D in (4.193), by Theorem 4.6 (with M = C(u)) and since Q(u) = Q(C(u)) (by (4.174) and (4.2)) and C(v) is closed for sup (by (1.68)), it will be sufficient to show that —

(G

Xx\G + d E C(v)

C(u), d E R).

E

(4.196)

To this end, let us first show (4.196) for d = 0, i.e., that —

(G

Xx\G E C(v)

(4.197)

E C(U)).

Take any nondecreasing continuous function k : R

R such that

—oc

if —oo À.

0

if À

0, (4.198)

k(X) =

Then k

E

+oo.

C. We claim that XX\G =

Indeed, if x

(k 0 XG)v

(G

E C(U)).

(4.199)

G = u(G), then, by (4.192), (XG)v(x) 0, whence, by (4.198), (k o x G )„(x) = k((x G ),(x)) = —oc. On the other hand, if x fz' G = u(G), then, by (4.192), (XG)v(x) > 0, and hence, by (4.181) (with f = XG), there exists Ti E C(V) such that h xG and -1;(x) > O. Then clearly ah- xG, (x)> 0 (a > 0). But a-1; = 1(1 0 h , where ki E Coo is the function k 1 (A) = aX (X E R) , whence, by 3', ah- E C(v) for all a > O. Hence, by (4.181), E

(x G ) v (x) = max h(x) ?. max (aTi)(x) =1;(x) max a = +-oo, a>0 a>0 hEc(v)

and therefore, by (4.198), (k o XG)v(x) = k((XG)v(x)) =- k(+00) = O. Thus (k o xG)v(x) =

f —oc

if X E G,

1 0

if x

EX

G,

which proves the claim (4.199). But, by (4.199) and (3.10), we have (—xx\G)v =7-(k o xG)„ v = (k o XG)v = — Xx\G for all G E C(U), which proves (4.197). Finally, for any d c R we have — Xx\G ± d = k2 o ( — Xx\G), where k2 c C oo is the function defined by k2(À) = À + d (X E R) . Hence, by (4.197) and 3°, we obtain (4.196), which proves (4.193) and (4.186). Thus 3° 1°.

4.8 Some Characterizations of Quasi-Convex Hull Operators among Hull Operators

Remark 4.28 (a) One cannot replace 1°-4° of Theorem 4.15 do not imply that (k o f), = k o f,

Coo by (f

171

K oo in (4.187), that is, conditions

E

le , k

E

(4.200)

Kœ )

as shown by the following example: Let X = R and u(G) = G, the (usual) closure of G in R, whence fq (u) = f for all f E R R (by Section 4.2e, for M = C(u)). Define f E R R and k E K oo , respectively, by

{ —x

if x < 0,

(4.201)

f(x) = x+ 1

if x

X

if —oc ( À

1

if 0 < À

0, 0,

k(X) =

(4.202) +oc.

Then { k(—x) = 1

if x < 0 I

(k o f)(x) = k(f(x)) =

—= 1, k(x + 1) = 1 if x

fq (u)(x) = f (x) =

if x

0

0,

if x > 0,

1 = (k o f) q(„ ) (0). whence (k o fq(0 )(0) = k(7(0)) = k(0) = 0 (b) One can show (using an argument similar to that of the proof of Lemma 7.1 below) that C = {k K = fk

E

E

R I7 I k is nondecreasing and continuous},

(4.203)

R T? I k is nondecreasing and lower semicontinuous). (4.204)

Chapter Five

Dualities Between Complete Lattices As in Chapter 1 in the present chapter we will place ourselves in the general framework of arbitrary complete lattices.

5.1

Dualities and Infimal Generators

Definition 5.1 Let E = (E,) and F = (F,) be two complete lattices. A mapping A : E —> F is called a duality if for each index set I (including / = 0) we have A (inf ) = sup A (x i ) iez iet

c E).

(5.1)

Remark 5.1 (a) Condition (5.1) for 1 = 0 means, by (1.1), that A (+co) =

(5.2)

where H- oo = oo E , —cc = — oo F Let us recall that a mapping A : E —> F is called a complete inf-antihomomorphism if for every index set 0 we have (5.1). Thus a duality A : E F is nothing else than a complete inf-anti-homomorphism satisfying (5.2). (b) Using the complete lattice F- = (F,) (i.e., F endowed with the "reverse" or the complete lattice E - = (E,), or both, the results on complete order inf-anti-homomorphisms also yield results on complete inf-homomorphisms, complete sup-homomorphisms, and complete sup-anti-homomorphisms, respectively. However, in the sequel we will not mention these consequences. Let us recall that a mapping A : E —> F is said to be antitone, if .

x,

eE,x

Y- -'= A(x) 172

A ().

(5.3)

5.1 Dualities and Infimal Generators

173

Lemma 5.1

Let E and F be two complete lattices. Then for a mapping A : E —> F the following statements are equivalent: 1°. 2°.

A is antitone. For any index set I A( inf x i ) (El

3°.

0 we have sup A (x i )

({xi}id g E).

(5.4)

({xi}id c E).

(5.5)

For any index set I 0 0 we have A( sup x i )

inf A (x i )

iEI

Proof 1° = 2° and 3°, by inf x 1 2° = 1'. If 2° holds and x we obtain

sup x (i E I) and (5.3). ier 7x, then, by (5.4) for / = {1, 2 } , x i = x, x2 =

A(x) = A(inf (x, .7))

xi

sup {A(x), A(Z))

A ( ).

Finally, the proof of the implication 3° = 1° is similar. Corollary 5.1

(a) Every complete inf-anti-homomorphism (and hence, in particular, every duality) A : E —> F is antitone. (b) Every complete inf-anti-homomorphism of E onto F is a duality (c) A mapping A : E —> F is a duality if and only if it is antitone and satisfies for any index set I, A (inf xi )

sup A (x i ).

(5.6)

iEl

Proof (a) This part is obvious from Lemma 5.1, implication 2° (b) Taking = +oo in (5.3), it follows that for every antitone mapping A : E —> F we have A (x) A(H-oo) (x E E), that is, A (-koo) = min { A(x) I x E E),

(5.7)

and hence, if A maps E onto F, so min{A(x) I x E E) —oc, then we must have (5.2). Thus, if A is a complete inf-anti-homomorphism of E onto F, then, by (a) above and Remark 5.1(a), it is a duality. (c) If A is a duality, then we have (5.1), whence (5.6), and A is antitone (by part (a) above). Conversely, if A is antitone and satisfies (5.6), then by Lemma 5.1, implication 1° = 2°, we have (5.1) for / 0 0. Finally, for ! = 0, formulas (5.6) and (1.1) yield (-I-oo) —oc, whence (5.2), and thus (5.1) holds also for I = 0.

174

Dualities Between Complete Lattices

An important tool in the study of dualities between complete lattices are the infi mal generators (see Definition 1.5). Indeed, we have: Theorem 5.1 Let E, F be two complete lattices and Y an infimal generator of E. Then for a mapping A : E F the following statements are equivalent: 1 0. 20 .

A is a duality. For every index set I (including I = 0) we have A( inf yi ) = sup iEz

(5.8)

({Yi},Et ç Y).

iEr

These statements imply: 30 .

We have A(x) = sup {A(y) ly

E

Y, y

x}

(x

E

E).

(5.9)

Proof The implication 1° 2' is obvious. 2° 3°. Assume 2' and let x E E, ly,) i , 1 =tYcYlyx)CY. Then, by (1.37) and (5.8), we obtain A(x) = A (inf {y E Y ly

x)) = sup {A(y)ly E Y, y

x}.

2° 1°. Assume 2°, and let I 0 0 and Ix, )/Et c E. Then, by the above, we have also 3°, and hence A is antitone. Also, by (1.37), we have

x i = inf Yi (i

E

I), inf x i = inf Yo ,

(5.10)

where = {Y E YIY

xi} (i E /), Yo = ty E Yly

inf jet xii•

(5.11)

x i }.

(5.12)

We claim that inf x i = inf U Yi = inf {y E Yi E I, y iEr

Indeed, since Yi g UjEi infY

C

inf

Yo (i E I), we have Yi

inf Yo

(i

E

I),

whence, by (5.10), we obtain inf x i = inf inf Y1 iEt iet

inf

Yi ' Et •'

inf Yo = inf x i , ICI

(5.13)

5.1 Dualities and Infimal Generators

175

which proves the claim (5.12). Then, by (5.12), 2°, and (5.3) (for x = xi , we obtain A(inf xi ) = A(inf {y e YI3 i e I, y te!

= y),

x i ))

= sup {A(y) I y E Y, 3i e I, y

xi }

sup [A(Y) I y c Y, 3 i E I, A(y) ( A (x)} ( suP A(x), jer

and hence, by 2° for I = 0, (1.1) and Corollary 5.1(c), A is a duality. Remark 5.2 (a) In general, 3° A. 1°. Indeed, if E is a complete lattice, then Y = E is an infimal generator of E, and a mapping A : E —> F satisfies 3° with Y = E if and only if it is antitone. However, there exist antitone mappings that do not satisfy (5.2) (and hence they are not dualities), such as any constant mapping A(x) o = E Fy\ {—co}

(5.14)

(X E E).

Let us also mention that there exist antitone mappings A : E —> F that are not complete inf-anti-homomorphisms and hence not dualities (e.g., see Sikorski [253], ch. 2, §18, example D). (b) If A : E —> F is a duality, then, by (5.1) and Proposition 1.6(a), we have A(x) = sup {A(y) I y e Y, H-oo > y

x}

(x e E).

(5.15)

Corollary 5.2 Let E, F be two complete lattices and Y C E an infimal generator of E. If A1, A2 E --> F are two dualities such that A1 y = A2 1 3„ then A1 =- A2. We have the following "extension theorem": Theorem 5.2 Let E, F be two complete lattices and Y an infimal generator of E. For a mapping A o : Y —> F the following statements are equivalent: 10

There exists a duality A : E F such that Al y = AO. 2°. A o is antitone, and for any index set I we have

sup {A0(Y) I y

E

Y, Y % inf Yi} = sup Ao(Yi) iE/ je!

3°. A o is antitone, and the mapping A : E A(x) = sup [Ao(Y) y E Y, y

is a duality.

({Yi}iEr

g Y).

(5.16)

F defined by x)

(x e E)

(5.17)


176

Moreover, in this case the duality A of 1° is uniquely determined, namely it is given by (5.17). Proof 1° = 2°. If 1° holds, then A is antitone, whence so is A o = A yO Furthermore, since A is a duality, by (5.9) and (5.8) of Theorem 5.1 we obtain for any family {YiliEt c Y, sup {A 0 (y) I yE Y, y

yi } = sup {A(Y) I y e Y, y inf ,E/

inf ier yi

= A (inf yi ) = sup A (yi) = sup Ao(Yi)• iEl iEl

2° E

iEl

3°. Observe that if Ao : Y F is antitone, then for the mapping A : F defined by (5.17) we have (even when A is not a duality) A(37) = sup

{NAY) I y e Y, y

= A0

(

that is, A l y = Ao. Hence, by (5.17) and (5.16), for any I A (inf yi) = sup {Ao(Y) I y e Y, y

E Y),

7)

0 and tyihEr g Y we get

inf y, } = sup Ao(Yi) = sup A(Yi), iei

tEl

and therefore, by 2° for ! = 0, (1.1) and Theorem 5.1, A is a duality. 3 0 = 1°. By the observation made in the above proof of the implication 2° o 3°, the duality A of 3° satisfies A y = A o . Finally, if A is as in 1°, then, by (5.9) and Al y = AO, we obtain (5.17). Remark 5.3 (a) If +oc ?' Y. then condition (5.16) is satisfied for I = 0 (by (1.1)). On the other hand, if +oc c Y, then condition (5.16) is satisfied for ! = 0 if and only if A0(+oc) = —oc.

(5.18)

(b) For I 0 0 and for every mapping A o : Y —> F, we have the inequality (5.16). Indeed, since infi e r yi yi (i c I), we have {Ao(Y) I Y E Y, y

in

inf y1} 3 AO(Yi)

JE'

whence the assertion follows.

5.2

Duals of Dualities

Definition 5.2 Let E and F be two complete lattices and A : E The dual of A is the mapping A' : F E defined by

F a mapping.

(z E F).

(5.19)

A 1 (z) = inf

E

E I A(x)

5.2 Duals of Dualities

Remark 5.4

177

By (5.2), we have +00 E {X E

E I A(x)

z} 0 0

(Z E

(5.20)

F).

Proposition 5.1 Let E and F be two complete lattices. (a) For any mapping A : E —> F, the mapping A'A : E —> E is contractive, that is,

A'A(x)

x

(X E

E).

(5.21)

z

(Z E

F).

(5.22)

(b) If A : E — > F is a duality, then A A l (z)

Proof (a) By (5.19) (with z = A(x)) and x e

A'A(x) = inf {Tic' e E I A(7c)

e E I A( )

A(x)}

x

A(x)), we have

(x

E

E).

(b) If A : E —> F is a duality, then, by (5.19) and (5.1), A A I (z) = A(inf {x

E

E A(x)

sup {A(x) Ix

=

E

Z))

E, A(x)

z)

(Z E

Z

F).

Proposition 5.2 Let E and F be two complete lattices. Then for any mapping A : E —> F the dual A' : F —> E is antitone. Proof Ifz, e F, z whence, by (5.19),

then {x

E

E I A(x)

z}

C tx E

E I A(x)

Corollary 5.3 Let E and F be two complete lattices. (a) For any mapping A : E —> F we have the implication A(x)

Z = A i (Z)

X

(X EE, zE

F).

(5.23)

F).

(5.24)

(b) For any duality A : E —> F we have the equivalence A(x) Proof (a) If x x

A'A(x)

E

E,

Ai(z).

z ZE

(z)

x

F, and A (X)

(x

EE, ZE

z, then, by Propositions 5.1(a) and 5.2,

178


F is a duality and x E E, z E F, A l (z) (b) If A : E Proposition 5.1(b) and Corollary 5.1(a), z A Al(z) A (x).

x, then, by

LI

Proposition 5.3 Let E and F be two complete lattices. A mapping A : E —> F is a duality if and only if it is antitone and satisfies (5.22). Proof If A : E —> F is a duality, then, by Corollary 5.1(a) and Proposition 5.1(b), A is antitone and satisfies (5.22). Conversely, assume now that A : E F is an antitone mapping satisfying (5.22), and let {x, }/ c E, Z = supiEi A(x). If / 0 0, then z A(x) (i E I), whence, by Corollary 5.3(a), A t (z) ( xi (i E I), and thus A t (z) infi E / xi. If +00 = infi E0 x i . Hence, by (5.22) and since A is antitone, we I = 0, then A t (z) obtain sup A(xi) = z jer

AA t (z)

A(inf x i ), ie!

and therefore, by Corollary 5.1(c), A is a duality.

Theorem 5.3 Let E and F be two complete lattices. If A : E —> F is a duality, then so is A' : F —> E, and for A" = (AT : E —> F we have A" = A.

(5.25)

Proof Let us first observe that, by Definition 5.2 and Corollary 5.3(b), we have A"(x)

(AT(x) = inf {z E F = inf{z E F I A(x)

(z) x}

z} =- A(x)

(x E E),

which proves (5.25). Hence, by Proposition 5.1(a), we obtain A'A"(x) = A'A(x)

x

(x E E).

(5.26)

Furthermore, by Proposition 5.2, A t is antitone. Consequently, by Proposition 5.3 (applied to A' : F —> E), A t is a duality.

Remark 5.5 Theorem 5.3 yields the following "duality principle for dualities": Each result on dualities can be "dualized," interchanging the roles of E, F and A, A t . For example, the "dualization" of (5.19) is A (x) = inf (z E FIA' (z)

x)

(x E E).

Corollary 5.4 If Al, A2 : E —> F are two dualities and Ail = A'2 , then Al = A2-

(5.27)


179

Proof By (5.25) and .6:1 = A'2 , we have Ai = (6:0' = (6/2 )' =

0

A2.

Definition 5.3 Let E = (E, () and F = (F, () be two complete lattices. A pair of antitone mappings (A : E ----> F, 8 : F ---> E) is said to be a Galois connection if the compositions OA : E ---> E and AC : F --> F are contractive. Remark 5.6 In the usual definition of a Galois connection (e.g., see [261) it is required that CA and AC should be expansive (i.e., OA(x) x and A0 (z) z for all xcE,zc F), so a pair (A, 0) satisfies the condition of Definition 5.3 if and only if the pair (A : E - ---> F- , 8 : F.- --> E- ) is a Galois connection in the usual sense, where E - = (E, ,), F- = (F,). The theories of Galois connections and dualities are equivalent, as shown by Theorem 5.4 Let E and F be two complete lattices. A pair of mappings (A : E ---> F, e : F —> E) is a Galois connection if and only if A is a duality, with A' = 8 (or, equivalently, 0 is a duality, with 0' = A). Proof If A : E ---> F is a duality, then, by Corollary 5.1(a) and Propositions 5.2 and 5.1, (A, A') is a Galois connection. Conversely, assume now that (A : E ---> F, 0 : F ---> E) is a Galois connection. We claim that for xc E, zc F we have the equivalence A(x) ( z F is a duality, then A'A : E --- > E is a hull operator, that is, we have (5.21) and x

Y= A'A(x) .., A'A(Y),

(5.30)

A t AAt A = A'A.

(5.31)

Proof By Corollary 5.1(a) and Proposition 5.2, we have x ,,, Y = A(x) ?. A(Y) = A'A(x)

which proves (5.30) (note that this holds for any antitone mapping A : E —> F). Finally, (5.31) follows from (5.29). 111 Definition 5.4 For any duality A : E —> F, we will call A'A : E —> F the hull operator associated to A. Remark 5.7 Note that although A'A is a mapping of E into itself, it involves implicitly also F (via A : E —> F and A' : F —> E). For any duality A : E —> F, let C(A'A) be the set of all AtA-convex elements of E, that is (by (1.68) for u = A'A), C(A'A) = ( x E Elx = A I A(x))-

(5.32)

Corollary 5.6 For any duality A : E --- > F we have C(A'A) = {A'(z) i z

E

(5.33)

n.

Proof If x c C(A'A), then x = At(A(x)), where A(x) E F. Conversely, if x = A t (z) for some z c F, then, by (5.29), A'A(x) = AtAAt(z) = A'(z) = x, so X E C(A'A). 0 We will now give some formulas for A t (z), A t A(x) and C(A'A) in terms of infimal generators. In the sequel we will denote by Y and T two infimal generators, of E and F, respectively. For simplicity we will assume that A : E --- > F is a duality. Note first that by dualizing (5.9), we obtain At (z) = sup {A'(t) It

E

T, t- z}

(Z E

F).

(5.34)


181

Proposition 5.5 Let E and F be two complete lattices, A :E—>- Fa duality, and Y an infimal generator of E. Then Ai (z) = inf ty e Y I A(y)

(5.35)

G F).

Proof By (1.37) and (5.24) we have for any z E F, (z) = inf fy eYly

0

A i (z)} = inf ly E Y I A(y)

Remark 5.8 The dualization of (5.35) is A(x) = inf {t E T

(t)

x}

(5.36)

(x E E),

where T is an infimal generator of F. Applying (5.19) and (5.35) to z = A(x), and dualizing, we obtain:

Proposition 5.6 We have for any x E E, z E F, A'A(x) = inf

El A ( )

A Al (z) = inf rz- E F I ATz- )

A(x)) = inf ty e

A(x)},

A(Y)

A'(z)} = inf (t E T I A t (t)

(5.37)

At (z)). (5.38)

Remark 5.9 (a) The expressions in (5.37) contain explicitly only E, Y, and A but not F. (b) We will not state separately the obvious dualizations of the subsequent results, but we will use them freely whenever necessary. Corollary 5.7 We have

C(A'A) = (xEEIxy(yEY, A(y) = Ix E E I A(x)

A(y) (y E Y, y

A(x))) x)}.

(5.39)

Proof We have x y for all y E Y with A(y) A(x) if and only if x inf fy E Y I A(y) A(x)} = A'A(x) (by (5.37)), so it remains to apply (5.21) and (5.32). LI The following result expresses A'A(x) and C(A'A) in terms of separation (in the sense of Remark 1.9(a)).

182


Theorem 5.5

We have A'A(x) = inf ty E Y1 13 t ET, x. A 1 (t), y = inf ly e Y I y C(A'A) = {x

E

ElVy

A l (t)}

A / (t) (t E T, A t (t) .. x)}

Y, y 0 x, 3 t

E

E

(x

E),

E

(5.40)

T, x ?. A i (t), y 0 A l (t)). (5.41)

Proof By Lemma 1.1(a) (applied for the infimal generator T of F), we have A (y) A(x) if and only if It E T 1 A(x) t) c ft E T I A(Y) t). Therefore, by (5.24), A(y)

A(x) •(=> ft

E

T I A' (t) .,. xl ç ft E T I 6.' (t)

y).

(5.42)

Hence, by (5.37) and (5.42), we obtain (5.40). Finally, from (5.39), using again (5.42), we obtain (5.41). I: Remark 5.10 One can also express A'A as a supremum. Indeed, by (5.34) for z = A (x) and (5.24), we have

5.3

A'A(x) = sup {A'(t) it

E

T, t

= sup {A'(t) it

E

T, A'(t) ,,, x)

A(x)} (x E E).

(5.43)

Relations Between Dualities and M-Convex Hulls

By Corollary 5.5 above, if A : E —> F is a duality, then A'A : E —> E is a hull operator, and hence, by Theorem 1.5, the set C(A'A) of (5.32) is a convexity system, with coc (A , A) = A'A .

(5.44)

Therefore, by Remark 1.12(c), there exists M c C(A'A) (e.g., one can take M = C (A: A)) such that C(M) = C(A'A), co m = A'A.

(5.45)

In this section we will show some important deeper conections between dualities and M-convex hulls. Theorem 5.6

Let E be a complete lattice, and let M c E. Then the mapping Am defined by AM(X) = {tne.Mim

x}

(X E

E)

:

E —> (2m , D)

(5.46)

5.3 Relations Between Dualities and M-Convex Hulls

183

is a duality, satisfying 1\'A4 ({m}) = in

(in

C(A im Am) = C(M), A im

(5.47)

E M),

Am = com,

(5.48)

and the restriction Am IC(M) is one-to-one. Proof For any I 0 0 and {xi hi c E we have, by (5.46) and (2.1), A m (inf x i ) = iE/

E .A4 I in

inf x i }=n{mEA4Inix i } tEl tEl

= sup Am(x i ); iEi

(5.49)

furthermore, by (5.46) and (2.1), Am (d-oo) = .A4 = —oo m , so Am is a duality. Also for each m E M we have, by (5.19) and (5.46), Am / ({m)) = inf

{X E

EI

Am(X) 3

which proves (5.47). Furthermore, let Tm (2 M , D) defined (see (2.21)) by TM = {{m} I

m} = inf {x E Elm x} = in , C (2 M , D)

be the infimal generator of

(5.50)

E

Then, by (5.43) (for A = Am, T = Tm), (5.47), (5.46) and Theorem 1.2, we obtain

A im Am(x) = sup {A im ({m}) I e M, Am(x) = sup {m E Mini x) = com x

3

MI (X E

E), (5.51)

which yields (5.48). Finally, if xi , x2 E C(M) and Am (xi) = Am (x2), then, by (1.3) and (5.46), we obtain x i = sup A m (x i ) = sup Am (x2) = x2, so A m c(m) is one-to-one. LII Definition 5.5 Let E be a complete lattice, and let M C E. The mapping Am : E (2M , D) defined by (5.46) is called the Minkowski-type duality associated to M. Remark 5.11 (a) The mapping AM C(M) : (C(A4), —> (2m , D) is called in [152] the "Minkowski duality." Although in [152] the word "duality" is not used in the sense (5.1) but merely in the sense of" simultaneous study of a pair of objects," let us note that (C(A4), is a complete lattice (by Proposition 1.11) and that the mapping A mIc04) is a duality also in the sense (5.1). (C( M ) ' ) (2M' D) Indeed, for any index set I 0 and {xi } i , / C E we have, by (1.66), (1.35), (5.48),

184


(5.29), and (5.49), • c( m) x i ) — A m (coc (m) infExi ) = Am (com inf Exi ) Am (inf l

iE/

EI

iE/

= A m A im Am(inf Ex') = Am(inf E xi ) = sup Am (x i ), tEl and for I = 0 the proof is similar, using also (1.1). (b) Theorem 5.6 shows one of the main uses of Minkowski-type dualities; namely they are used to decompose any hull operator co m : E —> E, expressing it as a composition of dualities A lm Am. (c) For each x E E the set Am (x) is nothing else than the carrier U(x) of x (in M), defined by (1.12). (d) Applying Theorem 5.5 to A = Am and T = Tm above, and taking into account (5.48) and (5.47), we obtain again Theorem 1.3. In the converse direction, we have: Theorem 5.7 Let E and F be two complete lattices, T an infimal generator of F, and A : E —> F a duality. Then, for M = M AJ C E defined by M = {A' (t) t E T},

(5.52)

we have M c C(A' A) and (5.45). Proof By (5.52) and (5.33), M c C(A / A). Also, by (5.32), (5.43), (5.52), and (1.3), we have C(A'A)=

E Elx = A'A(x)) =tx GElx= suP {A'(t) I t E T, A i (t)

=

x11

= sup{m E Mim x}}

whence, by (1.35) and (1.70), com = coc(m) = coc (A, A ) = A'A. Definition 5.6 Let E, F, and F2 be three complete lattices, and A i : E —> A2 : E F2 two dualities. We will say that A i and A2 are equivalent, and we will write A 1 — A2 if for any x, R E E, A i (x) = A 1 (') .#>. A2(x) = A2(').

(5.53)

Proposition 5.7 Let E, F1 , and F2 he three complete lattices and A 1 : E ---> Fi, A2 : E —> F2 two dualities. We have A1 — A2 AA1 = A'2A2,

(5.54)

5.3 Relations Between Dualities and .A4-Convex Hulls

185

that is, A 1 and A2 are equivalent if and only if their associated hull operators coincide. Proof Assume that A 1 — A2, and let x E X. Then, by (5.29), Ai A/1 A I (x) = = 6:1 A I (x), we obtain A2A /i A 1 (x) = A 2 (x). A (x), whence, by (5.53) for Consequently, by (5.21) (for A = A2), A /2 A2 (x)

= A /2 A2A /i A (x)

A /1 AI (x),

A (x) = A/2 A2 (x). and hence, by symmetry, AA 1 (x) A/2 A2 (X). Thus Conversely, assume that A'1 A 1 = A 2/ A2, and let x, e E, A 1 (x) = Al (i). '- and since A2 is antitone, we obtain Then, by (5.29), (5.21) (for A.A 1 ,x= x-) A2(x) = A 2 A/2 A2 (x) = A 2 A /1 A l (x) = A2AZI AIC-0 and hence, by symmetry, A2 (X) = A2 (). Thus A1 — A2. F a duality. Definition 5.7 Let E and F be two complete lattices and A : E We will say that M C E is a generating class for A if A — Am (of (5.46)). Remark 5.12 (a) Although M of Definition 5.7 is a subset of E, we call it a "generating class", in view of the important particular case when E = (2X, D), where X is a set (and hence M is a class of subsets of X), which will be considered in Chapter 6. This will lead to no confusion. (b) By Proposition 5.7, M Ç E is a generating class for A : E F if and only if A / A = AA4 Am,

(5.55)

A/ A = com .

(5.56)

or, equivalently (by (5.51)),

F admits a generating class (c) Theorem 5.7 shows that every duality A : E M C E, namely for any infimal generator T of F the set M = MA, 7- defined by (5.52) is a generating class for A, which we will call the standard generating class for A (corresponding to T). For the compositions A ---> M ---> Am and M mappings, we obtain:

—> M of the above

Corollary 5.8 F is equivalent to a Minkowski-type duality Am : (a) Every duality A : E (2M, E D), where M is the standard generating class for A corresponding to any infimal generator T of F. (b) For any M C E, the standard generating class for the Minkowski type duality Am associated to M, corresponding to the infimal generator TM of (2M, D) defined by (5.50), coincides with M.

186


Proof (a) If A : E F is a duality and T is any infimal generator of F, then for .A4 = MA, T of (5.52) we have, by Theorems 5.7 and 5.6, A'A = com = A'm Am,

(5.57)

and thus, by Proposition 5.7, A — Am. (b) If .A4 C E, then, by (5.52) for Am and TM of (5.46) and (5.50), respectively, and by (5.47), we have MAA4.Tm = tA i.A4({M}) in C

= M

(5.58)

.

5.4 Partial Order and Lattice Operations for Dualities Let E and F be two complete lattices with infimal generators Y and T, respectively. We recall that the natural order on FE , the set of all mappings A : E —> F, is defined pointwise; in other words, for A1, A2 : E —> F we write A A2 if A1

(x)

(x e E).

A 2 (x)

(5.59)

Now we will consider the set D = D(E, F) of all dualities A : E —> F, endowed with the partial order induced by the above (i.e., D = (D, and the lattice operations generated by this partial order. Remark 5.13 D has a smallest element 3,- m,n and a greatest element a max , that is, we have

A

Amin

(A c D),

Amax

(5.60)

—

where Amin e D and Amax

E

D are the dualities defined by

(x e E),

Amin(X) -= -

-Foe

(5.61)

if x < ±oo (5.62)

Amax (x) = if x = ±oo.

—oo Proposition 5.8 For A 1 , A2 G D we have A 1

A2 if and only if

Ai (Y)

(5.63)

(y e Y).

A2(Y)

Proof If (5.63) holds, then, by (5.9), A i (x) = sup {A i (y) y E Y, y

sup {A2(Y) y

Y,

x}

y .?; x} =

6. 2 (x)

(x

E).

5.4 Partial Order and Lattice Operations for Dualities

187

For the supremum, respectively, the infimum, of a family tA J ) JEJ c D, we will . A • A jEJ A J• " use the notations V jEJ instead of supiE j j and infi , j A 1 , which j (A j (x)) and infiE j (A j (x)), in the right-hand will permit us to avoid writing supjE sides of (5.66), (5.67), and so on, below. We recall that, by definition, •

JEJ

Aj

min (A

E

A A., = max (A

Dl A1

J))

({Ai}JEi g D),

(5.64)

A j (j e J))

(k\ i }j E j C D),

(5.65)

AO

e DIA

G

jef

provided that they exist in D. We have the following results (for the proofs, see (2721): Theorem 5.8 (a) D = (D,

is a complete lattice, and for any A j

(V Ai)

(x) =

jEJ

E

inf A 1 (x) JEJ (b) For A i 1°.

E

D (j

G

(5.66)

E),

x)

Y, y

(X E

(y) = inf A 1 (y)

(5.67)

E).

(y E Y).

jeJ

There exists a duality A

E

(5.68)

D such that

6.(y) = inf A 1 (y)

(y E Y)-

JEJ

(5.69)

For each family fyi L E/ c Y we have sup (inf A 1 (y) l y E Y, y JEJ

4°.

J) we have

We have

jeJ

3'.

E

J), the following statements are equivalent:

E

(A A

2'.

(X E

D (j

jef

sup (inf Ai(Y)IY jEJ

) (x)

(AA j J

sup A 1 (x)

E

The mapping A : E

jet

iEr

lei

(5.70)

F defined by

A(x) = sup {inf A j (y) I y jEJ

inf yi ) = sup inf A j (yi ).

E

Y, y

x}

(x E E)

(5.71)

188


is a duality. 5'. We have (A A J EJ

(x) = sup Da A 1 (y) ly E Y, y x} J EJ

1/

(x G E).

(5.72)

Remark 5.14 The inequalities in (5.67) may be strict even when J is finite. Indeed, if Y = E and the mapping A : E F defined by A(x) = inf

(x E E)

Ai(X)

jeJ

(5.73)

is not a duality, then, by Theorem 5.8(b), implication 5° = 2°, the first inequality in (5.67) is strict. If Y 0 E and (5.68) holds (e.g., see Remark 6.6), but A of (5.73) is not a duality, then, by Theorem 5.8(b), implication 1' = 4', the second inequality in (5.67) is strict.

Theorem 5.9 The mapping A A' (with A' of (5.19)) is a complete lattice isomorphism of D = D(E, F) onto D' = {A' I A E D) = D(F, E), the complete lattice of all dualities from F into E.

Proposition 5.9 For A i

E

D (j

E

(V A i jEJ

J) we have

(

A i (y) VA] (x) = inf ty e Y I sup J EJ jEJ

sup A i (x))

)

= sup A li (sup kJ JEJ

jEJ

Ak(X))

(X E

E).

(5.74)

Definition 5.8 For each X E E the quasi-complement of x in E (with respect to Y) is the element Tc E E defined by = .7(Y) = inf fy E Yly

(x

x)

E

E).

(5.75)

Remark 5.15 The quasi-complement .7 depends on the infimal generator Y (e.g., see Remark 6.7). In the sequel we will write 1 instead of .7(Y) when this will lead to no confusion. For results on the quasi-complement Tc, see [272]. We note here only the implication Xj, X2

indeed, if xi we obtain 36

GE,

x2, then (y GYly Y2.

xi

X2

Xi

Y2;

xi 1 c fy G YIY

(5.76) x2}, whence, by (5.75),

5.4 Partial Order and Lattice Operations for Dualities

189

Before defining the "quasi-complement" of a duality A, let us mention: Lemma 5.2 Let E, F be two complete latices, with infimal generators Y C E and T let A : E —> F be a duality. The mapping Ao : Y —> F, defined by (y

A0(y) = A(Y)

E

C

Y)

F, and

(5.77)

(where A(y) = A(y)(T)), is antitone if and only if A(y1) = A(Y2)

(Yi, y2 G Y, YI

Y2).

(5.78)

Proof Assume that Ao is antitone, and let y,, yz e Y, yi yz. Then, since An is antitone, A(y 1 ) = Ao(y 1 ) A0(y2) = A(y2)• On the other hand, since A is a duality, we have A (yi ) ?- A (y2 ) (by Corollary 5.1(a)), whence, by (5.76) (in F), we obtain A(Y i ) -. A(Y2)• Thus (5.78) holds. Conversely, if (5.78) holds, then Ao is obviously antitone. El Definition 5.9 Let E, F be two complete lattices, with infimal generators Y c E,TF, and assume that the mapping A o : Y ---> F defined by (5.77) is antitone. If there exists a duality A = A(Y, T) : E ---> F such that Al l, = Ao, i.e., such that (y

A(y) = A(y)

E

Y)

(5.79)

(where A (y) = A (y)(T)), then, by Theorem 5.2, it is (unique and) given by

A(x) = sup (Au') I y

E

Y, y

x}

(x

E

E),

(5.80)

and we will call it the quasi-complement of the duality A (with respect to Y and T). We will write A instead of A(Y, T) whenever this will lead to no confusion; also, instead of writing "A exists," we will simply write: A E D.

Chapter Six

Dualities Between Families of Subsets 6.1

Dualities A : 2x —>

2W

Where X and W Are Two Sets

Let us consider now the complete lattice E = (2X, D), where X is any set (see (2.1), (2.2)) which we will denote simply by 2x whenever this leads to no confusion. If F is any complete lattice, then, by (5.1) and (2.1), A : 2 x ----> F is a duality if and only if for every index set I we have

A(G) A (u G1) = sup icl

({Gdia c 2 x ).

(6.1)

iel

(a) If Y is the infimal generator (2.21) of E = (2X, D) and F is any complete lattice, then, by (2.1), formulas (5.8) and (5.9) of Theorem 5.1 become, respectively, Remark 6.1

A({x i 11 , / ) = sup A({xi l) j Er

A(G) = sup A({g))

((xibEr c E),

(6.2)

(G C X),

(6.3)

gEG

which are obviously equivalent. Thus in this case all three conditions in Theorem 5.1 are equivalent. (b) By Corollary 5.2 applied to this case, for two dualities A1, A2 : 2 X —> F we have A 1 = A2 if and only if Ai ({x}) = A2({x})

(x

E X).

We have now the following complement to Theorem 5.2: 190

(6.4)

2 w , Where X and W Are Two Sets

6.1 Dualities A : 2 x

191

Proposition 6.1

Let 2x = (2x , D), where X is a set, let Y be the infimal generator (2.21) of 2x , and let F be a complete lattice. Then every mapping A o : Y > F is antitone and can be extended to a (unique) duality A : 2 x --> F, namely A(G) = sup Ao({g})

(G c X).

(6.5)

gEG

Proof For Y of (2.21), every mapping A 0 : Y ---> F is antitone (since distinct

singletons (x), {5) are not comparable in (2 X , D)). Also -koo = 0 ig Y. Now let {xi } i , / c Y, I 0 0, and let x e X be such that inf ia {xi } {x}, that is, sup / A0({xi }), whence we obtain the inequality X G kiliE/. Then A0({x}) in (5.16), which, together with Remark 5.3(a), (b), yields (5.16). Hence, by Theorem 5.2, the conclusion follows. Let us consider now the particular case E = (2X, D) = 2x and F =(2W, D) = 2 w , where X and W are two sets. Then, by (5.1) and (2.1), A : 2x ---> 2 w is a duality if and only if for every index set I we have A

G

n

A(G i )

({G i } iEl c 2x ),

(6.6)

(G c X);

(6.7)

iE/

or, equivalently (by Remark 6.1(a)), A(G) = n A({g}) gEG

in particular, for I = 0, respectively, G = 0, this means, by (2.2), that A(0) = W.

(6.8)

Furthermore, by Proposition 6.1, if Y is the family (2.21), then every mapping Ao Y ----> 2w is antitone and can be extended to a (unique) duality A : namely the duality defined by

A(G) = n A o ({g})

(G c X).

(6.9)

gEG

The definition (5.19) of A' : 2 w —> 2x becomes now

A / (P)

(6.10) GCX

PCA(G)

and the equivalence (5.24) becomes P c A(G) G c (P)

(G c X , p c W).

(6.11)


192

Furthermore for Y of (2.21) and for T = {{w} w c W),

(6.12)

formulas (5.35) and (5.36) yield

(13 c W),

A i (P) = Ix E XIP c A((x)))

A(G) = fw c WIG ç Al ({w})}

(6.13)

(G C X).

(6.14)

Also formulas (5.37) and (5.38) now yield

(G C X),

6 = (x E XIA(G) c A({x})}

A'A(G) =

= tw E W I A l (P) ç A/ ({w})}

Az(P)

(6.15)

(P C W); (6.16)

i5cw A / (P)CA'(i;)

in particular, for the hulls of singletons they become

Since A

A'A({x}) =

c X I A({x)) ç Aft,71)}

AA/ ({w}) =

G W I A l ((W))

g B

is equivalent to A \ B

A i WZDI

(x c X),

(w c W).

(6.17) (6.18)

0, formula (5.39) yields

C(A / A) = IG c X I A(G) \ A (( Y)) 0 0 (Y E X \ G)).

(6.19)

Hence, in particular, the equalities

= (x)

(x E X)

(6.20)

hold if and only if for each pair x, i' e X, x Y, we have

(6.21)

A((x)) \ A({:x } ) 0 0. Theorem 5.5 says now that A(G) = Ix e X I /3 w E W, G C A'({w}), x c X \ A l ({w})}

= fx c X Ix c A'((w1) (w E W, G ç Al ({w}))) (G c X),

(6.22)

C(A'A) = IG c X IV x V G, 3 w c W, G C X e X \ A'((w))),

(6.23)

6.1 Dualities A : 2 x ---> 2 w , Where X and W Are Two Sets

193

where, according to Remark 2.2, the conditions G c A'({w}), x E X \ A1 ({w}) can be expressed by saying that A'({w}) separates G from x. From (6.23) it follows, in particular, that we have (6.20) if and only if for each pair x, .-,i' E X, x 3-"c., there exists tt) G W such that A 1 ({w}) separates x frorn : ic . Remark 5.10 says now that

A'A(G)

=

fl

A / ({ wl) =

wE A (G)

n

(G

A l ((w))

C

X);

(6.24)

WEW

GC A/Owl)

clearly the last term of (6.24) coincides with the last term of (6.22). Theorem 5.6 says now that if X is any set and M C 2x , then the mapping AM : 2 X ---> 2-A4 defined by (G C X)

Am(G) = IM E MIG c Ml

(6.25)

is a duality (called, by Definition 5.5, the Minkowski-type duality associated to M), satisfying (M E M),

(6.26)

C(A m / Am) = C(M),A m i Am(G)

n

= com G =

ivi

(G

C

X),

(6.27)

ME.A4

GCM

and the restriction Am l am) is one-to-one; here, by Remark 5.11 (c), the family Am (G) of (6.25) is the carrier of the set G (in M). Note also that, by (6.27) and Definitions 4.10 (for u = A m ' Am) and 4.1, we have

Q(A im Am) = Q(M).

(6.28)

In the converse direction, Theorem 5.7, with T of (6.12), says that if X and W are two sets, then every duality A : 2 x --> 2 w is equivalent to some Minkowski-type duality A : 2 x --> 2-A4 where M c 2 x , namely, for M = MA defined by M = {A / ({w}) I w G WI

(6.29)

we have M C C (A /A) and (5.45); here, by Remark 5.12 (c), the family M of (6.29) is the standard generating class .A4 A.T for A, corresponding to T of (6.12), which we will denote simply by MA (this will lead to no confusion). Note also that for this .M, by (5.45) and Definitions 4.10 (for u = A'A) and 4.1, we have

Q(Ai A) =

(6.30)

and the set Jm of (2.157) becomes J.A4 — { — Xx\A/({w}) I w G W).

(6.31)

194


Remark 6.2 For any sets X and W and any duality A :2 X —> 2 w , applying (6.13) to P = (w), we can write the standard generating class .A4 = MA of (6.29) in the form M = {tx e Xlw G A({x})}lw E W).

(6.32)

Using the equality (6.30), one obtains the following corollaries of some results of Section 4.1. Corollary 6.1 Let X and W be two sets and A : 2x —> 2 w a duality. Then for any f : X —> R we have fq(A'A)(X) =

AGA/0x)

inf

xEA , A(sd (f))

sup

=

d -=

inf

d

(x E X),

xEA'A(Ad(f))

inf f (X \ ((w)))

(x G X),

(6.33) (6.34)

wE W XEX\ A'({w})

S d(fq ( A' A )

A(St(f)) =

)

t>d

n

A(Ad(f))

(d E R). (6.35)

t>d

Proof For .A4 of (6.29) we have (6.30) and (5.45), whence, by (4.177), (4.11), Theorems 4.2, 4.3, and Corollary 4.3(a), we obtain (6.33)—(6.35). 111 Corollary 6.2 Let X and W be two sets and A : 2)( —> 2 w a duality. Then (a) For any set G C X we have ( — XX \ G)q(A'A) — (b) If 0

E

— XX\A'A(G)•

(6.36)

C(A / A), then for any set G C X we have (XG)q(A'A) — XA'A(G)

—

(

XA'A(G))q(A'A)•

(6.37)

Conversely, the first equality in (6.37) implies that 0 e C(ZSZA). Proof The proof is similar to the above proof of Corollary 6.1, applying now LI Theorem 4.5 to .A4 of (6.29). Remark 6.3 only if

(a) The condition 0

E

C(A / A) of Corollary 6.2(b) is satisfied if and

A / (W) = 0,

(6.38)

or, equivalently, AUx1)

W

(x c X).

(6.39)

6.1 Dualities A : 2 x ---> 2 w , Where X and W Are Two Sets

195

Indeed, 0 e C(A I A) if and only if A'A(0) = 0, which is nothing else than (6.38) (by (6.8)). Finally, (6.38) means (by (6.7) for A') that n wew A'({w}) = 0, which holds if and only if for each x E X there exists wx E W such that x g A'((wx/), or, equivalently (by (6.11)), w g A({x}). (b) For any set G C X, and its representation function 'la (defined by (4.40)), we obtain, similarly to (6.37) (using now (4.42)), that (6.40)

( 77G)q(A'A) —

here we do not need the assumption

0E

C(A'A). Also, by (4.43), (5.45), and (6.30),

G G C(A'A)

I GE

Q(A'A).

(6.41)

Corollary 6.3 Let X and W be two sets and A : 2 x —› 2 w a duality. Then G

RxI

sup

f=

(6.42)

d)).

Xx\A , ou,})

(w,d)EW>q? dinf (X\A' Owl))

Proof Apply (4.47) and Theorem 4.6, formula (4.46), to M of (6.29).

Corollary 6.4 Let X and W be two sets, A : 2 x ---> 2 w a duality and Q(A'A; x0 ) = If E R x 1f(x()) = fq(A/A)(xo)} we have Q(A / A; xo) = (f

G

X0 G

X. Then for

R x I Vd E R, xo g Sd(f), ]w E W, S11(f) g Xo E

X \ A'((w))).

Proof Apply Proposition 4.4, formula (4.53), to .A4 of (6.29).

(6.43) LII

Given a set X and M c 2x , often it will be convenient to replace the Minkowskitype duality Am : 2 x —> 2m by an equivalent duality A9 : 2 x 2 w obtained from AM with the aid of a parametrization (W, 0) of Nt (see Definition 2.8), as follows:

Theorem 6.1 Let X be a set, let M C 2', and let (W, 0) be a parametrization of ,A4 (so W is a set and 0 is a mapping of W onto .A4). Then the mapping A9 : 2 X —> 2 W defined by A0(G) = (w G WIG g 0(0}

(G

C

X)

(6.44)

is a duality, with the following properties: 1'. .A4 = ILY9 ({w})lw 20 . Ao Am.

G

WI (i.e., M is the standard generating class for AO.


196

3°.

The restriction Aol c(m) is one-to-one.

Proof By (6.13) and (6.44), we have A 01 ({w}) = {x c X lw c A9({x})} = Ix c Xix c 9(w)} = 0(w)

(W E W),

(6.45) whence, since (W, 0) is a parametrization of M,

.A4 = 19(w) I W G WI =

{A 10({W})

I W E WI.

(6.46)

Furthermore, by (6.24), (6.45), 0(W) = M, and (6.27), we obtain AZ0 A o (G) =

n

Yo({w}) =

n

n

wEw

wEw

mEM

GC64,((tv})

Gc0(w)

GCM

= 4/ Am(G)

M

(G c X),

(6.47)

and hence, by Proposition 5.7, Ao '-`-' AM. Finally, if G 1 , G2 C C(M), AO (G I ) = A9 (G2), then, by (2.12), (6.27) and (6.47), G1 = A N i t Am (GI) = 64) ,A9 (GI) = 64) A0(G2) = YAM(G2) = G20 Remark 6.4 (a) In general, the set W in (6.44) may be chosen to be a more natural "dual set" to X than the family M occurring in (6.25), and therefore the duality Ao may be more convenient than Am, to decompose com in the form (5.45). (b) By abuse of language, following Evers and van Maaren 1801, we will call Ao : 2X —> 2 w of (6.44) (for any parametrization (W, 0) of M c 2X ) a parametrization of the duality Am : 2 X —> 2m. Let us note that 0 (A9 (G)) = {0(w) I G g 0(w)) -= Am(G) w c {17) C W I 9(w) C 0(w)) = Ao (0 (w))

(G C X), (W E W).

(6.48) (6.49)

The importance of the parametrized Minkowski-type dualities Ao of (6.44) is also due to the fact that for any sets X and W, every duality A : 2 x --> 2 w is a parametrized Minkowski-type duality (6.44) for some suitable M C 2x and 0 : W ----> .A4 = 0(W). In other words, we have: Theorem 6.2 Let X and W be two sets and A : 2x —> 2 w a duality. Then there exist a family .A4 =M A C 2x and a mapping 0 = 0A of W onto .A4 (so (W, 0) is a parametrization of .A4), such that A = Ao (with A o of (6.44)).

(6.50)

2 w , Where X and W Are Two Sets

6.1 Dualities A : 2x

197

Proof Let M = MA C 2x be the family (6.29) (i.e., the standard generating

class for A), and define 9 = OA : W --> M by 0(w) = A l ({w))

(w

E

W).

(6.51)

Then 0 maps W onto M and, by (6.14), (6.51), and (6.44), we have for any G C X, A(G) = (we WIG

({w})) = tw e WIG C O(w)} -= Ao(G).

D

Remark 6.5 Combining Theorems 6.2 and 6.1, one obtains again that every duality 2M, namely A : 2x --> 2 w is equivalent to a Minkowski-type duality AM : 2x one can take M c 2' to be the standard generating class (6.29) for A (this has been observed before (6.29), as a particular case of Theorem 5.7). Finally, let us consider the partial order and lattice operations for dualities A : Az, becomes 2x --> 2 w . Formula (5.59), defining the partial order A 1 A 1 (G)

D

(G

Az(G)

C

X),

(6.52)

and the dualities Amin, -LLax, of (5.61), (5.62) are now (G c X),

Amin(G)= W

(6.53)

if G 0, A max (G) = { 0 W

(6.54) if G = 0.

Proposition 5.8 (with Y of (2.21)) means that AI

Az Ai ({x}) 2 A2(()0)

(x

E

X).

(6.55)

Definition 6.1 Let X and W be two sets. For any dualities A1, Az, A j : 2 w (j E J) we will denote the inequality A A2 by the containment symbol AI

(6.56)

Az,

D

and we will call the dualities of (5.64), (5.65), the intersection and the union of the A 1 's, respectively; in symbols,

n= v

J E,

J E,

Ai,

u

Ai

J E,

By (2.1) and Theorem 5.8(a), for any A i

E

A Ai-

(6.57)

J E,

D(2 x , 2 w ) (j

(G) = n A 1 (G)

(G

C

E

J) we have

X).

(6.58)


198

Remark 6.6 For Y of (2.21) and dualities A i : 2x --> F (j E J), where F is any complete lattice, we have (5.68), (5.72) (i.e., the first inequality in (5.67) becomes an equality); indeed, condition 2° of Theorem 5.8(b) is satisfied (by Proposition 6.1). Hence, in particular, for any dualities A : 2x ---> 2W (j E J) we have

U A JjE

J

(

(x e X),

((xl) = A j ({x})

(6.59)

jEJ

(U A i) (G) = jEJ

For Y of (2.21) (in E = (2X, becomes

nu

A i ((g))

(G

C

X).

(6.60)

gEG jEJ

D)),

formula (5.75), defining the quasi-complement,

G = G(Y) = U(x) = X \ G

(G

C

(6.61)

X),

xG

where G should not be confused with the closure of G (when X is a topological space). Remark 6.7 If we take the infimal generator Y1 = 2x of (2x D), then formula (5.75) yields

G (Y, ,\ = /

G'cx

if G 0 X,

=X

(6.62)

Gi\G(4

if G = X,

U0=0

which substantiates Remark 5.15. Note also that (2 X , D, G ---> G(Y)) (with G(Y) of (6.61)) is a complete Boolean algebra, while (2 X , D, G ---> -d(Y 1 )) (with -d(Y1 ) of (6.62)) is not even a complemented lattice, since G n G(Y 1 ) = G 0 0 for 0 0 G X; we recall that (E, x ----> C(x)) is called a complemented lattice if C : E ---> E is such that sup (x, C(x)) = +oc, inf (x, C(x)) = —oc

(x

E

E).

(6.63)

Let us pass now to the quasi-complement A of a duality A. Remark 6.8 For Y of (2.21) (in E = (2 X , D)) and any duality A : 2 x ---> F, where F is an arbitrary complete lattice with an infimal generator T C F, we have (in the sense of Definition 5.9) A

E

D

(A

E

D),

(6.64)

6.1 Dualities A : 2 x --> 2 w , Where X and W Are Two Sets

A(G) = sup A({g})

(A

E

D, G

199

(6.65)

X);

C

gG

indeed, this follows from Proposition 6.1 applied to Ao of (5.77). Hence, in particular, for any duality A : 2 x —> 2 w , where X and W are two sets, and for A = A (Y, T), we have, by (5.79), (6.61), and (6.65), A({x}) = W \ A({x})

A(G) = n (w

(x

A({g}))

E

(G

(6.66)

X),

C

X).

(6.67)

gEG

In the sequel we will consider only A = 3.(Y, T), with Y of (2.21) and T of (6.12). We shall denote (A)' by A'; note that, by (6.64) and Theorem 5.3, : 2 w —> 2 x is a duality. Propositon 6.2

For D = D(2x , 2 w ) we have A' = A'(T, Y) = KA'

E

D'

(A

E

(6.68)

D),

with D' of Theorem 5.9. Proof By (6.13), (6.66), (6.11), and (6.61), we have A ' (Iwl) = tx

Xlw e A({x})} = {x e Xlw e W\ A({x})}

E

= X \ A l ({w}) = A'({w})

(w

E

W).

(6.69)

Thus for each A E D there exists a duality from 2 w into 2, namely A', satisfying (6.69), whence, by Definition 5.9, we obtain (6.68).

Proposition 6.3 For any A

E

D(2x , 2 w ), we have

A(G) = {w

E

WIG n A 1 ({w}) = 01

fl

(G C X),

(6.70)

(x A'({w})

wEW GçX\A'({iv))

I

= X E XI A ({ X } ) C U A(1g1) gEG

(G

C

X).

(6.71)

200


Proof By (6.14) and (6.69), we have K(G)= {w E WIGc A l ({w}) = (w E WIGn(x\A / ((w))) = 01 = ( w E WIGn A f (1W)) = 01

(G c X).

Furthermore, by (6.24) and (6.69), A'A(G) =

n („

--, n A ({w}) = weW GÇA'((w))

(G c X).

\ Al ((w)))

weW

GcX\A'({w))

Finally, by (6.15) and (6.67), we obtain A / A(G) = (x E XI n (W \ A((g})) C W \ A({x})} geG

= PC E X I A({x}) c U A({01

(G C X).

0

gEG

Remark 6.9

By (6.71), we have, in particular,

A'A({x}) =REXI A({Y}) c A({x})}

(x E X),

(6.72)

whence, by (6.17), A'A({x}) n A'A({x}) = 1.-i- E X I Aa.,i- 1) = A({x})}

(x E X).

For various algebraic properties of D(2 x , 2 w ) and of the mappings A A —> —A-1 = A' (A E D), see [272].

6.2

(6.73) A and


In this section we will show that various examples of dualities A : 2 x —›- 2 w X and W are two sets, possibly endowed with some structures) existing in(wher the literature (many of which were introduced to obtain ad hoc "decompositions" u = A/A of certain hull operators u : 2x 2x without any explanation of the choice of W and A) are nothing else than either the Minkowski-type duality Am of (6.25), for a "natural" family M C 2, or the parametrization Ao defined by (6.44), of the Minkowski-type duality AM, for a "natural" family M c 2x and a "natural" parametrization (W, 0 : W . 11 4 = 9(W)) of M. Regarded from the converse point of view, i.e., that of Theorem 6.2, the examples of the latter group also show that for the usual dualities A : > 2 w the standard generating family .A4 __, M A C 2x of (6.29) is a "natural" family (and we know from Section 2.2 1 that these families M admit "natural" parametrizations). Some other dualites between families of subsets, namely dualities r : 2x x T? —> 2 w x -f? and r, : 2xxR _.„ 2wxR


201

used to derive mappings A : R x —> R w , will be considered in the Notes and Remarks to Chapters 7-9.

6.2a

Some Minkowski-Type Dualities

For the sets X (endowed with various structures) and families M c 2 x mentioned in Sections 2.2a—i, it is useful to consider the Minkowski-type duality Am : 2 x —> 2m of (6.25), since it gives the "decomposition" (6.27) for the corresponding M-convex hull operator com (of (2.8), (2.19)). For example, if X is a linear space and M is the family of all semispaces in X, then (6.25) becomes AM(G) = IM is a semispace in X1G c M}

(G c X),

(6.74)

and, by (6.27) and (2.36), we have Aim Am(G) = co G

(G c X),

(6.75)

where co G is the (usual) convex hull of G. Or, if X is a topological space and M is a base (intersectional) for the closed subsets of X, then, by (6.27) and (2.66), for the Minkowski-type duality Am : 2 x —> 2m of (6.25) we have Aim Am (G) = G, the closure of G, and so on. Let us also mention the case where X is a set, W c Te, and M = S(W x R) of (2.111), which was used to define the W-convex subsets of X (by (2.112)). In this case, the duality Am of (6.25) becomes the duality A s(w x R) : 2 X —> 2S(W x R) defined by As (w x R)(G) = (Sd(w) w E W, d c R, sup w(G)

(G c X), (6.76) where As(wxR)(G) is now the carrier of the set G (in the sense (2.7), for M = S(W x R)); by the above, As(w x R) satisfies (6.26) and (6.27) (with .A4 = S( W x R)). This example admits the following extension: Given any sets X, W and any x R) of (2.191) the coupling function go : X x W —› R, for M = Minkowski-type duality Am : 2 x —> 2m of (6.25) becomes d}

As(vxR)(G) = (Sd((P( . , w))Iw c W, d c R, supgEG go(g, w)

6.2b

d}

(G c X).

(6.77)

Some Dualities Obtained from the Minkowski-Type Dualities Am, by Parametrizing the Family »I

We have seen, in Remark 2.16(b), that most of the families M c 2 x considered in Sections 2.2a—i admit parametrizations (W, 0), with a natural "dual set" W to X. Therefore, by Remark 6.4(a), it will be useful to give now explicitly the expressions of the corresponding "parametrized" dualities A o : 2 x —> 2 w of (6.44).

202


For example, if X = R" and M is the family of all semispaces in R", then, by (2.39) of Remark 2.6(b), M can be parametrized by W = 14(R n ) x

((u, Z) E W), (6.78)

, 0(u, z) = ly E R n I u(Y) 214(Rn) defined by (6.81)

(G C X).

E U(R n ) u(g) 2 X* \ {(3} of (6.44) is the duality

A 9 (G) =

E X * \ {0} I sup 413(G)

0}

(6.85)

(G C X).

For the family M of all closed half-spaces in X, which do not contain 0, parametrized (using (2.50)) by

W = X* \ (0}, 0(1.) = M4) = Ix E X I (i) (X) 2 x* \ {()) is the duality

A 9 (G) = ((D e X* \ {0} I (1) (g) < 1 (g

E

(G C X),

G)}

(6.92)

and for the family M of Section 2.2c, part (3), a corresponding expression holds. For M of Sections 2.2d, parts (1), (2), and 2.2e, we obtain, respectively, the dualities

A ) (G) = f((1), d)

E

(X * \ {0}) x RI (I) (g) = d (g E G)}

A9 (G) = {(1) E X * \ {0} I (1)(g) = 0 (g E G)} = G1

A9(G) = ( (1), d)

E (X* \ {0})

x R I (1)(g)

0

\ { 0}

d (g E G)}

(G C X), (G c X), (G C X),

(6.93) (6.94) (6.95)

for which Aie Ao (G) = the closed affine subset of X spanned by G, the closed linear subspace of X spanned by G, and the evenly coaffine hull of G, respectively. If X is a metric space and M is the family of all closed balls in X, parametrized by (2.94), we obtain A o : 2x __> 2x x R , defined by A 9 (G) = {(a, d) GXxR±I sup p (a , g) gEG

d}

(G C X).

(6.96)


204

If X is a poset and .A4 is the family of all subsets M, of X of the form (2.71), where a c X, parametrized by (2.95), then (6.44) becomes the duality A o : >2 x definby A9(G) = (a E Xla

(G C X),

g (g E G))

(6.97)

which, by (2.72), is nothing else than A 9 (G) = X \ co m G

(6.98)

(G C X).

Similarly, for the family M of all subsets Ma of X of the form (2.76), we obtain A0 (G) = fa c X1g a (g E G)) = X \ co m G

(G C X).

(6.99)

If X is a poset and M c 2' is the family (2.77), parametrized by (2.96), then 2(x x {z})u({z} x X) defined by (6.44) becomes the duality A o : 2x A0 (G) = {(a, Z) E X X IZI a

g (g E G)) b (g E G)}

U {(z, b) G {z} x Xlg

(G c X).

(6.100)

If X is a complete lattice and .A4 c 2 x is the family of all subsets Ma of X of the form (2.79), parametrized by W = X, 0(a) =

= Ix E Xlx

a)

(a E X),

(6.101)

(G C X);

(6.102)

then, by (6.44), we obtain Ae(G) = fa G X I sup G

a)

note that, in general, A ( (G) 0 X \ co m G (by (2.81)). Similarly, for the family M of all subsets fiti of X of the form (2.84), we obtain A 9 (G) = fa E Xia

inf G}

(G c X),

(6.103)

and for the family M c 2x of (2.86), parametrized by (2.97), we obtain A9(G) = Ra, z) E X x

(z) I sup G

U ((z, b) E (z) x X lb

a)

inf G)

(G C X).

(6.104)

Let us also mention the case where X is a set, W c R x , and .A4 = S(W x R) (of 2w x R (2.111)) parametrized by (2.210). Then (6.44) becomes the duality A o : 2x defined by A 9 (G) = f(w, d) E W x RI sup w(G)

d} = Epi aG

(G C X), (6.105)


205

with aG of (2.146), and we have Ao AS(w x R) (of (6.76)), A'0 A9(G) = cos(wxR) X)). G = cow G (G Similarly, if W c -1?-x and .A4 = A(W x R) (of (2.178)), then (6.44) becomes the duality A o : 2 x —› 2 w x R defined by A 9 (G) = f(w, d) E W x RIw(g) < d (g E G)}

(G Ç X).

(6.106)

Finally, let us mention the following extensions of (6.105) and (6.106): Let X and W be two sets, yo : X x W R a coupling function, and let .A4 = S(V w X R) of (2.191) be parametrized by (2.213). Then (6.44) becomes the duality A o, : 2 x 14/ x R defined by 2 A(G) = ((w, d) E W x RI sup yo(g, w)

d}

gEG

= Epi

(6.107)

(G C X),

with aG,,p of (2.198), and we have Aoy, AS(Vw R) (of (6.77)), AA0,(G) = cow, (p G (G C X). Note also that, by (2.190) and (2.189), we have now C(A4) = C(S(Vw 4 x R)) = k(Vw, q)) = k(W , (P), and hence, by Theorem 6.1, Aow is one-to-one. Similarly, if we parametrize A(Vw 4 x R) of (2.218) by 14-7 - = W X R, Wç'(w, d) = Ad(v

then (6.44) becomes the duality A 0--, :

13) = Ad(0 . , w))

((w, d) c (6.108)

> 2 w x R defined by

A -9-,(G) = f(w, d) E Wx RI ço(g , w) < d (g E G))

(G C X).

(6.109)

Remark 6.10 The duality A0, : 2 x —› 2 14/ x R of (6.107) is the composition of the mappings a., (p : 2 x —> R w and g E R W —> Epi g E 2 w x R . Since the latter is one-to-one (by Proposition 3.3), identifying R w with its image in 2 w x R one can regard A 0, of (6.107) as being essentially the duality In the above examples, in which X is a locally convex space, one can replace X* \ (0) by X*, defining a suitable extension of 0 so as to conserve A'0 A0. Indeed, more generally, we will prove this for any set X and any W c R x (instead of X*). Definition 6.2 Let X, W, and WI be three sets and 0 : W 2', 0 1 : W 1 —› 2 x two mappings. We will say that (a) the pair (W1, 01) is an extension of the pair (W , 0), or, in symbols, (14/ 1 , 01) (W, 0), if W C W 1 , Od w = 0;

(6.110)

(b) the pair (W1, 01) is equivalent to the pair (W, 0), or, in symbols, (W1 , 01) — (W, 0), if for the dualities Ao : —> 2 w and Ao, : 2 x --> 2 14/1 defined by (6.44)


206

we have L.

Ao,

(6.111)

Remark 6.11 We shall use the above concepts for the case when (W, 0) and (W1 , 01 ) are parametrizations of families M c fc and M i c 2 x , respectively. Note that in this case, by Proposition 5.7 and Theorem 6.1, we have (W1, 01) (W, 0) if and only if com, = Yo, A o , = 46, 0 = com

(6.112)

Theorem 6.3 Let X be a set and M c (a) If (W , 0) is a parametrization of.M, with0 W defined by = W U (0), 01(w) = 0(w)

-1?-x , then the pair (W j , 01 ),

OD E W), 01(0) = X,

(6.113)

is a parametrization of MI = M U {X} such that (W 1 ,9 1 )

( W , 0),

(W 1 ,01 )

( W , 0).

(6.114)

(b) If (W x A, 0) is a parametrization of A/1, with OVWc W x and A C R, then the pair (IV , 0;) -. defined by

Vi = (W U {0}) x A , j'i (w , d) = 0 (w , d),

(w c W, d E A), (6.115)

(O, d) = X

is a parametrization of ..A.41 = M U {X} such that (IV1, WI)

(W x A, 0),

(17111, 4- 1)

(W x A, 0).

(6.116)

(c) If 0 c C(114) and (W x A, 0) is a parametrization of .A4, with OVWc R x and A c R, then the pair (W2 , 02) defined by W2 = (W U {O}) 02(w, d) = 0(w, d),

X

A,

(6.117)

02(0, d) = either X or 0 (w

02(0, do) = X,

02(0, di) = 0

E

W, d

E

A),

for some do, di

E

A,

(6.118) (6.119)

is a parametrization of M2 = M U {X, 0} such that

(W2, 02)

(W x A, 0),

(W2, 02) « (W x A, 0).

(6.120)


207

Proof (a) Clearly (W I , 0 1 ) of (6.113) is a parametrization of .A4 1 = .A4 U {X},

satisfying (6.110). Also, by (6.47) applied to (W, 0) and (W1, 01) and by (2.9), we have (6.112), so (W1, 01) — (W, 0). (b) The proof is similar to that of part (a). (c) Clearly (W2, 02) of (6.117)—(6.119) is a parametrization of .A42 = M U {X, 0}, extending (W x A, 0). Furthermore, by (6.47) applied to (W2, 02) and (W x A, 0) and by 0 G C(M) and (2.9), (2.10), 64)2 A02 ( G ) =

n

9(w, d) = com2 G

(w,c1)E(W xA)U({0}xA) Gc0(w,d)

(G C X),

= com (G) = A'0 A 0 (G) so (W2, 02) — (W x A, 0).

El

Remark 6.12 (a) By Theorem 6.3, in each of the above examples of Ao in which X is a locally convex space, one can replace X* \ {0} by X*, considering (X*, 01 ), or (X* x A, 01), or (X* x A, 02) of (6.113), or (6.115), or (6.117)—(6.119), respectively. Moreover it turns out that these 0 1 , 9 1 , and 02 coincide with the "natural" extensions of the mappings 0 : X* \ {01 —> M, respectively, 0 : (X* \ {0}) x (R + \ {0}) —›- M or 0 : (X* \ {0}) x R M of those examples, in the following sense: If we take (I) = 0 in 0 of (2.93) and (2.92), we obtain 0(0) = X = 0 1 (0) of (6.113) and, respectively, 0(0, d) -= X = 01(0, d) (d > 0) of (6.115) (with A = R + \ {01); if we take (I) = 0 in 0 of (2.91) and (6.90), then we obtain, respectively, X

if d 0,

0

if d < 0,

X

if d > 0,

0

if d

0,

so in these cases 0(0, d) behaves like 92(0, d) of Theorem 6.3(c) (which now applies, since the families M of Sections 2.2b, part (1) and 2.2c, part (1) satisfy 0 E C(M)). (b) For WI of (6.113) we have W 1 C R X and if 0 E K(W), then co w, = COw (by (2.126)). On the other hand, I;i71 and i-ii-72 of (6.115) and (6.117) (where A c R) are not contained in R x ,ww i.A',,widco- 2 m not defined. Note also that when W c R x , in general one cannot expect to have 6.'9 Ao = co w for Ao of Theorem 6.1, as in shown, for example, by Ao of (6.83)—(6.95). But it may also happen that Ai° Ao = co w , or, equivalently, com = co w , as is shown, for example, by A o of (6.82), (6.105), or the same examples with W replaced by W ± R (by Proposition 2.10). (c) If we adjoin 0 e X* to W = X* \ {0 }, as in Theorem 6.3(a), and to the set Ao (G) of (6.85), we obtain a weak* closed convex cone in X* with vertex 0 and, in the case of (6.94), even a weak* closed linear subspace of X* . (d) Some of the above sets A 9 (G) U {0} and A 9 (G) are "polars" of the set G, in the various senses used in functional analysis and optimization theory (see the Notes and Remarks). ,


208

6.3 Representations of Dualities A : 2x —÷ 2 w with the Aid of R Subsets St of X x W and Coupling Functions cp:Xx W Theorem 6.4 Let X and W be two sets. For a mapping A : 2 x —> 2 w the following statements are

equivalent: 1°. 2°.

A is a duality. There exists a set Qc XxW such that A(G) =

(W E W

Moreover the set Q = Q= Proof 1°

I (g, w)

(g

E

(6.123)

(G C X).

G))

of 2° is uniquely determined by A, namely we have

QA

QA=RX,W)EXXWIW1?'

> 2 w and for Q of (6.124), we have

2°. For any mapping A : A({x}) = lw

(6.124)

A({01.

E W I (X, W)

Q)

(X E

(6.125)

X).

Thus, if 1° holds, then, by (6.7) and (6.125), there follows (6.123). 2° = 1°. If 2' holds, then for any I A 0 and IGILEI C X we have lw E W I (X, W)

A

E

iE, Gi)

u iE,

n

E

Gi)I

W I (x, w) g Q (x E G i )) =

n

A(G i ).

iEr

Also, by (6.123) applied to G = 0 we have (6.8). Finally, if 2° holds, then, by (6.123) applied to G = Ix) we obtain (6.125), which proves (6.124) and the uniqueness of Q in 2°. > 2 w of (6.123) is Remark 6.13 (a) For any QcXx W the mapping A : called the duality associated to Q, and it is also denoted by AQ. Thus AQ(G) =

G

W I (g, w) g Q (g

E

G))

(G C X).

(6.126)

Note that, by (6.13) and (6.22) (for A = AQ of (6.126)), we have A'Q (P) = Ix A'Q A Q (G) = Ix

E

X I (X,

EX

W) 1?' Q (W C

I (x, w) g Q (w

E

P))

(P

C

147 ),

(6.127)

W, (g, w) g Q (g E G))) (G C X).

(6.128)

6.3 Representations of Dualities A : 2 x —> 2 w

209

(b) By (6.126) and (6.127), we obtain that the inclusions in (6.11) (for A = of (6.126)) are equivalent to GxPC (X x W) \ Q.

(6.129)

Consequently AQ (G) is the greatest subset P of W among those satisfying (6.129) (and, a similar statement holds for A(P) c X). (c) By Theorem 6.4, there exists a one-to-one correspondence between dualities A : 2 x —> 2 w and subsets Q of X x W (or, equivalently, binary relations xQw). For any duality A : 2 x —> 2 w the set Qp of (6.124) is called the subset of X x W associated to A. Clearly, for any duality Ao : 2 x —> 2 w and any subset Qo of X x W, we have AQA0

=

AO, QP Q0 = Q0-

One can show that the above mapping A —> QA is a complete Boolean algebra A —> A) onto (2 x x W , 9, Q —> (X x W) \ Q), with anti-isomorphism of (D, inverse Q —> AQ, of (6.126) (see [272], thm. 4.2). (d) For any Qc X x W the subset AQ (G) of W defined by (6.126) is called the Q-polar of G, and the mapping AQ of (6.126) is also called the polarity (between the subsets of X and the subsets of W) associated to Q. Then Theorem 6.4 shows that a mapping A : 2 x —> 2 w is a duality if and only if it is the polarity associated to some (unique) subset Q of X x W. Theorem 6.5 Let X and W be two sets. For a mapping A : 2 x —> 2 w the following statements are equivalent: F. A is a duality. 2'. There exists a coupling function ça : X x W A(G) = (w e W I ço(g, w)

—1 (g c G))

3'. There exists a coupling function cp : X x W that we have (6.130).

T? such that (G c X).

(6.130)

T? of type {0, —oo} such

Moreover ça = (pp of 3 0 is uniquely determined by A, namely P x w) = (PA(X,

(

(

'

—

tV) =

XX\A'(()))(X)

((x, W) E X x W).

(6.131)

Proof 1' 3°. For any mapping A : 2 x —> 2 w and for the coupling function ça: X x W —> R of (6.131), we have, by the definition (2.156) of XG, A({x}) =

E W

XW\A({x})(W)

— —00)


210

= (21) E W

I

= (w

I ço(x , w)

E W

-1}

XW\A({x})(W)

—1)

(x

E

X).

(6.132)

Thus, if 1° holds, then, by (6.7) and (6.132), there follows (6.130). The implication 3 0 2° is obvious. 2° 1°. If 2° holds, then for any I 0 0 and (Gi lEI c X we have

A U G , ) = / w c Wicp(x, w) (El = n{W E W

—1 x E

I go(x, w) —1 (x

E

G)} =

n

A(G i).

Also, by (6.130) applied to G = 0 we have (6.8). Finally, if 3° holds, then, by (6.130) applied to G = Ix} we have W \ A({x}) = lw

E

W I ço(x, w) > —1),

(6.133)

whence, since go is type (0, —oo), we obtain the first part of (6.131), which proves LII the uniqueness of ço in 3'. The last equality in (6.131) holds too, by (6.11). (a) For any coupling function cp:XxW —>T?, the mapping A : 2x —> 2 w defined by (6.130) is called the duality associated to go, and it is also denoted by Ag,. Thus Remark 6.14

A(G) = fw

E

W I go(g , w)

—1 (g

E

G)) = S_ 1 (o-G.(p )

with o-G4 of (2.198). Note that, by (6.13) and (6.22) (for A = have A(P) = Ix c X I ço(x, w) AA(G) = fx

E X

I (p(X,

W)

(P C W), (6.135)

—1 (w c P)) —1 (w

E

A 95,

(G c X), (6.134) of (6.134)), we

W, sup ço(g, w)

— 1))

geG

(G C X).

(6.136)

In other words, formula (6.135) says that Aiço is the duality associated to go', i.e., (6.137) where go' : W x X —> R is the coupling function defined by

go' (w, x) = go(x , w)

E

W, x

E

X).

(6.138)

6.3 Representations of Dualities A : 2 x —> 2 w

211

In particular, for W c R x and the natural coupling function (of (2.184)), we obtain, by (6.130), (6.135), and (6.136),

A mi (G) = 01) c W I sup w(G) A'90. (P) = Ix

E

—1)

X XW

ÇOnat

(6.139)

(G c X),

(p c W),

X I sup p(x)

(6.140)

pc P

A:Nat A cprim (G) =

E

X I w(x)

—1 (w E W, sup w(G) (G

C

—1)) X);

(6.141)

note also that if X is a locally convex space and W = X* \ (0), then A cpn,, of (6.139) is nothing else than the duality A 9 of (6.87). (b) Clearly we have A(X) = 0 if and only if sup (p(x, w) > —1

(w c W).

(6.142)

XEX

(c) By Theorem 6.5, there exists a one-to-one correspondence A ço z, between dualities A 2 x —> 2 w and coupling functions go : X x W —> R of type (0, —oo), with inverse (p —> A ço (of (6.134)). For any duality A : 2 x —> 2 w the function ço A of (6.131) is called the coupling function (of type [0, —co)) associated to A. Clearly, for any duality Ao : 2 x —> 2 w and any coupling function (po : X x W —> R of type [0, —oo), we have v., A0 = AO, (pA ço0

(6.143)

= (P0.

) and Some algebraic properties of the above correspondence between D (2 x , (0, —00) x x w have been given in [272]. For example (see [2721), we have the equalities max(ço i 42)

A inf

j

jeJ

PV A•

(

jeJ

= A(jo i A

=

wxxw),

({(pi) iE c RXxin,

V A

(6.144) (6.145)

jeJ

= inf

ÇOA

jEJ

({A i } jej c D),

(6.146)

({A i } jEj c D),

(6.147)

J

ÇOA A i = SUp (pp j jeJ

((pl, (p2

Ago2

jEJ

but, in general, only the inequality A sup (pj jcJ

(6.148)

Aço, jEJ

(d) If t is the embedding w —> (w, —1) of W into W x R, extended to 2 w , then : 2x —> 2 w x R is a duality, satisfying tA q,(G) = {(w , —1)EWxRIsup(p(g,w) ( gEG

c A 0 (G)

(G C X),


212

where A o‘p : 2 x —> 2 w ' R is the duality (6.107). (e) The above representations A = A ço and A = AQ, for dualities A : 2 x —> 2 w , are closely related. Indeed, for any set Qc XxW define the coupling function of type {0, —oo} associated to Q by (6.149)

goQ = — XQ,

and for any coupling function go : X x W —> R define the subset of X x W associated to go by (6.150)

Ro = {(x, w) e X x WI ço(x, w) > —1).

Then R0 ,20 = Q0 (Q0 c X x W) and (A-2,0 = — xs40 (= (po, when (po is of type {0, —oo)). Also, by (2.156), for any Qc Xx W, go : X x W —> R and (x, w) E X x W, we have the equivalences (x, w) gi Ro go(x , w)

(x, w) V Q ;=>. goQ (x, w) . 2W x R of

Ao :

(6.165)

A0

(6.166)

A 2 w

215

whence, by Proposition 5.7, we obtain (6.170). On the other hand, if ça satisfies (6.168), then, by Proposition 6.5 applied to Vw, 90 c RA', we have, again, (6.169) and (6.170).

Remark 6.15 (a) Since Awp : 2 x —> 2 w x R of (6.107) has been derived from the parametrization (2.213) of .A4 = x R), and since for Os° of (2.213) the {0, —oo}-type coupling function ciao : X x (W x R) —> R of (2.203) becomes (2.214), from Proposition 6.5 and Remark 6.14(g) we see that ifyo:XxW—>liis either of type {0, —oo} or satisfies (6.168), then A y,

(6.172)

Awp — Alfro‘p

with .1fr.o(p : X x (W x R) —> R of (2.214). (b) Since G C co w G CA A(G) (by (6.167) applied to Vw, cp Ç 7Rx ), from Propositions 5.2 and 5.4 we obtain Aso (G) = A cp (co w, (p G)

(G

C

(6.173)

X);

note that this also follows from (6.134) and the extension of Remark 2.25(c) to of (2.198) (which is a consequence of (2.196)). j7? x 147 of Theorem 6.5 suggests to introduce an equivalence relation in the familyx as follows: alcoupingfts, : X x W —> T? are Definition 6.3 We say that two coupling functions yo, , if the associated dualities (see Remark 6.14 a)) equivalent, and we write ça = A * , that is, A(G) = A * (G) (G C X). satisfy Remark 6.16 By Remark 6.1 b), we have ça A vi ({x}) (x E X), that is, {w

I

E W go(x , w)

I

—1} = {w

E W Vf(X, W)

Ir if and only if &({x)) =

—1}

(x E X).

(6.174)

Clearly (6.174) holds if and only if (x

E X

I ÇO(X,

W)

=

E X

I *(X, W)

—1 }

(w

E

W). (6.175)

Proposition 6.6 Each coupling function ça : X x W —> R is equivalent to a unique coupling function 0 of type {0, —oo}, namely

ça

= — X((x,w)Exxw I so(x.w)>-I}•

(6.176)

Proof By Remark 6.14(a), for any coupling function yo : X x W —> R we have a unique associated duality A s, : 2 x —> 2 w given by (6.134). By Theorem 6.5, for this A99 there exists a unique coupling function ça iof type {0, —oo} such that A 99 = Ao,


216

given by (Pi (x , w) = = — Xiii-,Ewiox,)>-1)(w)

((x, ZE) E X x W),

(6.177)

Ell

which is nothing else than (6.176).

Remark 6.17 (a) By the above results we have a one-to-one correspondence A —> kol between the dualities A E D(2 x , 2 w ) and the equivalence classes kol E Tex W1 and each equivalence class Rol contains a unique representative yo l of type which we will denote by (So)io. Clearly (49 1)koi = (0)A, and [(49 1)( goil = (0, —ool, [ça]. If A —> [ça], we will denote kol by [ça] and A by A. Thus [(PA] = kolA, and A m = A * for any ik E [(p]. Furthermore, by the above results and Remark 6.14(e) (or, alternatively, by the above results and Remark 6.13(c)) we also have a one-to-one correspondence Q —> [ço] between the subsets Q of X x W and the equivalence classes [go] E Te xi41 / (b) By part (a) above, the composed maps ÇO -> A 90 -> [yo] and (la Rp —> [(p] xiT? xw _), Texiiv ,. are nothing else than the canonical mapping

6.4


6.4a Representations with the Aid of Subsets 12 of X x W One can give representations of various dualites A : 2 x —> 2 w with the aid of subsets Q of X x W, using formulas (6.124) and (6.123). For example, if X is a set and M c 2x , then for the Minkowski type duality Am : 2 x —> 2N1 (of (6.25)), formulas (6.124) (with W = .A4) and (6.123) yield QA m = {(x, M) E X XMIMV Am(tX)))

(6.178)

={(X,M)EXXMIXØM),

Am(G) = {111 E M I (g, M) g Q A, (g E G))

(G C X). (6.179)

If X is a locally convex space and A9 : 2 X -> 2 X* \ 1°) is the parametrized Minkowski-type duality (6.83), then (6.124) (with W = X* \ (0)) and (6.123) yield QAp = i(x, (1) ) E X x (X* \ {0}) I ct(x) > 1), A9(G) =

[CD

E X * \ {0} I (g, 4:13.) g Qp, (g E G)).

(6.180) (6.181)

If X is any set, W c /ix, and A9 : 2 X -> 2' is the duality (6.105), then (6.124) (with 14-7 - = W x R instead of W) and (6.123) yield QA0 = {(X, (W, d)) E X X (W X R) I w(x) > d),

A 9 (G) = {(w, d) e 141 x R i(g, (w, d)) g Q A , (g E G)).

(6.182) (6.183)


6.4b

217

Representations with the Aid of Coupling Functions ço:XxW—*R

One can give representations of various dualities A : 2 x —> 2 W with the aid of coupling functions go : X x W —> R of type {0, —Do}, using formulas (6.131) and (6.130). For example, if X is a set and M c 2 x , then for the Minkowski-type duality AM : 2 x —> 2-^4 (of (6.25)), formulas (6.131) (with W = M), (6.26), and (6.130) yield (PA,(x, M) — — Xx\A',({M))(x) = — Xx\m(x)

((x, M) E X x M),

Am (G) = {M E M I goA m (g, M)

—1 (g E G)).

(6.184) (6.185)

Definition 6.4 Let X be a set, and let M c 2. Then the coupling function ÇOA m : X x M —> R of (6.184) is called the normal coupling function. Remark 6.18 (a) Using the normal coupling function goAm , the family Jm of (2.157) can be expressed as JM =

{M('

M) 1M E M},

(6.186)

which is nothing else than V w,cp of (2.188), for W = M, go = goAm . Also, if X x Jm —> R is the natural coupling function (2.184) (with W = Jm ), then ()Nat P M Cr, M ) = SO n at(x, — X x \m)

(

((x, M) E X x M).

(6.187)

(b) As a complement to the concepts of convex sets G c X with respect to a family M c 2x (Definition 2.1), or to a set W c -fix (Definition 2.9), and convex functions f : X —> R with respect to a set W c R x (Definition 3.1), one could try to introduce, using the normal coupling function yo of (6.184), the concept of a convex function f : X —> T? with respect to a family M c 2 )( , as follows: Let us say that a function f : X —> R is M-convex if f is convex with respect to the set Jm of (6.186), that is, f E C(Jm) = C(Vm 4 ) with Vw, cp of (2.188) (for W = M, cv = indeed, this corresponds to the fact that a subset G of X is M-convex if and only if it is fm -convex (Theorem 2.8). Note that, by (4.113) and Theorem 4.8, every M-convex function is M-quasi-convex (but, in general, the converse is not true). However, since M-convexity of f : X —> R means that

f=

sup

mEm — xx\m

(—xx\m) = — inf Xx\m, mEm

(6.188)

flx\m A -

we see that this is a rather restrictive notion, since an M-convex function f can assume only the values 0 or —oo.

218


>2x* \ {()) Similarly to (6.184), (6.185), if X is a locally convex space and Ao : is, for example, the duality (6.83), then (6.131) (with W = X* \ {0}) and (6.130) yield (since A'0 ({(1)}) = ly e X I (I)(y) 1}) q) Ao( X, (ID) — — X(yeX

E X, w E

(13(),)>II(X)

A 0 (G) = {(1) E X * \ {0) I (pA „(g, (I))

W),

—1 (g E G)}

(6.189) (G c X). (6.190)

Remark 6.19 The representation (6.130) of a duality A : 2 )( —> 2 w with the aid of a coupling function yo : X x W —> R is useful also for coupling functions cp that are not of type (0, —oo}. For example, if ço = (Pnat (of (2.184)), A q, of (6.134) becomes A iat of (6.139), which, for a locally convex space X and W = X* \ {0}, reduces to the duality A o of (6.87). Also, if X is a locally convex space, W = X* \ {0} and —

(Pnat

—

2,

(6.191)

then A,p of (6.134) becomes the duality Ao of (6.83), while if = (Pnat

then A w of (6.134) becomes A o of (6.85).

—

1,

(6.192)

Chapter Seven

Dualities Between Sets of Functions 7.1

Dualities A

:

Rx

kW Where X and W Are Two Sets

where X is any set (see (3.1), Let us consider now the complete lattice E = (Rx , (3.2)), which we will denote simply by R x . For any mapping A : R x —> F, where X and F are two sets, and any f G R X , we will denote A (f) by f A• Thus, when F is a complete lattice, a mapping A : R x —> F is a duality if and only if for every index set I we have sup fi°

(inf fi ) ° iE/

(7.1)

ç R x ).

({fi}iEf

iEr

Remark 7.1 (a) If A : -1–?x —> F is a duality, then, by (3.17) and (7.1), we have f A = (inf xex{X(x)

f(x)1A

f (x)}) ° = sup {x{,} xEx

(f c R x ).

(7.2)

(b) For E = (Rx ,

and Y of (3.18), and for any complete lattice F, Theorem F is a duality if and only iffor all {xi LEI C X 5.1 yields that a mapping A : Rx and {di i iEl c R we have [Ulf (x{x iE/

d)] ° = suP (X{x,} iE/

di) ° ,

(7.3)

(f c R x ).

(7.4)

and if this holds, then

fA

sup

(Xix}

d)

(x,d)ExxR f

Moreover, by Remark 7.3(a) below, A : R x —> F is a duality if and only if for all x E X and {d i } iEl c R we have (7.3) with x i = x (i E I) and (7.2). 219

220

Dualities Between Sets of Functions

(c) If A : R x —> F is a duality, then, by the obvious equivalence f

x(x)

(f E R x , x c X, d e R)

d

d . f (x)

(7.5)

(for d c R, this is nothing else than (3.32)), and since A is antitone (by Corollary 5.1 (a)), we obtain, for any f : X —> R, • A sup (xix1 + d) =

oc,c0ExxR

sup (x/ x ) +• d) A = sup (xix}

(x)) ° ,

xEx

(x.d)ExxR

(x)‹d

f

which is a direct proof of the equality of the right-hand sides of (7.4) and (7.2). (d) By Corollary 5.2 applied to this particular case, if A 1 , A2 : R X —> F are two dualities, we have A 1 = A2 if and only if (X{x}

d) °1 = (x{x}

d) A2

(x E X, d

E

R).

(7.6)

We have now the following complement to Theorem 5.2.

Proposition 7.1 Let X be a set, Y the set (3.18), and F a complete lattice. For a mapping 6, 0 : Y —> F consider the following statements: 10.

2°.

There exists a duality A : R x —> F such that Al y = Ao. For any index set I we have

(Xixi

inf di) °" = sup (X{x} iEt 'Et

d)60

(x

E

X, (diiiEt g R).

(7.7)

Then 1 0 = 2° and the duality A of 1 0 is uniquely determined, namely it is given by

f ° = sup (X{.0

(f

f (x)) °'

E

R x. ).

(7.8)

XEX

Conversely, if2° holds and if we define A : R x —> F by (7.8), then Ao is antitone and AIY = Ao . Proof 1 0 = 2°. If 1 0 holds, then, by Remark 7.1(b), A satisfies (7.3), and hence, by (2.156) and Al y = Ao (with Y of (3.18)), we obtain (7.7). Also, by (7.2) and Al y = Ao, we have (7.8). In the converse direction, let us first show that if 2° holds, then A o is antitone. Indeed, if xi x ,1 d1 d2, then either d2 = +oc, or d2 < +oc and xi = X2 Xix2 i (since if d2 < +oo and x i x2, then xi x o (x2) j- d 1 Xix21(x2) + d2), whence d2. But, if d2 = +pc, then, observing that (+00) ° " = —oc (by (7.7) for I =- 0), we obtain Ao (X{x,} j- d

00

( x{x21

d2 )A0 ;

7.1 Dualities A : R x —> R w , Where X and W Are Two Sets

d2 < +Do, then, by 2° for I = {1, 21, we

on the other hand, if x1 = x2 and di have (Xix,1

di ) ' = (Xix2 1

221

inf {d{, d2}) °"

= sup {(Xix2 1

di) °° , (Xix2 } 4 d2) °0 )

d2) °0 ,

(Xix2 )

so Ao : Y —> F is antitone. We claim that (Xixi

d) A " = suP (Xril ± X{ x }

-x-Ex

Indeed, by xix i antitone, we have (Xixi

d

d) °`)

X{Y} ± X{x}

(Xi,vi 4- xi x i

d) A " d

d) ° "

(x E X, d E R).

G X, d E R) and since Ao is

(x ,VE X, dE R),

with equality for = x, which proves (7.9). Now define A : R x —> F by (7.8). Then, by (7.8) (for f = x{ x } we have (X[x)

d) A =

d) A() = (X{x}

(X { x-} ± X{x}

SUP VE X

(7.9)

d) A( '

d) and (7.9),

(x E X, de R), (7.10)

so Al y = A o . Let us consider now the particular case E = (Wx , = Te and F = (R w , , = R n', where X and W are two sets. By (3.2) (with X and fi replaced by W and f/ respectively), A : RX —> Tel is a duality if and only if for every index set I we have (7.1) pointwise on W, that is, (inf fi ) ° (w) = sup fi° (w) tel

(1i) E W);

(7.11)

iEl

in particular, for I = 0 this means that (+oo) A = — oc.

For any mapping A : R x becomes g 6:

= inf {h

(7.12)

R w the definition (5.19) of A' :

E

Rx Ih°

g}

(g E R w ),

>Rx

(7.13)

that is (by (3.1), (3.2)),

g A' (X) =

inf

hERx /IA (u)Kg(iv) (ivE

h(x)

(g E R w ),

(7.14)

222


and the equivalence (5.24) becomes f A . . . ‹. . g . g A '

(f c -1?-x , g e R w ).

(7.15)

If A : R x —> R w is a duality, then for Yo of (3.25) and

(7.16)

To = {x {w} ± rlw e W, r e R}, formulas (5.34)-(5.36) yield, using the equivalence (7.5), g °' = sup {(x{ u,} + r) ° '1 w c W, r E R, g(w) g °' (x) = inf {d e RI(x{ x } + d) °

g)

f NO = inf fr c R I (X(}D) + O A' < f)

(g E R w ), (7.17)

r)

(g c --k w , x c X),

(7.18)

(f e R x , w c W).

(7.19)

Also, formulas (5.37) and (5.38) yield now, denoting A'A (f) and A A 1 (g) by f °° ' and g ° ' ° , respectively,

= inf {h(x) I h E R X , h A

fA )

= inf {d e RI(X{x} + d) ° -.. f A l g A' A (W) = inf {s(w)

IS

G

WW ,

S A'

= inf fr E R I (X{w} + r)°'

(f E R X , x e X),

(7.20)

(g e R w , w c W).

(7.21)

g °) g°')

Formula (5.39) yields now, using the equivalence (7.5), f

_ fAA'

f (x)

d

(x c X, d c R, (x{x} + d) °

f A ),

(7.22)

and Theorem 5.5 yields, using the equivalence (3.32), f AA' = inf {x{x) +dlx c

f

X, d c R, /3(w,r)eW x R,

(x{w} + r) °' , (X{w} ± r) °' (x) > d}

(f c R x ),

(7.23)

f f °° ' Vx E X, Vd E R, f(x) > d, ]w E W, ]r E R,

f ?- (x(w) + O A' , (X{w} + r) °'(x) > d,

(7.24)

where, according to Remark 3.3(a), the conditions

f ?- (x{,} + r) °' , (X{w} ± O A ' (x) > d can be expressed by saying that (X{ zv i ±

O A ' separates f from x{x } +

(7.25) d.

7.1 Dualities A : R x —> R w , Where X and W Are Two Sets

Remark 5.10 means now, using (7.5), that for any f

f

sup {(x{ w ) = sup {(x{ w )

E

r) °' I w E W, r c R, f(w) OA' I w

r

G W,

E

Iv I

v

r}

R, (X{w} + 0 °'

Theorem 5.6 says now that if X is any set and W A w : (R x , —> (2 W , D) defined by f A w = fw E

223

(7.26)

C -kx,

then the mapping

le)

(7.27)

(j. E

f)

fl.

is a duality (called, by Definition 5.5, the Minkowski-type duality associated to W), satisfying (7.28)

(w E W),

AW({w}) = w

(f c R x ),

C(A'w Aw) = C(W), f °w°'w = 1.03(w)

(7.29)

and the restriction A w c(w) is one-to-one; here, by Remark 5.11(c), the set f A w of

(7.27) (i.e., the set of all minorants of f, belonging to W) is the carrier of the function f (in W).

Remark 7.2 (a) For the dual AW : (2 w ,

(R X ,) of the duality A w of

3) —>

(7.27) we have, by (5.19), AW(P) =

inf h = hERx hADp

inf

h=

hERx {wEw I wo,}Dp

inf

h = sup P

(p c vi

)

hERx sup Ph

(7.30) (or, alternatively, by (6.3) for AW and (7.28), AW (P) = supwEp 4({w}) = sup P), whence, using (3.11), we obtain again (7.29), since f A w Ai w (x)

= A 'w ({ w E wIt

f })(x )

sup w(x) = fco (w)(x) wEvii

(f

E

RX ,

X

e X).

zuf

(b) The above result shows that for any W c R x , fco(W) can be expressed as f ° "'w , with the duality A w : R x —> 2 w of (7.27). Moreover there exists also a duality X w : R x R w such that = fco(w)

(f

E

R x ),

(7.31)

namely the duality defined by f

= X(wEw

(7.32)

224


Indeed, by (7.20), (7.32), (7.27), and (7.29) we have for any f E R X , inf h= hW

Rx f -4

h

inf heT?x X(Ive W

11,h )-- R w (see also Theorem 8.7 and Remark 8.14). In the converse direction, Theorem 5.7 (with To of (7.16)) says that if X and W are two sets and A : R x —> R w is a duality, then for M C R x defined by

M = {(X{w} + t) °' I w E W, r c R},

(7.33)

we have M c C(A 1 A) and (5.45); here, by Remark 5.12(c), the set .A4 of (7.33) is the standard generating class for A (corresponding to To of (7.16)). Finally, let us mention the partial order for dualities A : R x —> R w . Formula (5.59), defining the partial order A 1 z A2, becomes f and the dualities

Amin, Amax

(f

E R X ),

(7.34)

of (5.61), (5.62) are now oc

f Amax —

(f { +00

if f

E R X ),

(7.35)

+cc (7.36)

—00

if f

+DO.

Proposition 5.8 (with Yo of (3.25)) means that for A1, A2 E D = D(R x , R w ), we have the equivalence A1

A2 (X{,c)

d) AI

(X{x} + d) A2

(x e X, d c R).

(7.37)

We omit the specializations, to this case, of the lattice operations (5.64), (5.65) for dualities A i : R x —> Tel (j e J) and of the subsequent results of Section 5.4.

7.2

Representations of Dualities A A x F, Where X Is a Set and (A, C (R, and F Are Complete Lattices :

Rather than working with (R x , where X is a set, we will consider, more generally, (Ax,), (i.e., A the complete lattice where A is a subset of R such that (A, endowed with the order induced by the natural order (3.1) of R x ) is a complete

▪ 7.2 Representations of Dualities A : A x

225

F

lattice. This will have the advantage that the results can be applied not only for A = T? (so Ax' = R x ) but also for other subsets A of R, for example, A = [0, +oo} (note that in this case —oo A = 0, +oo A = +oo) or A = [0, 1] (so —co" = 0, +oc A = 1), instead of (2.156) we A = R. + = [0, +oc], and so on. For the case of (A x , introduce, by abuse of notation (even when 0 V A), the functions lifx,a = X{x} ± a : X —> A (x cX, a v A) defined by

l

a

if

= x,

if

x.

(7.38)

(X{x} ± O(î) = +00A

Then, extending the proof of Lemma 3.1 to this case, we obtain for any f : X —> A, f = inf

1)(1x)

xEX

(7.39)

f (X)}1

where E = (A', } F, where X is a set and (A, are complete lattices, and any f E A x , we will denote A (f) by f ° . Then conditions (5.1) and (5.2) become, respectively, (infE fi) A = suPf ° je!

({

fdiel c Ax),

(7.43)

je!

(+ 00 E ) A — —00 \

,

(7.44)

where E = (Ax, Note that if A C R is closed for inf, then E is closed for inf in (Rx, and hence a complete lattice (e.g., see [27], ch. V, thm. 3) and infE = inf, +oc E = +co in (7.43), (7.44).

226


and Y of (7.40). Indeed, first, if One can extend Remark 7.1 to E = (Ax, A : A x —> F is a duality, then, by (7.39) and (7.43), we have f (x)D A = sup (x(x) xEx

f t = ( xEx infE tx(x)

(f c A x ).

f (x)) °

F is a duality if and

Furthermore, Theorem 5.1 yields that a mapping A : A x only if for all {x }j C X and (a }, j c A we have

ai \-1A ll = sup (x {x,)

rinf E (X{x,}

•1

iEi

jE!

(7.45)

)6,

(7.46)

ai, ,

and if this holds, then fA

sup (x,a)EXxA

(f

(X(xl

(7.47)

E A X ).

f ‹,xod -"Fa

Also, using the obvious equivalence xfx)

f

a

f (x)

a

(f

E

A X , x E X, a

G

A)

(7.48)

(for A = R, this is nothing else than (7.5)) and Corollary 5.1(a), one obtains directly the equality of the right-hand sides of (7.47) and (7.45) (for a duality A : A x —> F). Now we will prove the following more complete result: Theorem 7.1

Let X be a set, and let (A, () C (R, and F be complete lattices. For a mapping A : A x —> F the following statements are equivalent: 1°. 2°.

A is a duality. There exists a mapping y:X x A —> F, satisfying for every index set I,

y (x , inf A ai ) = sup y(x, ai ) iel

(x

E

A),

X,

(7.49)

iel

and such that f ° = sup y(x, f (x))

(f

E

(7.50)

A X ).

xeX

Moreover, the mapping y of 2° is uniquely determined by A, namely we have Y(x, a) = (X1x}

a) °

(x

E

X, a

Proof 1° = 2°. Let us observe that, by (7.38), for I •

X{x}

•

r.A.

i€1

ai — i nfE (x {x} iEl

ai )

(x

E

X,

E

A).

(7.51)

0 there holds A),

(7.52)

7.2 Representations of Dualities A : A x —> F

227

Hence, if 1 0 holds, then for y defined by (7.51) we have, by where E = (A x (7.52) and (7.46) with x = x (i e I), ino a)''

Y (x , inf A ai) = (X{x) ie!

tel

= iE/ E (X[x} = sup (x{ x }

ai ) ° = sup y(x, ai )

(x

E

X, {ai}i E l

C

A).

icl

E

Also, by (1.1), (7.51) for a = 4-oro 4 , (7.38) and (7.44), we obtain (7.49) for I = 0. Finally, by (7.45), for y of (7.51) we have

(f c A x ).

f ° = sup (x{x) 4 - f (x)) ° = sup y (x , f (x)) XEX

X EX

2° = 1°. If 2° holds, then for E = (A x (7.49),

and any set I we have, by (7.50) and

(inf E fi ) ° = sup y(x, inf A fi (x)) = sup sup y(x, fi (x)) tel LE! xEx xEx iEt = sup sup y(x, ict xEx

(x)) = sup fi° jet

((AEI g A x ).

Finally, to prove the last statement, let y be as in 2°. Then, by (7.49) for I = 0, we have (x c X),

Y(X, +00 A ) = —00

whence, by (7.50) applied to f = x{x} (X{x}

(7.53)

a of (7.38),

a) A = sup y(, (X{x)

a)(3 ))

EX

= max ty (x , a), = y(x, a)

sup y(±cc A )) eX\(x) (x

E

X, a c A).

LI

Remark 7.3 (a) By Theorem 7.1, A : A x —> F is a duality if and only if for all X e X and fa i i iEl C A we have (7.45) and (X{x}

• •

) — sup (x(x} + ai) A • int-A ai -A il iEz

(7.54)

Hence, by (7.52), A : A x —> F is a duality if and only if for all x E X and fai } i , 1 c A we have (7.46) with x i = x (i E I) and (7.45). not (b) Theorem 7.1 remains valid for an arbitrary complete lattice (A, necessarily contained in (W, (see the Notes and Remarks to Section 7.3).

228


In order to replace (7.49) by suitable equivalent conditions, let us first prove: Lemma 7.1 (R, and F be complete lattices. For a mapping h : A —> F the Let (A, following statements are equivalent: 1 0.

For every index set I we have h(inf A ai ) = sup h(ai ) ici jEt

20 .

({a}i c A).

(7.55)

(y

(7.56)

h is antitone and all "level sets" S y (h) = {a

E

y}

A I h(a)

E

F)

M C S y (h) and -FooA =

are closed for inf A (i.e., inf A M e S y (h) for all 0 inf A 0 E S v (h)).

If A C T? is closed (i.e., closed in for the natural topology of — R), these statements are equivalent to 3°.

h is antitone and all Sy(h) (y

E

F) are nonempty and closed.

Proof 1° 2°. Assume 1°. Then, applying (7.55) to {al , a2} c A, where al a2, it follows that h is antitone. Furthermore, let y e F, 0 M C Sv (h). Then, by (7.55), h(inf A M) = sup h(a)

y,

acM

so infA M e S y (h). Finally, by (7.55) for I = 0, we have h(±00A ) y (y e F), whence inf A 0 = +00 A E Sy (h) (y E F). 2° = 1°. Assume 2', and let faiLEI c A, 1 0 0. Then infiA€i ai whence, since h is antitone, h(inf A icl

)

sup h(a ).

_ 00 F

ai (i

E

I),

(7.57)

ici

To prove the opposite inequality, note that for yo = sup, E , h(ai ) e F we have ai E S o (h) (i e I), whence, since S 0 (h) is closed for inf A , we obtain inf / ai vo (h), that is, S h(inf A ai ) tEl

yo = sup h(a),

(7.58)

ii

which, together with (7.57), yields (7.55) for I 0 0. Furthermore, by 2° we have e (7.55) holds also -Fcc A = infA 0 E S y (h) (y e F), whence h ±( 00 A ) for / = 0. Finally, assume that A c R is closed.

7.2 Representations of Dualities A : A x

F

229

2° = 3°. If 2° holds, then -Foo A E Sy (h), so Sy (h) 0 0 (y E F). Furthermore, assume 2°, and let y E F, {an } C Sy (h), lim n , a, = a G A. If there exists no such that an„ a, then, since an„ E Sy (h) and h is antitone, we have y ?„ h(an) ) h(a), so a E Sy (h). On the other hand, if no such no exists, then a = inf„ an , and hence, by (7.55) for I = (1, 2, . . .) (which holds by the equivalence y, so 1° - 2°, proved above), we obtain h(a) = h(inf, an ) = supn h(an ) a E Sy (h). 3° = 2°. Assume 3°, and let y E F, 0 M c Sy (h ). Then, since A and Sy (h) are closed, we have inf A M = inf M E Sy (h). Finally, since h is antitone h(a) y, so and Sy (h) 0 0, say, a E Sy (h), a +00 A , we have h(±oc A ) +00 A E Sy (h).

Combining Theorem 7.1 and Lemma 7.1 (applied to hx = y(x, .)), we obtain:

Theorem 7.2 C (Te,) and F be complete lattices. For a !napping Let X be a set, and let (A, A : A x —> F the following statements are equivalent:

1°. A is a duality. 2°. There exists a mapping y:XxA F, such that all "partial mappings" y(x, -) : a ---> y(x, a) (x E X), from A into F, are antitone, and all level sets Sy (y(x, .)) = {a E AI y(x, a)

y}

(x E X, y E F)

(7.59)

are closed for inf A , and such that (7.50) holds.

If A

C

R is closed, these statements are equivalent to

3°.

Same as 2°, with "closed for inf A " replaced by "nonempty and closed."

Moreover in these cases y is uniquely determined by A, namely we have (7.51).

Remark 7.4 (a) By (7.53), formula (7.50) of Theorems 7.1 and 7.2 can be replaced by the equivalent formula f °. =

sup y(x, f (x)) xEdom f

(f E /VC ),

(7.60)

where dom f = (x E XIf (x) < -Foo A ), the domain of f E A x . (b) If inf A (A \ {—oo A } ) = —oo A , then, by (7.41), (7.43) and (7.51), for any duality A : A x —> F we have f° =

sup (X{x) (x,ci)EEpi f

a) ° =

sup (x,a)EEpi f

y(x, a)

(f E A x ).

(7.61)


230

7.3

Dualities A : A x B w, Where X Is a Set and (A, (B,) C (R,) Are Complete Lattices

Theorem 7.3 Let X and W be two sets, and let (A, C (R, C (R, and (B, be two complete lattices. For a mapping A : A x —> B w the following statements are equivalent: 10 .

2°.

A is a duality. There exists a function e:XxWxA > B satisfying for every index set

e(x, w, infA ai ) = supB e(x, w, ai ) tEl

(x

E

X, w

E

W, fai h Ei

C

A),

icl

(7.62)

and such that f ° (w) = sup B e(x, w, f (x)) xEx

(f G Ax, w

E

W).

(7.63)

Moreover the function e of 2° is uniquely determined by A, namely we have e(x, w, a) = (X{x)

(X E

a) ° (w)

X,

WE

W,aE A).

(7.64)

Proof 1° = 2°. If 1° holds, and y :Xx A —> B ' = F is as in 2° of Theorem 7.1, then for e:Xx Wx A B defined by e(x, w, a) = y(x, a)(w)

(x

E

X, w

E

W, a

E

A),

(7.65)

we have (7.62) and (7.63) (by (7.49), (7.50), and since the sup in B w is defined pointwise). 2°1°.Ife:XxWxA—>Bisasin2°,thenfory:XxA—>B w = F defined by (7.65) we have (7.49) and (7.50), and hence, by Theorem 7.1, implication 2° = 1°, A of (7.63) is a duality. Finally, (7.51) and (7.65) imply (7.64). Combining Theorem 7.3 and Lemma 7.1 (applied to hx , u, = e(x, w, •) and F = c (R, we obtain: (B,

Theorem 7.4 For X, W, A, and B as in Theorem 7.3, and for a maping A : A x —> B w , the following statements are equivalent: 1 0 . A is a duality. 2°. There exists a function e:XxWxA > B such that all partial functions e(x, w, .) : a —> e(x, w, a) (x G X, w G W) from A into B are nonincreasing and

7.3 Dualities A : A x

Bw

231

all level sets Sy (e(x, w, •)) = (a

E

A I e(x, w, a)

(y

y)

E

B)

(7.66)

are closed for inf A , and such that we have (7.63). If A

C

R is closed, these statements are equivalent to

Same as 2 0 , with "all level sets (7.66) are closed for infA " replaced by: all e(x, w, •) (x E X, w E W) are lower semicontinuous and 3'.

e(x, w, +oo A ) = —oo B

(X E

X, w

W).

E

(7.67)

Moreover the mappings e of 2° and 3 0 are uniquely determined by A, namely we have (7.64). Definition 7.1 Let X, W, A, and B be as above. (a) For any function e: X x Wx A ---> B satisfying (7.62), the duality A : Ax ---> B W defined by (7.63) is called the duality associated to e, and it is denoted by A(e). Thus f A( e) (w) = sup B e(x , w, f (x)) xEx

(f

E Ax ,

w

E

W).

(7.68)

(b) For any duality A : A x —> B w the unique function e:XxWx A ---> B of Theorems 7.3, 7.4 (i.e., the function (7.64)), is called the function associated to A, and it is denoted by e. Thus for any duality A : A x —> B w , f ° (w) = sup B e (x , w, f(x))

(f

G Ax, W G

W).

(7.69)

xEx

Remark 7.5 (a) By (7.63) and (7.67) (or (7.60) and (7.65)), we have p.(e)(w)

sup' e(x, w, f (x))

(f

E Ax,

w E W).

(7.70)

xEdomf (b) If inf A (A \ {—oo A ))

f i(e)

=

= —ooA, then, by

sup' e(x, w, a)

(7.61) and (7.65), we have

(f e Ax, w

E

W).

(7.71)

(x,a)eEpi f

From Theorem 7.4 and Remark 7.5(a) we obtain the following result on the representation of dualities A : A x —> B ' with the aid of functions eo :Xx Wx (A \ {-koo A }) —> B: Corollary 7.1 For X, W, A, and B as in Theorem 7.3, with A C R closed, and for a mapping A : Ax ---> B w , the following statements are equivalent:

232


P. A is a duality. 2°. There exists a function eo :X x Wx (A \ {-koo A }) --- > B such that all partial functions eo(x, w, .) : a ---> eo(x, w, a) (x E X, ID E W) from A \ {±oo A } into B are nonincreasing and lower semicontinuous, and such that we have f A (w) = f °(e()) (w) =

supB eo(x , w, f (x)) xEdom f

(f G Ax, w E W). (7.72)

Moreover the function eo of 20 is uniquely determined by A, namely we have (with e:XxW xA--- >Bof (7.64)) (7.73)

eo = (eo) A -= elx xw x(A\H-00A))•

Proof 1° = 2°. If 1° holds, then for eo of (7.73) (with e of (7.64)) we have 2° (by Theorem 7.4). 2° = 1°. If e0 is as in 2°, define e : X x WxA —> B by (7.73) and (7.67). Then, by 2°, e satisfies condition 3° of Theorem 7.4, and hence the mapping A (e) : Te ---> R w defined by (7.70) is a duality. But, by (7.70), the definition of dom f, (7.73), and (7.72), we have A(e0 ) -=-- A(e),

(7.74)

so A = A(e0) of (7.72) is a duality. Finally, by the above argument and by the uniqueness part of Theorem 7.4, eo of 2° is unique and satisfies (7.73). El From Theorem 7.4 and Remark 7.5(b) we obtain the following result on the representation of dualities A : A x ---> B w with the aid of functions e l :X x Wx (A \ {—oo A , -1-oo A }) —> B:

Corollary 7.2 For X, W, A, and B as in Theorem 7.3, with A closed and satisfying inf A (A \ —oc A , the following statements are equivalent: 1°. A is a duality. 2°. There exists a function e l :XxWx (A \ {—oo A , +oo A }) ---> B such that all partial functions e i (x, w, .) : a ---> ei (x, w, a) (x E X, 21) E W) from A \ {—oo A , -FooA } into B are nonincreasing and lower semicontinuous, and such that we have f A (w)

Tmeo(w)

sup B e i (x, w, a)

(f E Ax,

ti) E

W). (7.75)

(x,a)eEpi f

Moreover the function e l of 2° is uniquely determined by A, namely we have (with e:XxWxA--->Bof (7.64)) e l = (el)A = e lxxwx(A\{-00A.+00^})•

(7.76)

7.3 Dualities A : A x -9- B w

233

Proof 1° 2'. If 1' holds, then for e l defined by (7.76) (with e of (7.64)) all partial functions e l (x, w,.) (x E X, w E W) are nonincreasing and lower semicontinuous, and we have, by (7.41), (7.43), (7.64), the definition of Epi f, and (7.76), f(w) =

infAx (X{x}

a)) ° (w) =

(x,a)EEpi f

sup B e 1 (x, w, a)

suP B (X(x)

(x,a)EEpi f

c) A (w)

(f E A X , w G W).

(x,a)EEpi f

2 0 = 1 0 . If e l is as in 2°, define e:Xx Wx A —> B by (7.76), (7.67), and e(x, w, — oc/') = limB e i (x, w, a)

(x E X, w E W).

(7.77)

a—>—ooA

Then, by 2', e satisfies condition 3 0 of Theorem 7.4, and hence, similarly to the above proof of Corollary 7.1 (using now (7.71)), we see that A(el) = A(e)

(7.78)

and that A = A(e l ) of (7.75) is a duality. Finally, as in the proof of Corollary 7.1, e l of 2° is unique and satisfies (7.76). El Remark 7.6 If the function el is such that all e i (x, w, .) are nonincreasing, then it is already uniquely determined by A of (7.75), and it satisfies (7.76); indeed, then for each x E X, w E W, we have (X{x)

a)A(w) =

sup B

e l (y, w,

a)

(a E A \ {—oo A + 00A ))

= e i (x, w, a)

a < ±oo)). (because Epi (x( x ) a) = {(x, a) la For a duality A : A x B w , the definition (5.19) of A' : B w —> A x becomes g A' = irif Ax 112 E A X I h A

g)

(g G B W ),

(7.79)

and, by Theorem 5.3, A' is a duality. Theorem 7.5 c (R, Let X and W be two sets, (A, c (R, and (B, two complete lattices, A : A x —> B w a duality, with dual A' : B w —> A x (of (7.79)), and e = e A :XxWxA—> B, e' = e : Wx X x B —> A the functions corresponding to them by Theorem 7.3, i.e., the unique function e satisfying (7.62), (7.63), given by (7.64), and, respectively, the unique function e' satisfying, for every


234

index set I, et (w, x, infB bi ) = supA e' (w, x, b i ) iEt iEt

(w E W, x

E

X, {bi } ia

C

B),

(7.80) g °' (x) = supA (w, x, g(w))

(g

E

BW, x

E

(7.81)

X),

wE W

given by (w E W, x

b) °' (x)

(w, x, b) = (X{w}

X, b

E

E

(7.82)

B).

Then we have ei (w,x,b) = min {a

G

Ale(x, w, a)

(w

b)

G

W, x c X, b

B).

E

(7.83) Proof Let Y be the set (7.40), and let

(7.84)

blw E W, b E B).

T = {X{w}

Then, by (7.82), Proposition 5.5, (7.38), (7.5), and (7.64), we have (w, x, b) = (X{w}

b) ° ' (x)

-=- inf A {(x{7} infA {a

E

a)(x)

A I (X{x)

E

X, a

a) ° (w)

= inf A {a c A I e(x, w, a)

A, (x{,v)

E

a)A

b)

x{ w }

b) (w

b)

E

W, xE X, bE B). (7.85)

But from (7.85) and (7.62) we obtain e(x, w, e' (w, x, b)) -=- sup' {e(x, w, a)l a

E

A, e(x, w, a)

b)

b,

so the last infA in (7.85) is attained for a = e'(w, x, b) c A, which proves (7.83).

Remark 7.7 (a) From Theorems 7.3-7.5 it follows that for any function e : X x Wx A —> B satisfying (7.62), if we define e' :Wx X x B - A by (7.83), then A(e) / = A(e').

(7.86)

(b) If e" = (e')' is the function corresponding (by Theorem 7.3) to the duality A" = (AT : A x —> B w , then, by (5.25) and the uniqueness part of Theorem 7.3,

7.3 Dualities A : A x --÷ B's'

235

we have

= e.

(7.87)

Propostion 7.2

For X, W, A, and B as above, let e:Xx Wx A ---> B be a function such that all partial functions e(x, w, .) : A —> B (x E X, w E W) are nonincreasing, and let :W xXxB > A be the function (7.83). Then we have the equivalence e(x, w, a)

b (w, x, b)

a.

(7.88)

Proof If e(x, w, a) b, then, by (7.83), e'(w, x, b) a. Conversely, assume now that e'(w,x,b) a. Then, since e(x,w,.) : A ---> B is nonincreasing, we obtain, using also (7.83), that e(x, w, a)

e(x, w, (w, x, b)) = e(x, w, min {a E A I e(x, w,

bl)

LI

b.

Corollary 7.3 Under the asssumptions of Theorem 7.5 we have

f ° (w) = min {13

E

B I et (w, • , b)

f}

(f c A x , wE W).

(7.89)

Proof By (7.63) and (7.88), we have

-= min (b E BI f A (w)

f

= min fb

E

B I ei (w,

b) = min fb E B I e(x , w, f (x))

f

(f E A X , W E W).

b (x E X))

LI

Proposition 7.3 Let X, W, A, and B be as above, e : X x W x A --- > B a function satisfying (7.62), and e' :WxXxB —> A the function (7.83). For two functions f E A x and g E B w the following statements are equivalent: 1°.

We have e(x, w, f (x))

20 .

g(w)

W).

(7.90)

(x E X, w E W).

(7.91)

E X, ID E

We have x, g(w))

30. g

fA

4'.

g °(e)'

f

(

e

)

f(x)

236


Proof The equivalence 1' A x . By (7.81) applied to g = f° , we have f AA ' (x) = sup B e' (w , x, f ° (w))

(f e A x , x E X).

(7.94)

(f E Ax, X E X).

(7.95)

wE W

Theorem 7.6 Under the assumptions of Theorem 7.5 we have

sup B e'(w, x, b)

f AA ' (x) = WE

W, bEB


237

Proof By (5.43) applied to the infimal generator (7.84) of B w and by (7.82), we have (7.95). El

Remark 7.10 (a) Formula (7.94) implies that

e , (w , x, f A(e) (w))

tmewer (x)

(f

E A x , tt) E

W, x

E

X).

(7.96)

Note that (7.96) is equivalent to the generalized Fenchel-Young inequality (7.93). Indeed, clearly (7.96) implies (7.93) (by f)' f); conversely, if (7.93) holds, then, applying it to f' (instead of f) and using the fact that f A(e)A(eY A(e) ___ f A(e) , we obtain (7.96). (b) By Remarks 3.25(d) and 8.8(a), the functions e' (w , • , b) : X ---> T? (ti) E W, b E B) occurring in (7.95) may be called the elementary functions associated to (W, e), or, briefly, the (W, e)-elementary functions.

7.4


Leaving to the reader the cases A = ii and A -=- B = — R of Section 7.1, we give here some consequences of the above results for other particular choices of A (c --/-i) and, respectively, A = B(c T?). For any f : X ---> A( T?) we will use the notation (f) = {x

7.4a

E Xlf

(x) = 0).

(7.97)

The Case Where A = {0, +oc }

Let us consider now the particular case where A = {0, -Foo}.

Theorem 7.7 Let X be a set and F a complete lattice. For a mapping A : {0, -Foo) x ---> F the following statements are equivalent: 1°. 2°.

A is a duality. There exists a mapping X : X —> F such that f ° = sup X(x) xEç(f)

(f

{0, -1-oo) x ).

E

(7.98)

Moreover in this case A is uniquely determined by A, namely we have )1/4-(x) = (x{x1) °

(f

E

{

0, +00

}X)

(7.99)

Proof 1° = 2°. Assume 1 0 , and let y :X x A ---> F be as in Theorem 7.1 with A = {0, +oo). Then, using (7.50), (7.53), (7.97), (7.51) (with a = 0), and defining A by (7.99), we obtain for any f E {0, +00} x ,

f A = sup y (x , f (x)) = xEX

sup y (x , 0) = sup (x{x}) ° = sup A(x); xE(f)

xE(f)

xEÇ(f)


238

note that the equality f ° = supxE((f) y(x, 0) follows also from (7.60), since for any f G {0, H-oo) x we have dom f = (f). 2° = 1°. If 2° holds, then, by (7.98), for ! 0 we obtain sup

(inf fi ) ° -= /Et

sup

X(x) =

xE(inff)

X(x)

xEU "(.);)

f€/

ICI

= sup sup X(x) = sup iel xe - (f,)

fi°

(iniEt g_ {0, ±oo} x );

iel

also, by (7.98) for f -koo and (1.1) we have (7.44), so A is a duality. Finally, to prove the last statement, assume that X is as in 2'. Then, by (7.98) applied to f = )(pc ), we obtain (X{x)) ° =

(x e X).

sup X(.7v) = X(x) 3c- e((xl,))

Remark 7.11. One can also give the following direct proof of the implication 1° = 2', with X of (7.99): We have

f=

X“f) =

inf

X{ x }

xe - (f)

whence, by 1°, we obtain, for any f

E

(f

E

{

0, +00} X ),

(7.100)

(0, +oo} x ,

f ° = ( inf x{ x }) ° = sup (Xix}) ° = sup A(x). Xe(f) X “-(f) Xe(f)

In the particular case where A = ( 0, -koo) and F = Bw , , there holds:

Theorem 7.8 Let X and W be two sets and let (B, c (R, be a complete lattice. For a mapping A : ( 0, +oo) x —> B w the following statements are equivalent: P. A is a duality. 2°. There exists a function y9 : X x W —> B such that f ° (w) = sup B yo(x, w)

(f

E 10, +00} X , W E

W).

(7.101)

xEw)

Moreover in this case ço is uniquely determined by A, namely (,0 x, w) = (X{xJ) A (

(x c X, w c W).

(7.102)

Proof 1° 2°. If 1° holds and if X : X —> B w = F is as in Theorem 7.7, then for yo : X x W —> B defined by ço(x, w) = X(x)(w)

(x e X, w G W),

(7.103)


239

we have 2°. 2° = P. If 2' holds, define A :X--->B w =Fby (7.103) and apply Theorem 7.7, implication 2' = 1 0 . Finally, (7.99) and (7.103) imply (7.102).

Remark 7.12 One can show that the particular case B = {0, +pc)} of Theorem 7.8 is equivalent to Theorem 6.4, using the fact that for any set X the mapping f ---> (f) (of (7.97)) is a complete lattice isomorphism of ({0, +oo} x , onto (2 x , D), with xm • inverse M

7.4b

The Case Where A = B = [0, 1]

We recall that given a set X, each f E [0, I] called a fuzzy subset of X, and a fuzzy subset fl is said to contain a fuzzy subset J2* if f l f2 (in ([0, 11 x , Although [0, 1] is neither closed for inf nor closed for sup (in T), ([0, 1], is a , +00 [0, t =_ 1, 00 [0.11 , 0 complete lattice, with _ and we have sup [" M = sup M, inf" M — inf M

(0 M

C [0, 1]);

(7.104)

also [0, 1] is closed. Hence from Theorems 7.3 and 7.4 we obtain:

Theorem 7.9 Let X and W be two sets. For a mapping A : [0, 11 )( —> [0, 1 r the following statements are equivalent: 1 0 • A is a duality. 2'. There exists a function e:XxWx [0, 1] index set I 0 0, e(x, w, inf ai ) = sup e(x, w, ai )

[0, 1]„ satisfying for every

(x E X, w E W,

fail1€1

c [0, 11), (7.105)

e(x, w, 1) = 0,

(7.106)

and such that f A (w) = sup e(x , w, f (x)) .xeX

(f E [0, 1 1 X ,

tV E

W).

(7.107)

3°. There exists a function e:XxWx [0, 1] [0, 1], satisfying (7.106), such that all partial functions e(x, w, •) : [0, 1] —> [0, 1] (x E X, w E W) are nonincreasing and lower semicontinuous and such that we have (7.107). Moreover in these cases e is uniquely determined by A; namely we have (7.64) with A = [0, 1] (where x{ x } a E [0, 11 X is defined by (7.38), with A = [0, 1], +oo A = 1).


240

Remark 7.13 One can show that Theorem 7.9 is equivalent to the particular case A = B = R of Theorems 7.3 and 7.4, using the homeomorphism of [0, 1] onto

R defined by (00 = tg (n- a — —) 2

7.5

if 0 < a < 1, V f(0) = —00, *(1) = +Do.

(7.108)

Strict Dualities A : A x --+ B w

(B, Let X and W be two sets and (A, C two complete lattices. We say that a duality A : A x ---> B w is (a) strict at (xo, wo) E XxW if, for the function e: X x W x A ---> B defined by (7.64), e(x o , w o , -) is a strictly decreasing function that maps A onto B(xo, wo) = {e(xo, w o , a) a E A} c B. (b) strict, if it is strict at each (xo , wo) E X x W. Definition 7.4

Proposition 7.5 A duality A : A x ---> B w is strict at (x0, w o ) EXx W if and only if there exists ) fx o : X \ {x0 } ---> A such that the function a ---> (f11 ) A (wo) is strictly decreasing, where fxao E A x is defined for each a E A by

f.:0 (x) =

i

f(x)

if x E X \ {xo}, (7.109)

a

if x = x0 .

Proof If A : A x ---> B w is a strict duality at (x 0 , w o ), then for fx () a 1 , a2 E A with al < a2, we have, by (7.63), (7.109), and (7.67),

+00 and

(f':') A (wo) = sup B e(x, wo, f: )1 (x)) xEx = e(xo, WO, al) > e(xo, w 0 , a2) = (f.:02 ) A (w().

Conversely, if the condition is satisfied and if a1, a2 e(x o , w o , al ) = e(xo , w o , a2 ), then

E

A are such that

(f..,T ) A (wo) = sup B e(x, wo, f caol (x)) xEx

= max ( sup

e(x, w o , f ) (x)), e(x o , w o , a i ))

XEX\kd

= max ( sup xEx\fx () )

e(x, wo, fx 0 (x)), e(xo, wo, a2)) = (fxa02)A(wo),

whence, by our assumption, al = a2, which proves that e(xo, wo, .) is strictly decreasing (since it is nonincreasing, by Theorem 7.4).

7.6 Dualitylike Mappings A : A x. --> B w

241

Some examples of strict dualities and of dualities that are not strict at any (x0, tvo) and some applications of strict dualities will be given in Chapter 10 (see Remarks 10.8(b) and 10.21(c), respectively).

7.6

Dualitylike Mappings A : A x --÷ B w

Let X and W be two sets and (A, ,), (B, ) c (7R, ) two complete lattices. A mapping A : A x' ---> B w is called a dualitylike mapping if there exists a function e:X x Wx A ---> B such that we have (7.63). In this case we will call A the dualitylike mapping associated to e, and we will denote it by A (e). Definition 7.5

Remark 7.14 (a) Given a mapping A : A x —> B w , for the function e defined by (7.64) we need not have (7.63), and the function e:Xx Wx A —> B satisfying (7.63) need not be uniquely determined by A (only sup, Ex e(x, w, a) = (L) A (w) is uniquely determined by A for each (w, a) E W x A, where fa (x) = a for all x E X). However, as is shown by the first part of the proof of Proposition 7.5, a function e:Xx Wx A —> B satisfying (7.63) and (7.67) is uniquely determined by A, namely it is nothing else than (7.64). (b) The inequality (7.92) (with f ') of (7.68)) remains valid for any function e:Xx Wx A —> B, that is, for any dualitylike mapping A(e) : A x —> B w . (c) For any X, W, A, and B as in Definition 7.5 and any function eo : X x W x (A \ {-Foo A }) --> B, the mapping A(e 0 ) : A x —> B w defined by (7.72) is a dualitylike mapping (since if we extend eo to e:X x Wx A --> B by (7.67), then A(e) of (7.68) coincides with A(e0)), which we may call the dualitylike mapping associated to eo.

Chapter Eight

Conjugations R w where X and W are In the preceding chapter we studied dualites A : R x two sets, defined by condition (7.1). In the present chapter we will study a particular class of dualities c : R x —> R w , called conjugations, which are important e.g. for the definition of various "dual objective functions" (i.e., objective functions of dual optimization problems). Conjugations are mappings c : R x R w defined by (7.1) (for A = c), and a certain "second condition" involving the upper addition + in R x and lower addition + in R w (defined by extending (3.13)—(3.15) pointwise) of a function and a constant function.

8.1

Conjugations c : Rx

R w , Where X and W Are Two Sets

Definition 8.1 Let X and W be two sets. A mapping c : R x ---> R w is called a conjugation, if it is a duality, that is, if for each index set I (including I = 0) we have

(inf fi )c = sup fT

(8.1)

({f),Er ç Te),

tE l

and if (f

E

,d

E

R),

(8.2)

where we use the same notation for the elements of R and for the constant functions with values in R, that is, we identify each d e R with the function hd E R x defined by hd(x) = d (x E X). Now we will show that in (8.2) it is enough to consider d

E

R.

Lemma 8.1

We have a ± —oo = inf (a ± d) dER 242

(a

E

(8.3)

8.1 Conjugations c :

W , Where X and W Are Two Sets

fix

a + +oo = sup (a — d) dE R

(a

E

R).

243

(8.4)

Proof If a E RU {— oc}, then both sides of (8.3) are —oc. If a = +oo, then both sides of (8.3) are +oo. The proof of (8.4) is similar. Proposition 8.1 > R w is a conjugation if (and only if)

Let X and W be two sets. A mapping c : we have (8.1) and

(f +

(f

= fe — d

Proof If (8.1) and (8.5) hold, then for all f

E

E

R x , d E R).

(8.5)

ire we obtain, using Lemma 8.1,

(f — oc) c= (inf (f + d))c = sup (f + d)c = sup (fc — d) = f ± +oo . dER dER dER

On the other hand, by (8.1) with I = 0, we obtain for any f

E

RX,

(f +oo)c = (+DO' = —oo = fC+ —oc,

and thus we have (8.2). Remark 8.1 (a) If a mapping c : R x --> R w satisfies only (8.1) for every I 0 and (8.5), then (+DO' (w) = +co for all w E W; indeed, by (8.5) applied to (w) = (+oo)c(w) — d for all w E W f +oo, we have (+oo)e(w) = (+oo + and d E R. (b) For any conjugation c : R x R w we have, by (7.2) (with A = c) and (8.2), (f E R x ).

fc = sup Rxtx))c ± — f (x)) xEx

(8.6)

(c) The duality A : R x ---> R w (where W c R x ) of (7.32) (related to the Minkowski-type duality A w : R x ---> (2 w , D) of (7.27)) does not satisfy (8.5), so it is not a conjugation. Let us consider now the dual c' : -1—? w ----> le of a conjugation c : R x ---> R w (defined by (7.13), with A = c). Theorem 8.1 Let X and W be two sets. If c : 17e —> R w is a conjugation, then so is its dual R w —> R x

Proof By Proposition 5.2, c' is antitone. Also, by (5.26) (applied to A = c : — R x —> — R w ), we have f fil e '

=

f CC /

f

(f

244

Conjugations

Hence, by Proposition 5.3 (applied to A = for any index set I we have

(inf (El

: R w —>

c' is a duality; that is,

= sup 81'' iEr

Note that it is also easy to show (8.7) directly (see [267]). Finally, by (7.13) and (8.5) we have for any g E .WW and d E R, (g ± d)c' =

inf hETe h' R x to a conjugation c : R x w may be called the dual conjugation to c. R We will now specialize formula (7.17) and the subsequent ones to conjugations A = c : R x —> R w . We recall the following well-known rule of computation with 2k and ± on R (e.g., see [206], p. 119, prop. 3(c)).

Lemma 8.2 For any a, b, d E — R, we have the equivalence

ad+b.#a+—bd.

(8.8)

Remark 8.3 In other words, formula (8.8) says that from the inequality a b we can "move b to the other side," replacing by + —. As has been mentioned in [206], the rule (8.8) can be remembered by noting that the symbol of upper addition occurs on the side of the greater member. We will use the following corollary of Lemma 8.2:

Corollary 8.1 For any a, b, d E R, we have the equivalence a + —b d a + —d Proof By (8.8) and

b = b d, we have

b.

(8.9)

8.1 Conjugations c :

--+ k w , Where X and W Are Two Sets

245

Now, if X and W are two sets and c : R x —> kW is a conjugation, then, by (7.17), (7.18), (8.5), and (8.9), we obtain gr c' = sup {(x{ ii, } ) el — r 1w E W, rER,g(w) gri (x) = inf {d E R (X{x))` — d

g} = inf {d E

= sup ((x{x}) c + — g)(W) Similarly, if c : Rx (8.9), we obtain

r}

(g E

k W ), (8.10)

I ()(( x )) c ± —g

d)

(g E R W , x E X).

(8.11)

R w is a conjugation then, by (7.19), Theorem 8.1, and

f c (w) = sup ((X{w})c/

(f E R X , 11.) E W).

—f )(X)

(8.12)

In particular, from (8.11) (applied to g= J.') and (8.12) (applied to f = gei ), or, from (7.20), (7.21), it follows that if c : Rx ---> R w is a conjugation, then fc ci(x )

g C/ C

( w )

=

sup sup

( (x po r ± — f c) (w) (( x{ w

Ci

g el

)

(f E R X , x E X), (g E R W ,

)

w

E

(8.13)

W).

(8.14)

Furthermore, if c : R x ---> R w is a conjugation, then from (7.22), (7.26) and the above methods, we obtain for any f E Rx, f = fcc i f (x)

(x E X, d

d

E

R, (X{x}) c

—fc

f ee' = sup {x{„,}) ci —rlw E W, r E R, fc (w)

d),

(8.15)

fl.

(8.16)

r}

= sup {x {u)) )`) —rlw E W, r E R, (x { ,, j )c — r Formulas (7.23) and (7.24) yield now for any f E R X , f cc/ =

inf {x{x} +dixe R, d E R, /](w, r) E W x R, f ?-• (x{w))c' — r, (Xful) cl (x) — r > d} ,

f = fcc' - V x f

E

(8.17)

X, Vd E R, f(x) > d, ]w E W, ]r E R,

(x{wi)c / — r, (Xfo) c) (x) — r > d.

(8.18)

The "separation" condition occurring in these formulas (i.e., (7.25) for a conjugation A = c : R x ---> R w ) is equivalent to (X{wi) ci (x) — d > r

(X{w}) c' ± —f.

(8.19)

Remark 8.4 If we use the infimal generator Y of (3.18) and the corresponding infimal generator T = {x{ w } ± r- Iw E W, r E R)

(8.20)

Conjugations

246

(instead of Yo and To of (3.25) and (7.16), respectively), then in formulas (7.17)—(7.26) we can replace d, r E R, and + by d, r E R, and +, respectively, and hence (using (8.2) instead of (8.5)) in (8.10), (8.11), and (8.16)—(8.19) we can replace d. r E R, and — by d, r E R, and + —, respectively. Indeed, this follows from (8.9) and the equivalence a + —r > d r

(a, d, r

G

R),

(8.21)

which is an obvious consequence of (8.9). R w the standard generating class M defined by For a conjugation c : R x (7.33) becomes

(8.22)

M = {(Xfulfi —rlw E W, r E RI.

Finally, from (7.37) it follows that, for any two conjugations CI, c2 : Rx —› R w , we have the equivalence C1

8.2

c2 (x{x}r

(x

(x{x})c 2

E

(8.23)

X).

Representations of Conjugations c : R x —> R w with the Aid R of Coupling Functions ço : X >C1V

We recall the following well-known rules of computation with + and + on R (e.g., see [2061, formulas (4.8), (2.1), and (2.3)):

Lemma 8.3 For any set X, any function f : X --> R, and any a, b, d

E

R, we have

sup (f (x) ± a) = a + sup f (X), xEx

(8.24) (8.25)

a + (b

d) = (a + b)

(8.26)

d

We have the following main representation theorem:

Theorem 8.2 Let X and W be two sets. For a mapping c are equivalent:

2°.

R-X

--> R w the following statements

c is a conjugation. There exists a coupling function y9 : X x W > R such that fr(w) = sup {cp(x, w) ± — f (x)} xEx

(f

E — RX

, WE

W).

(8.27)

8.2 Representations of Conjugations c : Rx

Rw

247

Moreover yo of 2' is uniquely determined by c, namely (P(x, w ) Proof

= ( x po Y (w)

(x e X, w E W).

(8.28)

1° = 2°. If 1° holds, then, by (8.6), for 99 of (8.28) we have (8.27).

2° = 1°. If 2' holds, then, by (8.27) and Lemma 8.3, we have for any I 0 0, ckX, f E Te, and d E (inf f1 )°(w) = sup {cp(x, w) iEt

— inf fi (x)) =- sup {cp(x, w) + sup (—f i (x))) iEt iEt xEx

xEx

sup {cp(x, w) ± — f i (x)) = sup ff(w) xEx,iEi 1E1 (f

(IV E

W),

d) c (w) = sup {cp(x, w) + —(f (x) -i- d)) xEx

= sup fy9(x, w) ± (—f (x) + —d)) xEx

= sup {69(x , w) ± — f (x)) + —d) xEx (w E W).

= fc (w) + —d

Also, by (8.27) and (3.15), (+00)C = —Do. Finally, if 2' holds, then, by (8.27) applied to f = (X{x)) c (w) = suP{OY w)

— X{x}(Y)) = go(x, w)

we obtain (x E X, w E W),

yEX

which proves (8.28) and the uniqueness of ço in 2°.

8.5 (a) The implication 1' = 2' also follows from Theorem 7.3 or 7.4, since, by (7.64) (for A = c) and (8.2), we have Remark

(x E X, w E W, a E R).

e(x, w, a) = (X{x}) c (w) + —a

(8.29)

Note also that, by (8.29) and (8.28), we have e(x, w, a) = cp(x, w) + —a

(x E X, w c W, a c R),

(8.30)

(x E X, w E W).

(8.31)

whence, in particular, yo(x, w) = e(x, w, 0)

(b) Since the elements x o E X with go(x0 , w) ± — f (x 0 ) = —oo may be disregarded in the supxEx of (8.27), we have sup

fe(w)= x Edom

(—q;1(•,

w))Cldom f

w) ± —f (x)}

(f E

RX , w

E

W); (8.32)

Conjugations

248

here dom (-0, w)) n dom f may be "smaller than" X (of (8.27)), but it depends on go (• , w) and f. (c) As shown by the representation (8.32), we have f c (w o ) = —oc for some Wo E W if and only if dom ( — (P( . , wo)) n dom f = 0.

(8.33)

Hence, when dom ( - 0, wo)) = X (i.e., when go(x , wo) > —00 for all x e X), we have fe (wo) = — 00 if and only if f -- +00, or, equivalently, fe(w)17:- —oc for all w E W. On the other hand, if there exist ID() E W and ro < +oo such that 0, wo) + —ro .„ f, then, by Corollary 8.1, we have 0-, wo) + — f ro < +co, whence, by (8.27), fe(w o ) < +oo. Thus, if also dom (-0, w 0 ) ) = X and f 0 +cc, then fe (WO) G R. (d) Using (3.23), (8.1), (8.5), and (8.28) (instead of (3.17), (8.1), (8.2), and (8.28)), we obtain for any f E RA', f c (w) =

sup

{cp(x, w) — d)

(W E

W);

(8.34)

(x,d)EEpi f

here only d E R is used (versus — f (x) e R U {-koo} in (8.32)), but we have Epi f c X x R (versus dom f c X in (8.32)). Both (8.32) and (8.34) are useful for applications.

Definition 8.2 Let X and W be two sets. (a) For any coupling function ço : X x W > R, the conjugation c : R x ---> R w defined by (8.27) is called the (Fenchel-Moreau) conjugation associated to go, and it is denoted by c(cp). Thus for any coupling function go : X x W —> R, —

fc((p)( w ) = sup {0x, w) + — f (x)) xEx

(f

E

RX ,

W

e W).

(8.35)

(b) For any conjugation c : R x —> R w the unique coupling function (p : X x W —> R satisfying (8.27), that is, the function (8.28), is called the coupling function associated to c, and it is denoted by (pc . Thus for any conjugation c : R x ---> R w , , fc(w) = sup ((Mx, w) ± — f (x)} xEx

(f e -1?-x , w e W).

(8.36)

Remark 8.6 (a) The particular case where X is a set, W c W x and go = (Pnat (defined by (2.184)) is important for applications. In this case fe (0 of (8.35) reduces to the so-called (generalized) Fenchel conjugate off, and it is denoted by f*: f* (w) = sup {w(x) + — f (x)) xEx

(f e P, w e W).

(8.37)

8.2 Representations of Conjugations c :

—>

249

In particular, if W c R x (and ço = ço„„ t ), then the symbol + — of (8.37) becomes simply —, that is, f* (w) = sup (w(x) — f (x))

(f E

,w

E

W).

(8.38)

XEX

The "usual" Fenchel conjugation occurs in the particular case where X is a locally convex space and W = X* (c R x ) ; in this case there hold some additional results involving the special structures of X and W = X* (studied in "usual" convex analysis). (b) Actually the historical development of the concept of conjugation has followed a different way, namely from (8.38) (defined first for X = R, next for X = R', then for an arbitrary locally convex space X), to (8.35) for two arbitrary sets X, W and an arbitrary coupling function ço, then back to W c R x and ço = (Pnat, that is, to (8.37), and then, finally, to Theorem 8.2, characterizing the mappings f —> fc of (8.27) as "conjugation operators" (8.1), (8.2), with cp = ço,. uniquely determined by c. (c) From Theorem 8.2 it follows that if X is a set and W c R x , a mapping c : R x --> R w is the (generalized) Fenchel conjugation * (of (8.37)) if and only if c satisfies (8.1), (8.2) (or, equivalently, (8.1), (8.5)) and ( X {x} ) c ( w )

= w (x )

(x e X, w c W).

(8.39)

(d) In the case where X is a set and W c R x , every conjugation c generates a conjugation C.' : R x --> R w +R such that

flw—fc

(f G R x ).

(8.40)

Indeed, let us define for any f c (w E W, r E R) .

(w + r) = f`. (w) — r

(8.41)

Then, since c : R x --> R w satisfies (8.1) and (8.2), so does Cs." : j X ----> R W R , and (8.41) for r = 0 yields (8.40). Note also that, by the uniqueness part of Theorem 8.2, we have for the associated coupling functions (see Definition 8.2 (b)), (x c X, wc W, r E R) .

(8.42)

The converse of the above remark is also true; that is, given any conjugationZ' : R x --> the mapping c : R x --> R w defined by (8.40) is a conjugation, satisfying (8.41) (since + r) = (w) — r = (w) — r for all w E W, d G R). Thus for W c R x the theories of conjugations c : R x --> R w andF : X > W +R are equivalent. For conjugations -6" : Rx ----> R W +R see Remark 8.14(b) below. (e) Similarly to Remarks 2.33 and 3.25, the general case of conjugations c((p) : ---> R w , where X and W are two sets and yo : X x W ---> R, can be reduced to the case * : R x —> R v , choosing V = Vw,,p c R x defined by (2.188). Indeed, by (8.35), (2.187), and (8.37) (mutatis mutandis), we have for any X, W, cp:XxW --> ,

Conjugations

250

7?- and f

E R X , W E W,

fe ( P) (w) = sup {yo(x , w) ± — f (x)} = sup {v 4 (x) ± — f (x)} = f* (v w 4 ). x eX xEX

(8.43) Thus the theories of conjugations c(v) : R x —> R w , where X and W are two sets and yo :XxW—> R, and conjugations * : R x —> R". where V c R x , are equivalent. Again, while the latter is simpler, the former is more versatile due to the flexibility in the possible choices of W and cp. (f) By (e) above, one can obtain results on general conjugations c(v) : R x —> R w from the results on conjugations * : R x --> R w , where W c R x , simply by replacing w(x) by (p(x, w). Let us return now to the general case where X and W are two sets and yo : X x W ---> R. Remark 8.7 (a) By Theorem 8.2, we have a one-to-one correspondence between conjugations c : R x —> R w and coupling functions yo :XxW —> R. Hence each statement on conjugations has an equivalent counterpart on coupling functions, and vice versa. Note also that for any conjugation co : R x ---> R w and any coupling function (po : X x W ---> T?, we have c(400)

=

co , (Pc(R)

(8.44)

= (Po.

Therefore each result on a pair (yo, c(v)) can be also formulated, equivalently, as a result on a pair (c, (pa). In the sequel we will state the results only in one of the two forms, but we will also use their equivalent reformulations (in terms of the other pair) whenever necessary. (b) The pairs (x0, w) e XxW such that the supx , x in (8.35) is attained at x = x o are of particular importance (see Chapter 10). (c) By (8.23) and the definition (8.28) of yoc , we have for any conjugations c1, c2 : R x ---> R w , c1

C2 •#' (Pc,

(8.45)

(Pc2•

We will denote by C — C (R x , R w ) the set of all conjugations c : R x ----> R w . Since C c D = D(Rx , R w ), we have a natural partial order on C induced by the one of D, that is, by (7.34). One can prove several algebraic properties of the above one-to-one correspondence (see [272], §5) of which we mention here only the following one (see [272], Theorem 5.1): Theorem 8.3 Let X and W be two sets. For any ço V JEJ AjEJ c(p 1 ) are taken in D) v c(yoi) EJ fJ

=

E RXxW

sup fc (0 = fc(suPiEJ (Pi) JEJ

E

E

J) we have (where

EX),

(8.46)

8.2 Representations

(X{x}

d)JEJ

c(cp,)

of Conjugations c :

--->

Ftw

251

= inf (x {x } + d)c 4/ )

JEJ

d)c(infiEJ

= (X{x}

(x

(Pi)

E

X, d

E

R),

(8.47)

A

I (x, d)

= sup {inf. (x{x } + d)°j

f JEj

J

fe

EJ

(pj)

(f

E

EPi f

E

(8.48)

R x ).

Hence C is a complete sublattice of D = D(Rx , R w ) (and thus Vc = V D, Ac = AD), and the one-to-one mapping c (. ) : ---> c((p), defined by (8.35) (with inverse c ---> (pc of (8.28)) is a complete lattice isomorphism of R x x w onto C. Returning to representations of conjugations with the aid of coupling functions, we have: Proposition 8.2 If c : R x ----> R w is a conjugation with associated coupling function yoc : X x W —> R, then for any f E R x we have inf

d =

inf

d

dER

dET?

(w

E

W).

(8.49)

40,( ., w) — d“

Proof By (8.36) and Corollary 8.1, we have (10 =

inf dER

d =

inf dER

d =

inf

d

(w e W).

dET? (19( ( • ,11))+ — cif

The proof of the second equality in (8.49) is similar, applying to a = f c(w) the formula a = inf Ici E Ria

d}

(a

E

R),

(8.50)

which is obviously true for all a E R; furthermore, if a = —oo, then inf (d E R a d} =- inf R = —oc, while if a = +oo, then inf {d E Ria d} = inf = +00, and thus (8.50) holds for all a E R. Let us consider now the dual conjugation c' : R w ---> R x (see Theorem 8.1 and Remark 8.2(b)). Theorem 8.4 Let X and W be two sets. If c : R x ----> R w is a conjugation with dual conjugation c' : Rw —> R x , then for the coupling functions (p c : X x W —> R and (pc, :

252

Conjugations

W x X —> R associated to these conjugations, we have Ve(w, x) = (X{w}) c) (x) = (Pc(x, w)

(x c X, w E W),

= (X{x)) c (w)

(8.51)

(g E R W , x E X).

gc'(x) = sup {(p c (x, w) ± —g(w)}

(8.52)

W

Proof By (8.12) and the uniqueness part of Theorem 8.2, we have the last two equalities in (8.51). Also from (8.11), and since (pc, is unique (by Theorem 8.2 applied -f?x), we obtain (p c, (w, x) = (X{T}) c (w) to the conjugation c' : Rw G W, X E X), which proves (8.51). Finally, by (8.11) and (8.51), we have (8.52). Remark 8.8 (a) For a direct proof of (x{„,})(x) = (x{ x }r(w), using only the definition (7.13) of c', see [267], thm. 4.2. Alternatively, the equality (pc, (w, x) = ve (x, w) follows also from Theorem 7.5, since by (7.83) applied to ee of (8.30) and by (8.9), we have ec, (w, x, b) = min {a c R I yo c (x , w) + —a

= min {a E

(P c (x, w) ± —b

b) a)

= vc (x, w) ± —b = ec (x, w, b)

(1.V E W, x E

X, bE T?),

(8.53)

whence, by (8.31), yo,s(w , x) = ee(w, x, 0) = ec(x, w, 0) = Vc(x, w) Note also that the functions (8.53) are the (W, e) -elementary functions of Remark 7.10(b) and the (W, (pc )-elementary functions of Remark 3.25(d)). (b) From the above it follows that for any coupling function cp:XxW—>R,if we define yo' :WxX-->R by (6.138), then

(8.54)

c((p) I = c((p').

Indeed, by (8.52), for c = c((p), (8.44), (6.138), and (8.35) (mutatis mutandis) we have for any g c R w and x c X, g cM 1 (x) = sup ((pcm (x, w) ± —g(w)) = sup {yo(x , w) wEw wew

—g(w))

= sup {(w, x) ± —g(w)) = g c(V ) (x). W

(c) In the particular case where W c le and ç = (Nat (of (2.184)), formula (8.52) yields, denoting gc (`Pr by g*, (x) = sup [w(x) + —g(w)} wEw

(g E

RW, XE

X),

(8.55)

8.2 Representations of Conjugations c : /ix

253

which corresponds to (8.37). However, while in (8.37) we had W C R x , in (8.55) we need not have X c R w (thus the symbol g* is an abuse of notation). For W c R x we can identify X (canonically) with a subset of R w if and only if the canonical mapping K of X into R w , defined by (2.140), is one-to-one (or, equivalently, W separates the points of X). In other words, the asymmetry between the roles of X and W c R x can be also expressed as follows: The relations WI, tv2 c W, çOnat wl = çOnat (*, w2) imply that w 1 = w2, but the relations X1, x2 c X, 49 nat(xi , .) = çonat(x2, .) need not imply that xi = x2. Of course, whether or not K : X R W is one-to-one, we have K (X) c R w , and hence we can consider the natural coupling function 0nat W X K (X) —> R defined (replacing in (2.184) X and W by W and K (X), (7 respectively) by (pnat(w, K(x))

=

(w c W, xc X); (8.56)

K(x)(w ) = w(x) = Sonat(x , w)

then we have ge ( r.dt ) (K(x)) = g* (x) (of (8.55)) for all g G R w , x c X. Proposition 8.3 Let X and W be two sets and cp : X x W R a coupling function. For two functions f c RA' and g c R n" the following statements are equivalent: 10

f

2°.

We have

g c((p)'

f (x)

3°. g

g(w)

yo(x, w)

(x c X, w e W).

(8.57)

f'.

Proof 2° 3 0 . By Lemma 8.2 and (8.35), we have the equivalences (x G X, wc W)

2° g(w) yo(x, w) ± — f (x) . g(w)

sup {(x, w) ± — f (x)} = f` (`P) (w)

(w c W).

XEX

Finally, the equivalence 1° •#. 3° holds by (7.15) (with A = c(0). Proposition 8.4 For any yo : X x W —> R, f E R x , X G X, and w G W, we have the following inequalities that are equivalent to each other: (x, w) ± — fc (° (w)

f (x)

(x, w)

fc ((P) (w)

40 (x, w) ± — f (x)

f (x) fcM (w).

(8.58)

Proof The last inequality of (8.58) holds by (8.35), and the inequalities of (8.58) are equivalent to each other by Lemma 8.2.

254

Conjugations

Remark 8.9 (a) Alternatively, applying Proposition 8.3 to g = fc(9') , we obtain the second inequality of (8.58). (b) Propositions 8.3 and 8.4 also follow by combining Lemma 8.2 with Propositions 7.3 and 7.4, respectively, applied to e and e' of (8.30), (8.53). By Remark 7.8, applied to these e, e', only the first one of the (equivalent) inequalities (8.58) should be called the Fenchel-Young inequality (for A(e) = c(v)). In addition to Remark 7.8 for the particular cases of the first and third inequalities of (8.58), let us mention that some authors, such as Moreau [205], have called the second inequality of (8.58) "the Fenchel-Young inequality." Definition 8.3 Two functions f E W x and g E W x are said to be (upper) dual (to each other, with respect to yo) if (8.57) holds. Remark 8.10 (a) Propositions 8.3 and 8.4 show that fc (`P) is the least element of the set fg E R W I f and g are dual]. Similarly, given g c , ge (`P ) ' is the least element of If E R x I f and g areduall. This observation is similar to Remark 6.13(b) (formula (8.57) "corresponds to" (6.129)). (b) Definition 8.3 and (a) above are, essentially, the particular case A (e) = c() of Definition 7.3 and Remark 7.9. Let us pass now to fa'' for a conjugation c : R x ---> R w , that is (by (8.52) for g = fC), to f cc'(x) = sup {yo c (x, w) ± — f c (w))

(x c X).

wEw

(8.59)

Theorem 8.5 Let X and W be two sets. (a) If c : R x ----> R w is a conjugation with associated coupling function (p r. : X x W > R and dual conjugation c' : R w —> R x , then for any f E R x we have f cc (x) = sup { inf ( f (y) we w yEX

—(p c (y, w)} ±yo c (x , w)}

(x E X).

(8.60)

(b) Under the same assumptions we also have for any f E R X , fC C'

sup

{q(.. w)

r} =

sup

{q(., w)

r).

(8.61)

wEW, rER

wEW, rET?

ço,(.,w)trf

Proof (a) By (8.59), (8.36), (8.25), and the commutativity of +, we have for any

fc

fcc' (x) = sup (yoc (x, w) ± —fc(w)} wEw

= sup (cpc (x, w) ± — sup koe (y, w) ± — f (y)}} wEw

yEx

8.2 Representations of Conjugations

= sup koc(x, w) + inf ( - 149c(Y, w) • yEX

wEw

sup {

wew y EX

c

:

—> R w

255

- f (y)1)1

{ f (y) + — (10(.(y , w)) ± goc(x , w)}

(x E X).

(b) By (8.16) and (8.51), we have for any f TT'

sup

[(pe (. , w) — r),

wEW, TER 49 (( ., w) - rf

whence, replacing r E R by —r E R, we obtain the equality of the first and third terms of (8.61). The first equality in (8.61) follows similarly, using also Remark 8.4 (applied to (8.16)). 0

Remark 8.11 (a) In the particular case where W c R x and çpc = Vnat (hence C = c(yonat ) = *), formulas (8.59) and (8.60) for any f E R X yield, respectively (denoting, by abuse of notation, fe@Pm.'"Pr.tr by f **), f ** (x) = sup

{w(x) ± —f

*(w)}

X),

(8.62)

(x G X).

(8.63)

(X E

WEW

f**(x) = sup {

,»Ew VEX { f (y)

— w (y)} ± w(x))

(b) The notation f* = f c 4nca ) E —1?-w of (8.37) can be used for any W c Te, but the notation f ** = f C (,Onal )C ((Nat E X or (8.62), (8.63) can be used only if W (c Te) is fixed unambiguously, since otherwise it may lead to confusion; indeed, although f —> f ** of (8.62), (8.63) is a mapping of T?x into T?x , it depends on W. Therefore, whenever it will be necessary to specify W, we will use, instead of f* = fc ( Pnat ) , the notation f e (w ''Pnat) . The notations f w , g w and f ww for f*, g* (of (8.55)) and f**, respectively, used in [266], may lead to confusion, since then gw = g k(X) (g E R W ); therefore they will not be used here. r

Finally, let us give an application of conjugates and biconjugates to (W, ço)-support functions crG, (p and (W, ço)-convexity of sets G C X (see Section 2.6).

Proposition 8.5 Let X and W be two sets and yo : X x W —> R. (a) For any set G C X there holds aG,(p = (XG) c(çQ) • (b) We have G

E iC(W,

(X0

(

8.64)

ço) if and only if Y (x) > 0

(x G X \ G).

(8.65)

256

Conjugations

Proof (a) By (2.198), (2.156), and (8.35), we have for any G C X and w E W. CFG,ço(W)

= sup gp(g , w) = sup (go(x, w) ± — xc(x)1 — gEG

(XG) c4) (W)-

xcX

(b) By (8.59) (for c = c(0) and (a), we have for any G C X and X E X, c(99)c(99Y (x) = sup MX, w)

± —

XG)

(

9) (W))

(XG)

WEW

= sup {v(x, w) + — sup go(g , w)), wEW

gEG

whence, by (2.197) and (8.21), we see that G E 1C(W , 0 is equivalent to (8.65). 10 Remark 8.12

In the particular case where W c R x and go = (Nat, (8.64) becomes ac

=

(

(8.66)

XG) * ,

and Proposition 8.5(b) says that G E 1C(W) if and only if (XG) ** (x) > 0

(x E X \ G).

(8.67)

8.3 Biconjugates and Abstract Convex Hulls Theorem 8.6 Let X be a set, W c Te , and cp = gona t : X x W ---> R, the natural coupling function (2.184). Then, denoting fc (`P) c ( P ) ' by f** (see (8.62) and (8.63)), we have f** — fc0 ( w+R) = fc0(w+R)

(f E R x ).

Proof This formula follows from (8.61) and (3.11).

(8.68) 0

Corollary 8.2 Let X be a set, and let W C Rx . Then for a function f G R x the following statements are equivalent: 1°. 2°.

f E C(W -1-- R). f E C(W + R).

30 .

f = f**.

40 .

We have f = sup {w + —r(w)}. • ivEw

(8.69)

8.3 Biconjugates and Abstract Convex Hulls

257

Proof The equivalences 1 2° 71 ?- P defined by

f A (p) = sup e(x, p, f (x))

(f

E

Rx , p

E

P),

(8.76)

XEX

is a duality, satisfying (8.72). Proof By (8.75), we have for any x

e(x, p, al)

E

X and p

E

P,

(ai, az E R, at

e(x, p, a2)

a2),

0,

and hence for any index set I

e(x, p, inf ai )

sup e(x, p, ai )

iEr

(8.77)

g R).

On the other hand, by (8.75) and (8.74), sup

W

b (X) < P

(x

a

E

X, pE P, a

E

R),

(8.78)

bER wr,(x)a

whence, since the mapping b —> w bp (x) is antitone (by (8.74)), sup,, / e(x,P,aj )

e(x,p,a,) ( wp

WP

and hence sup id e(x, p, a,) c lb c T? I

e(x, p, inf ai) = inf (b LEI

E

Wp b (X)

R I w b (x)

(i

ai

E /),

infia ai ). Therefore, by (8.75), inf ai ) tEl

sup e(x, p, ai ), iEl

which, together with (8.77), (8.75) for a = +Do and Theorem 7.3, proves that A : R x —> TV) of (8.76) is a duality.

259

8.3 Biconjugates and Abstract Convex Hulls

Furthermore, by (8.78) and since b a

w ep('P'a) (x)

w bp (X)

is antitone, we have

(a E R , e(x, p, a) R P and by (8.73), we obtain for any

f E WX ,

f

b = sup {w ipE P, b E R5 w" p

J. } = sup w =

III

fco(W)•

w EW w(

Remark 8.14 (a) If P = W in (8.73), then A : R x —> R w of (8.76) is a duality satisfying (8.72). Given any W C R X , it is always possible to parametrize it in the form (8.73) (even with P = W), since one can take

P = W, w = p

(p EP= W, bE R).

(8.80)

In this case (8.75) and (8.76) become, respectively, —oc

if p(x)

a

e (x, p, a) =

(8.81) if p(x) > a,

°

1

-00

if P

f

if P

f.

(8.82)

f(p) =

Note that (8.82) (with P = W) means that f A = 77{ tv€ W lw } (with r) G of (4.40)), so it is similar to (7.32). (b) In some cases (in which T? c W) one can take the parametrization W = P R, w p" = p

—b

(p E P, b€ i).

(8.83)

Then (8.75) becomes, using Corollary 8.1, e(x, p, a) = inf {b E R I p(x) = p(x) ± —a

—b

a} = inf {b E R I p(x) ± —a

(x E X, p E P, a E R),

b) (8.84)

260

Conjugations

and hence (8.76) reduces to f ° (p) = Sup (P(X) ± — f (x)}

(f E R x , p E P),

(8.85)

XEX

that is, to (8.37) with W replaced by P c W (of W = P + R). (c) Let us mention that if W c R x , then fco(w) can be also expressed as a "mixed" second dual of f, namely, by composing A w : R x ----> R w of (7.32) with * = c(çonat) / : Rw R x of (8.55); i.e., there holds (f °w ) * = fc0(w)

(f E R x ).

(8.86)

Indeed, by (8.55), (7.32), and (3.11), we have (f °w ) * (x) = sup {w(x) + — f °w (w)} = sup (w(x) + — X(5-jEw153(f}( 101 WEW

=

W EW

sup w(x) = fc0(w)(x)

(f E T?X , X E X).

wEW w(f

Theorem 8.6 and Corollary 8.2 can be extended to the more general case of two arbitrary sets X, W and an arbitrary coupling function ço :XxW----> R. Indeed, from Theorem 8.6, Corollary 8.2, and Remark 8.6(e), we obtain, respectively:

Theorem 8.8 Let X and W be two sets and ço : X x W ----> R a coupling function. Then, with Vw , v C R x of (2.188), we have fcc(p)c(0' _ fco(vw, v +R) = Lo(Vw,+R)

(f E R X ).

(8.87)

Corollary 8.5 Let X and W be two sets and go : X x W —> R a coupling function. Then for a function f E R x the following statements are equivalent: f E C(Vw, (p ± — R). 2°. f E C(Vw, cp R). 3° . f _ fe(oc((p)' .

4C.

We have f = sup tço(-, w) ± —if ( P) ( w)1.

(8.88)

WEW

Remark 8.15 (a) The equivalences 2° - R to the coupling function (po : X x (R n ) * —> R defined by E X = 10, 1} ' , (13 E (R n ) * )

(,0 0(X , 4) ) = (I) (X)

Then for the conjugation c((p0) (13 E (Rn)*, we have f*(431 x ) = sup xEx

Rx

FDI x (x) — f (x)I =

kir) * and for any f : X —> R and

sup ((Mx, (I)) — f (x)} = f (") (0), xEx

whence, by (8.93), f (xo) =

sup fw (x0) — f* (w)} wE(R")lx

fc (')'('")/ (x0)

f(4)

(8.96)

sup ko 0 (x0,

(D) —

( f G R X , xel E X),


263

and thus f(x 0 ) =

sup {(I)(xo) — f 4") (41))1 4)E(Rn)*

(f G R x , xo G X).

(8.97)

Hence, using again (8.94), we obtain, as above, that f (xo) =

(f E R X , x0 E X = {0, 1}"), (8.98)

max IDE(Rn)*

with the max being attained for (13 = (13f (x()) of (8.94). (c) Now we can also give a different proof of Theorem 3.13, as follows: Let f e R X and for each xo E X = {0, 1}", choose (1)R x„) E (R")* satisfying (8.94). Define :Rn ----> R by

,

T

1(x) = max (cDf(x (,)(x — xo) xo efo m"

(x c R").

f (xo)}

(8.99)

Then lis a polyhedral convex function on R" , and taking xo = x E (0, ir in (8.99), we obtain

1(x)

(x E X = {0, 1}").

f (x)

(8.100)

On the other hand, for the conjugation c((p0) : R x ---> R w of (b) above, we have f (xo) — (13.

f (x 0 )(X0)

f C 4° (q) f (x( ) ))

(f e R x , xo e X).

(8.101)

Hence, by (8.99), (8.101), and (8.97), we obtain .r(x) =

max { (1) Rx 0 )(x) — f C (") ) (q) px () )))

x000m"

= f (x)

sup

1.4)(x) — f c (') (q))1

(f E R X , X E X),

which, together with (8.100), yields :fi x = f.

8.4b

The Case Where X Is a Metric Space, W = X, and ço =

Let X -= (X, p) = W be a metric space, 0 < a Pa,N:XXX -->R by

1, 0 < N < +cc, and define

(

çoa,N(x,w) = — Np(x, w)a

(x, w e X).

(8.102)

Then, by (8.35) with ço = ço,, N we have T ( aN ) (w) = sup {—Np(x, X

— f (x)}

(f E R X ,

E X).

(8.103)

EX

Remark 8.17 By Remark 8.6(e), an equivalent method would be to use only Vw N C Rx of (2.188), i.e., Va ,N of (3.123), and (p = (Pnat : X x Va ,N —> R,

264

Conjugations

which would have the notational advantage of working with f* , f** instead of fP) and f c (0`.(`P)/ . Note that there is a one-to-one correspondence y v y = —Np (. , y)a between X and Va, N C R X ; indeed, the relations yi, y2 E X, — NP(x, yir = —Np(x, y2)a (x E X) imply (taking x = Yi) that yi = y2 • However, the main advantage of starting with W = X and (8.102) is that both conjugations c(cpc,,N) and c(goa,NY are mappings of R x into R x itself (versus * -= c(Sonat) : R x ----> R V'''N and c(çonatY : R V " ----> R X , with Va,N of (3.123)). Moreover, if W = X, then for any coupling function ço : X x X —> R satisfying ço(x, w) = ço (w, x)(x E X, w E W) we have, by (8.52) and (8.35), gc@PY (x) = sup {cp(x, w) ± —g(w)) = g (*° ) (x) wEx

(g E R X , x E X).

(8.104) Proposition 8.6

There holds f

fe((/9a.N)

(f

(8.105)

R x ).

Proof Take x = w in the right-hand side of (8.103). Now we can give the following complement to Theorem 3.14: Theorem 8.10

For a function f : X -- R the following statements are equivalent:

R), with 1°. f E C(Vci ,N functions of the form (3.124)). 2 0 . We have

Va,N

of (3.123) (i.e., f is the supremum of a set of

f C((Pcr.AI)

30.

f

(8.106)

We have

(--Pc ( " ) = f.

(8.107)

Proof 1' 2°. If 1 0 holds, then, by Theorem 3.14, either f *Do or f (X) C R and f is a-Holder continuous with constant N. If f ±oo, then, by (8.103), we have (8.106). Assume now that f (X) C R and f is a-Hiilder continuous with constant N. Then, taking x i = x and x2 = w in (3.120), we see that —Np(x,

— f (x) — f (w) = —Np(w,

— f (w)

(x, w G X);

thus the supremum in (8.103) is attained at x = w. Hence we obtain (8.106).


265

2' = 1°. If 2° holds, then, by (8.106), (8.105), (5.21), and (8.104),

f

_

c((P.,N)c(q3..N)

=f

(_f)CQ0.,N)

whence f = fc (`P. N"P") . Consequently, by (8.104), Corollary 8.5 (implication 3° = 2°), and V W,ÇOa N = Va,N, we obtain P. 1° 3°. This equivalence follows from 1' .R by (pa (x, (y, N)) = —Np(x,

1, and define

(x,yE X, ORis the coupling function (8.102). (b) By Remark 8.6(e), an equivalent method would be to use only Vw, ç,,o, c Rx of (2.188), that is, Va of (3.133) and (p = (pnat : X x V a ---> R. However, it will be more convenient to start with W and go, above (while using also Va = Vw, (pa )• (c) For go, of (8.111) we have (with (pa,N of (8.102)) f ( Pa) (y, N) = sup —N p(x, y) a — f (x)} = f c(çacr " ) (y) XE

(f E R x , y E X, N > 0), (8.113) eq'aY(x) =

sup (—Np(x, y) a — g(y, N))

(g E R W , x E X).

(8.114)

()),N)EW

Now we can give the following complement to Corollary 3.4: Theorem 8.11 For any f

E

Te and xo E X we have (with the notation Ha of Section 3.2f) fc0(k)(xo) = sup inf (f (x)

NP(x,

X

N >0 xEX

(8.115)

Or ) •

Proof By (3.149) and Theorem 8.8, there holds fco(Ha) =fc(c.)c((Pa)'

(8.116)

(f c R x ).

=

But, by (8.114) (for g = fc(ç'- ) ), (8.111)—(8.113), (8.104) (for (p f ( (P. N ) ), and (8.108), we have fc (g)a ) c (''' (x0) =

sup

Iço.(xo,

go a.N, g =

(Y N)) — fY , N)}

(y,N)EW

= sup sup ko ce,N (xo, y) —

fc(P-N) ( y )} =

N>0 yEX

= sup inf N >0 x

EX

sup

f

c( o.N)c((p..N) (x0 )

•

N>0

{f(x)

N P(x

which, together with (8.116), yields (8.115).

X

Or }

(f

E

W X , X0 E X),

El

8.5 The Conjugate of f d —h, Where f, h -

8.5

E

Rx

267

The Conjugate off —h, Where f, h E R x

In the present section we will assume that under addition, that is, satisfying

X

W W

is a set and W is a subset of R x , closed

C

W

(8.117)

,

and we will study the problem of expressing (f 4- —h)*, where f,h E R x and where * : R x --> R w is the conjugation (8.38), with the aid of f* and h*. We will use the following simple lemma:

Lemma 8.4 Let X be a set, and let W C Rx satisfy (8.117). Then for any f, h E Te and w, y e W we have

—h)*(w) = — inf ((f — w)(x) xEx

(f

—h(x)},

(8.118) (8.119)

(f — w)*(v) = f *(w + v). Proof By (8.38), we have for any f, h E R x and w E W,

(f

—h)*(w) = sup {w(x) — { f (x) XE

—h(x)}}

X

= — inf {—w(x) xEx

If (x)

—h(x)}}

= — inf (( f — w)(x) 4- —h(x)}. xex

Furthermore, by (8.117), f*(w y) is well defined for any f Thus, by (8.38), we obtain

(f — w)*(v) = sup {v(x)—(f — w)(x)} = xEx

E

R X and w, y

E

W.

sup {w(x)+v(x)— f (x)} = f*(w v). xEx

—h)* to the study Remark 8.19 (a) Formula (8.118) reduces the study of (f of the infimum of f —h, and formula (8.119) expresses (f — w)* for any w E W, with the aid of f* (also, it motivates the assumption (8.117), since f* is defined on W). Therefore we will first give some duality results for the optimization problem occurring in the right-hand side of (8.118) (even for any W c Te), and then we will combine them with Lemma 8.4 to obtain the desired results on (f (b) If 0 E W, then formula (8.118) for w = 0 yields

(f

—h )* (0) = — inf ( f (x) xEx

—h(x)},

(8.120)

268

Conjugations

and hence in this case one can also proceed in the converse direction, that is, applying the results on (f —h)*(w) to w = 0, one can obtain results on infxEx (x) —h(x)}; however, in the sequel we will not pursue this converse direction. Note that if 0 E W, then W=W+OC W+ W, and hence in this case (8.117) is equivalent to W + W = W. Proposition 8.7 Let X be a set and W

inf If (x) xex

C Rx.

—h**(x)} =

Then

inf {h* (w)

(f, h E R X ).

— f* (w)}

(8.121)

Proof Observe first that, by (8.24) and (8.25), for any set X, any function f : X —> R and any a E R we have

inf (f (x) 2F a) = inf f (X)

xex

(8.122)

a.

Thus, by (8.62), (8.25), (8.122), the associativity and comutativity of +, and (8.37), we obtain for any f, h G R x ,

inf If (x)

xex

—h**(x)} =

inf { f (x)

— sup {w(x) ± —h*(w)}}

xEx

= inf If (x) xEx

inf {h*(w) + —w(x))} wE w

= inf inf {f(x) xex wew

= inf inf {h*(w) wEw xEx

= inf {17*(w)

{h* (w)

—w(x)}}

{f (x)

— w(x)}1

inf If (x) 2F —w(x)}}

wew

xEx

inf {h*(w) wE

—f * (w)}.

Corollary 8.9 Let X be a set, and let W

inf { f (x)

xEx

C

—h** (x)} =

Ire. Then inf {f**(x) 2F —h**(x)}

xEx

(f, h G R x ).

(8.123)

Proof Applying Proposition 8.7 to f replaced by f** and using that (f **)* =f (by (5.29)), we obtain

inf { f** (x) 2F —h**(x)} = inf {h(w)

xEx

IvEw

— f * (01

which, together with Proposition 8.7, yields (8.123).

(f, h E R A'),

(8.124)

8.5 The Conjugate of f j- —h, Where f, h

E

269

Rx

Proposition 8.8 Let X be a set, and let W c R x . For a function h equivalent: 1 0.

E Rx

the following statements are

h = h**. We have

2'.

inf { f (x) xEx 30.

—h(x)} =

inf { f (x) xEx

(f E R x").

—h** (x)}

(8.125)

We have

inf { f (x)

—h(x)} =

xEx

4°.

inf {h* (w)

— f * (w)}

(f E R X ).

(8.126)

inf {f**(x)

—h** (x)}

(f E R x ).

(8.127)

tuEw

We have

inf { f (x)

—h(x)} =

xEx

xEx

5 0• There exists a function (ph : R x --> R such that inf { f (x)

xEx

(f E R x ).

(8.128)

(f E R x ).

(8.129)

— h(x)} = n(f ** )

There exists a function Ilfh : R w --> R such that

6".

inf { f (x)

xEx

— h(x)} = *h(f * )

We have

7'.

inf {f (x)

—h(x)} =

)(Ex

inf f**(x)

xEx

—h(x)}

(f c R x ).

(8.130)

Proof The implications 1 0 = 2° and 4 0 = 5° = 6° are obvious. 2° = 3°. If 2° holds, then, by (2° and) Proposition 8.7,

inf If (x)

xex

—h(x)} =

inf {f (x)

xEx

= inf

wEw

{h* (w)

—h** (x)}

— f*(w))

(f E R X ).

3° = 4°. If 3 0 holds, then, by (3° and) (8.124), we obtain 4°. 6' = 7°. If 6° holds, then, by f*** = f* , we obtain

inf { f (x)

xEx

—h(x)} = Ilfh(f * )

inf {f ** (x) = *h(f ***) = xEx

— h(x))

(f E Rx).

270

Conjugations

7' = 1'. 1f7° holds, then, since a + —a obtain 0

inf {h(x)

E -k),

from 70 for f = h we

—h(x)} = inf {h**(x) xEx

xEx

whence h**(x) + —h(x) with (5.21), yields 1°.

0 (a

0

E X).

Thus, by (8.8), h**

h, which, together

0

Remark 8.20 Since h*** = h*, the implication 1 0 = 3 0 of Proposition 8.8 is equivalent to Proposition 8.7. Let us pass now to the study of (f + —h)*:

Proposition 8.9 Let X be a set, and let W C R x satisfy (8.117). Then

(f

4

— h**)*(w) = sup { f* (w +

(f, h E R X , W E W).

—h* (v)}

vEw

(8.131) Proof By Lemma 8.4, Proposition 8.7 and (8.25), we have (f

—h**)*(w) = — inf {(f — w)(x) xEx

—h**(x)}

= — inf {h*(v) + —(f — w)*(v)} vEw

— inf {h*(v) + — f*(w + v)} ve W

= sup {f*(w + y) ± —h* (v)}

(f, h E R X , w E W). LI

VE W

Corollary 8.10 Under the assumptions of Proposition 8.9 we have (f

— h**)* = (f**

—h**)*

(f, h E

).

(8.132)

Proof Applying Proposition 8.9 to f replaced by f** and using that f*** = f*, we obtain (f** + —h**)*(w) = sup {f*(w + y) ± —h* (v)} vEw


(f, h E

WE

W),

(8.133) LI

8.5 The Conjugate of f + —h, Where f, h E R x

271

Proposition 8.10 Let X be a set, W c Te and Xo E X. Then (x{x„}

(h

—h )*(w) = h(x0) ± w(x0)

E

RX ,

e W).

W

(8.134)

Proof By (8.37) and (2.156), we have — h) * (w) = sup {w(x) xEx

(Xtx„)

— h(x)))

— (X{x0}(x)

= max ku (x0) + h(x0), — Do)

= h ( x0 ) ± w ( xo )

III

(h e — R x , w W).

Theorem 8.12 Let X be a set, and let W c R x satisfy (8.117). For a function h statements are equivalent: 1 0 . h = h**. 20 . (f + —h)* = (f —h")* 3°. We have

(f

—

(f

E

E

Te the following

R x ).

h) * (w) = sup { f* (w + y) ± —h* (v)}

(f

E

WX ,

WE

W).

VE W

(8.135) 4".

(f

—h)* = (f**

—h**)*

(f

E

R x ).

These statements imply the following statements, equivalent to each other: 5'.

There exists a mapping ah : X --> R w such that (f + —h)* = ah (f**)

6°.

(f

E

(8.136)

R x ).

There exists a mapping Ph : R w —> R w such that (f

— h) * = Ph(f* )

7'. (f + —h)* = (f** + —h)*

(8.137)

(f c R x ). (f

E

R x ).

Moreover, if — and only if — the constant function 0 statements 1' — .. . — 7' are equivalent.

E

RA' satisfies 0**

0, all

Proof The implications 1° = 2° and 4' = 5 = 6' are obvious. 2' 3'. If 2' holds, then, by (2" and) Proposition 8.9, (f

— h) * (w) = (f

— h ** ) * (w)

= sup ff*(w + vE w

—h* (v)}

(f

E

RX ,

WE

W).

272

Conjugations

3° = 4°. If 3° holds, then, by (8.133), we obtain (f + —h)*(w) = sup { f*(w ± y) + —h*(v)} VEW

(f e R x , w

= (f** + —h**)*(w)

W).

E

4° = 2'. If 4' holds, then, by Corollary 8.10, we obtain (f + — h) * = (f** + — h ** ) * = (f -.k — h ** ) *

2° = 1'. If 2' holds, let xo E X and w W(X0) E R, and 2° for f = xixo ), we obtain

E

(f

R x ).

E

W. Then, by Proposition 8.10,

h(x0) + w(xo) = (X{x 0 } --.k — h) * (w) = (X{x 0 } + — h ** ) * (w) = h** (xo) + w(xo), whence, since X0 E X has been arbitrary, h = h**. 6° 7°. If 6' holds, then, by f*** = f*, we obtain

(f

(f + — h) * = fi(r) = fih(f*** ) = (f** + —h) * The implication 7 0 = 5' is obvious. 7° = 1°, if 0** = 0. By (5.21), 7° with f = h, a + —a and 0** = 0, we obtain h** + —h

(h** + —h)** = (h + —h)**

E

0 (a

Te).

E

R), (5.30),

0** = 0,

whence, by (8.8), h** h, which, together with (5.21), yields 1'. 0** = 0, if 7' = 1'. If there holds the implication 7° = 1 0 , then, since h = 0 satisfies 7 0 , it must satisfy 1° as well, that is, 0 = 0**. CI

Remark 8.21 (a) The main part of Theorem 8.12, that is, the implication 1° = 3°, remains also valid for any W c R x satisfying W± WC W (see the Notes and Remarks). (b) By Theorem 8.6, one can replace in the above results f** and h** by f,„(w +R) and h co( w+R), respectively. For example, formula (8.123) can be written in the equivalent form inf { f (x) + — hco(w+R)(x)} -= inf { feo(w+R)(x) + — hco(w+R)(x)) xex xEx (f,h e R x );

(8.138)

8.5 The Conjugate of f ± —h, Where f, h

E

Rx

273

taking in (8.138), in particular, W = J.A4 of (2.157) (where M C 2 X ), we obtain, by Corollary 4.9 and formula (4.102), —144 04) (x)} = inf { fq 0,0(x)

inf If (x)

— hci(m ) (x)}

)(Ex

xEx

(f, h E R x ).

(8.139) Also, by Corollary 8.2, condition h = h** in 1 0 of Proposition 8.8 and Theorem 8.12 is equivalent to h E C(W R), and condition 0** = 0 in Theorem 8.12 means that O E C(W R). Note that if 0 E W (in particular, if we have (8.117) and W = — W), then OE W±Rc C(W R), so 0** = 0. (c) The above results can be extended to the case where X, W C R X (satisfying (8.117)), and * : R x R w are replaced, respectively, by two arbitrary sets X, W, R is a coupling function and c(p) : R x —> R w (of (8.35)), where (p : X x W such that Vw

Vw, g)

C Vvv.ço ,

(8.140)

with Vw4 c R x of (2.188); indeed, by Remark 8.6(f), this can be done replacing everywhere w(x) by ço(x , w). Condition (8.140) means that for any w 1 , w2 E W we have ço•, w1) çoe, w2) E Vw 4 , or, in other words, that there exists a binary operation on W, say, ± : W x W W, such that

(x

(P(x w1 + w2) = So(x , w1) + 49 (x w2) ,

E X, WI, W2 E W).

(8.141)

R x and f, h E — R X , one (d) Under more restrictive assumptions on X and on W c — also has a different formula for inf , Ex {f (x) —h(x)), or, equivalently (via Lemma —h)*, namely, 8.4), for (f

inf {f (x)

—h(x)} = — min { f*(w) weW

VEX

(f

—h)*(w) = min { f*(w — y)

vEiv '

(—h )*(v)}

(—h)*(—w)), (w E W),

(8.142) (8.143)

where of course we have to assume that —W = W and (8.117) (since (_h )* and f* are defined on W). For example, the classical Fenchel-Rockafellar theorem says that if X is a locally convex space, W = X* and f, —h E R x are proper convex functions such that one of them is continuous at some point xo e dom f n dom (—h ) (or, in another version, if f and —h are proper convex lower semicontinuous functions with dom f n Int dom (—h ) 0), then (8.142) holds; for proofs and other versions, see e.g. 12331, [156], [236]. In the case where X is a Banach space and W = X*, another (obviously less restrictive) sufficient condition for (8.143) to hold, due to Attouch and Brézis [7], is that f and —h are proper convex functions such that gdom f — dom (—h)) is a closed linear subspace of X.

(8.144)

The right-hand side of (8.143), with inf instead of min, is usually called the infconvolution (or epi-sum) of the functions f* and (_h )*, and, when for each w E W

274

Conjugations

this inf is attained (as in (8.143)), this inf-convolution is said to be exact. For some extensions of the Fenchel-Rockafellar theorem to a set X and W c R x , with f* of (8.38), see Martfnez-Legaz [176]. (e) Corresponding to the equivalence of (8.126) and (8.135) (for W C R X satisfying (8.117)), we have the equivalence of (8.142) and (8.143), under the assumption that X is a set and W is a subset of R x satisfying W = —W and (8.117). Indeed, if (8.142) holds, then, by Lemma 8.4, (8.142), and our assumptions on W, we have (f

—h)* (w) = — inf ((f — w)(x) .1-EX

= min (( f — w)*(v) VEW

—h(x)} (—h)*(—v)}

= min {( f * (w v)

(— h)* (— v)}

= min {(f *(w — y)

(—h)*(v)}.

vew

vEw

Conversely, assume now (8.143). Since by our assumptions 0 = w W W C W, from (8.143) for w = 0 and W = —W we obtain —h)* (0) = min (( f * (-1))

(—h)* (v)} = min { f* (v)

VEW

(—w) E

(—h)* (— v)} ,

VEW

(8.145) which, together with (8.120), yields (8.142).

8.6

Conjugations of Type Lau

The relation of equivalence of coupling functions, introduced in Definition 6.3, and the above one-to-one correspondence between conjugations and coupling functions (see Remark 8.7(a)) induce a relation of equivalence for conjugations, defined as follows: Definition 8.4 Let X and W be two sets. We will say that two conjugations — ci, c2 : RA' R w are equivalent, and we will write c i c2 if for the associated coupling functions ço,, , (pc., : X x W —> R we have co, c i q)(-2, or, equivalently (by Remark 6.16), E

wl(pc,(x,w)

—1} = fw C W I (Pc2(X,

—1}

(x E X).

(8.146) Remark 8.22 (a) Since every conjugation c : RA' > w is, in particular, a duality between the complete lattices E = (Rx , R w2 (where ço : X x W1 : X X W2 —> R) provided R, by Definition 5.6, or, more conveniently, by Proposition 5.7; namely c(p) c(*) as

8.6 Conjugations of Type Lau

275

dualities if and only if =

fc((p)c(0 1

fe(*)c(*)/

(f E R X ).

(8.147)

Even when W1 = W2 these two concepts of equivalence are different. Note that if c(*) as dualities, then, in particular, we have c((p) (XG) c(w)c(w)‘ = (XG) c(*)c(*)/

(G c X),

(8.148)

whence, by Proposition 8.5(b), we obtain

(8.149)

1C(W (P) = 1C(W *). Hence, in particular, if we have

A iw A ço = co w, * , A /0 * = co w, *

(8.150)

(e.g., if ço and Vi satisfy the second assumption of Proposition 6.5), then Aiço A * = A * , and thus, by Proposition 5.7, A * — A * ; however, this does not imply that — c(*) as conjugations Jr (in other words, that A * = A * ), i.e., that c(q) (e.g., take a locally convex space X, W = X* ± R, (p of (6.191), and * = — 1). In the converse direction, if c((p) c(*) as conjugations, that is, if (p *, and if both (p and * are of type (0, —cc}, then, by Proposition 6.6, we have ço = whence not only (8.147) but even c(q) = c(*). If ço or * is not of type (0, —cx)} and c(*) as conjugations, then A * = A * , so A'w A * = 4; hence, if (8.150) c(q) c(*) holds, then we obtain (8.149) (which does not imply (8.147), i.e., that c((p) as dualities). To avoid any confusion, in the sequel by c l c2 we will always mean equivalence in the sense of Definition 8.4 above, and for the case of equivalence of c((p) : R x R w ' and c(*) : R x —> R w2 as dualities (i.e., for (8.147)) we will write c((p) c(*) (or, that c((p) and c(*) are 6-equivalent). (b) For the equivalence classes [c], [c( -p)] E C(Te, Te)/ "' we have, by Definition 8.4 and (8.44), [c] = {c1 I coci "' (Pc} =

[c((P)1 = {c(*)

(Pc},

— (P } = {c I (Pc — (P).

(8.151) (8.152)

From Proposition 6.6 and (8.44) we obtain: Proposition 8.11 (a) Each conjugation c : Rx —> R' is equivalent to a unique conjugation c(0) : R x R w , , with (p i :X x W—> R of type 10, —00, namely (p i of (6.176) for ço = (pc (i.e., (p i = (0)[wei of Remark 6.17(a)). (b) For any coupling function ço : X x W R, the conjugation c((p) of (8.35) is equivalent to a unique conjugation c((p1 ), with (p i : X x W of type (0, —oo), namely (p i = (çoi ) [w] (of (6.176) and Remark 6.17(a)).

276

Conjugations

Remark 8.23 By the above proposition, each equivalence class [c] of conjugations contains a unique representative c(yo l ) with 'pi of type {0, —04 Now we will obtain explicit formulas for c(yo l ). First, for the conjugations c(p1) of Proposition 8.11(b), we prove: Theorem 8.13 For any coupling function ço : X x W —> R, the conjugation c(ço i ) : R x —> R w , where ço i = (0)E0 (the unique coupling function of type (0, —oc), equivalent to ço), coincides with the mapping L((p) : R x —> R w defined by fL((p)( w )

inf

xEx

(f E R X ,

f (x)

V)

E

W).

(8.153)

-I

Proof By (8.35), (pi (X x W) C (0, —001, (6.176), and (8.153), we have for any f E R X and w E W, fe ("'' ) (w) = sup {ço i (x, w) + —f (x)} = xEx

=

sup {O± — f (x)) xEx wi (x,0=o

sup {0 + — f (x)} = — X Ex

ço(x,o>

-1

inf f (x) = f L( q)) (w). xEx p(-1- .70> -I

(8.154)

,

0

Corollary 8.11 If ço 1 : X x W -- > R is any coupling function of type (0, —oc), then c(go i ) = L(yo l ), that is, fC(S0 1)( w ) _ fL(4 0 1)( w ) _

inf X EX

f (x)

(f E R X , w E W). (8.155)

(pi (x,w)>- I

Proof Apply Theorem 8.13 to yo = 'pi .

D

Remark 8.24 It will be useful to look at Theorem 8.13 also in the converse direction. Namely Theorem 8.13 shows that for any coupling function yo : X x W —> R (not necessarily of type (0, —oc)), the mapping L(yo) : Ttx --> Rw, defined by (8.153), is a conjugation; this follows also directly, verifying (8.1) and (8.2). Definition 8.5 (a) For any coupling function ço :XxW—> R, we will call the mapping L((p) : R x —> R w defined by (8.153) the conjugation of type Lau associated to cp. (b) We will say that a mapping c : R x —> R w is a conjugation of type Lau if there exists a coupling function ço : X x W —> R such that c = Op). Remark 8.25

(a) In Definition 8.5(b), ço is not unique (see Corollary 8.13 below).


277

(b) If X is a set, W c -R- x and go = (Pnat (of (2.184)), then

f L(Çonat)( w ) — —

inf f (x) xEx w(x)>-1

(f

E

RX , w

E

W).

(8.156)

(c) The form (8.35) of fc (0 shows the usefulness of working with coupling functions ço = (p i of type {0, —00 rather than "of type {0, +001." Indeed, if R is "of type {0, +oo}" (e.g., if goo = yoA0 of (6.155)), then, (Po : X x W by (8.35), we have fc (v)" ) (w) = sup {(po(x , w) ± — f (x)} xex

(f

E

T?x,

WE

W),

(8.157)

and hence f°(w) = +oo whenever there exists Xo E X with (po(xo, w) = +oo and f(x0) < +oo or Xi E X with (po(x , w) = 0 and f(x 1 ) = —oc. Let us mention that one can replace the definition (8.35) of f c (0 , and hence, correspondingly, the definition (8.153) of f', by other concepts of "conjugate of f with respect to (p," for which the coupling functions of type {0, +oo} are more useful (see the Notes and Remarks). (d) For (p i of type {0, —ool, we have, for example, {x c X I (p i (x , w) > —1} = {x

E

01

X I (pi (x , w)

(w

E

W), (8.158)

and hence, for any such (p i , (8.155) coincides with f c ( So i) ( w )

inf f (x) xEx sol(r,ti)o

(f

E RV,

w

E

W).

(8.159)

Replacing here (p i by an arbitrary coupling function (p, we arrive at the mapping RA' --> R w defined by

fi':(49

)

inf

w )

x

f (x)

(f

E

WE

W),

(8.160)

soc

which is easily seen to satisfy (8.1), (8.2). This conjugation has also been used in the literature (see the Notes and Remarks). Continuing to look at Theorem 8.13 in the converse direction (see Remark 8.24), let us give:

Corollary 8.12 For any conjugation of type Lau L(co) : le --> W , there exists a unique coupling function coi of type {0, —oc} such that L((p) = c(coi) (or, equivalently, such that L(co) = OM), namely, (Pi = (q) i)kol. Proof By Theorem 8.13 and Corollary 8.11, for coi = (0) k„i we have L(co) = c((p i ) = L(cp i ). On the other hand, if (p i and *I are two coupling functions of type

278

Conjugations

{0, —oo} such that c(ço i ) = c(* 1 ), or, equivalently (by Corollary 8.11), such that L(çoi) = L(*1), then, by the uniqueness part of Theorem 8.2, we obtain ça i = * 1 . Corollary 8.13 For any coupling functions ço,* : X x W if ço * (or, equivalently, c((p) c(*)).

R, we have L((p) = L(*) if and only

Proof Let çoi = (Sot)H, Vfl = )[vii. Then, by Corollary 8.12, we have L(v) = L(*) if and only if c(ço i ) = c(*1 ), which, by the uniqueness part of Theorem 8.2, is equivalent to (p i = . But, by Proposition 6.6, the latter equality holds if and only if ço * (which, by Definition 8.4, is equivalent to c(p) c(*)).

Remark 8.26 (a) Conversely, Corollary 8.12 follows also from Proposition 6.6, Corollary 8.13 applied to V/ = (p i = (0) m , and Corollary 8.11. (b) Let us also mention the following direct proof of Corollary 8.13: If ço then, by (6.175) and (8.153), we have L (ço) = L ( p. ) . For the converse part, observe that by (8.153) applied to f = x{,} we have for any x E X and w E W, (X{x})

L(0

(w) =

inf

yEX

X{x}(y) = — Xlyex I gy,w)>-1)(X)•

(8.161)

Thus, if L ((p) = L(*), then (6.175) holds, and hence, by Remark 6.16, yo . (c) By the above, we have a one-to-one correspondence between equivalence classes of coupling functions [(p] and conjugations of type Lau L (VT), where VT is any fixed representative of [o] (in particular, we may choose * = ((p i ) m ); therefore we can introduce the notation Lagoi) = L(*),

(8.162)

with * as above. (d) For any coupling function ço : X x W —> R, the equivalence class of conjugations [c((p)] (see (8.152)) contains a unique conjugation of type Lau, namely L((p) (which coincides with L(ço i ) = c(çoi ) of Corollaries 8.12, 8.11, and Proposition 8.11(b)). Thus c((p)

L((p),

(8.163)

and for any coupling functions ço,*:Xx W —> R we have L(ço) L(*) (if and) only if L((p) = L(*). (e) Using (8.44) and the above results, one obtains their equivalent counterparts for conjugations. Thus (see Theorem 8.13) for any conjugation c : Rx —> R w the unique representative c(ço i ) of [c], provided by Proposition 8.11(a), coincides with the conjugation of type Lau L(cp,), where (p c is the coupling function (8.28) associated to the conjugation c. Also (see Corollary 8.13))for two conjugations c 1 , c2 : RA' Rw we have c i c2 if and only if L(ço e ,) = L(cpc2 ). Furthermore (see (d) above), each equivalence class [c] of conjugations contains a unique representative of type Lau,


279

namely L(go c ). Therefore one can also denote L ((pc ) by L (c) or L ([c]); then, by (8.44), we have L((p) = L(c(go)) = L([c(q))]). (f) Similarly to Section 6.3, one can introduce notations such as (o),, ((P)N, and ((POL(0, with obvious meanings, and one can observe some relations, such as (goi)[ci = ((pi)kod, and ((PO/Jo = (S01)IL(01 = (49 1)Ec(oi = ((POR,01•

We have the following characterizations of conjugations of type Lau:

Theorem 8.14 — x —> R w the following statements Let X and W be two sets. For a mapping c : R are equivalent: 10 .

c is a conjugation of type Lau (i.e., there exists cp :Xx W—> R such that c = L(go), of (8.153)).

2°.

There exists a set Qc XxW such that fc (w) = —

inf X

(x)

(f E R x , w E W).

(8.164)

EX

(x,w)Es2

3°. c is a conjugation such that the coupling function go, associated to c is of type (0, — 001 (i.e., (Pc E 1 0 , —00 } XxW ). Moreover the set Q = Q, of 2° is unique, namely we have Q = {(x w) EX x WI (X[xl) c (w) = 01.

(8.165)

Proof 1' = 2°. If c = L(go), then, by (8.153), we have 2°, with

Q = {(x, w) e X x WI cp(x , w) > —1). 2"

(8.166)

3°. If 2° holds, then, by (8.164) and (2.156), we have

RIO = —

inf

xEx c ov)EQ

f (x)

= sup {—xax, w) ± —f (x)1

(f e R x , w E W),

xEx

and hence, by Theorem 8.2, c is a conjugation, with associated coupling function (Pc = — XQ,

(8.167)

which is of type (0, —oo}. 3° = 1°. If 3° holds, then, by (8.44) and Corollary 8.11 (applied to coi = go,), we have c = c(go) = L((pc).

280

Conjugations

Finally, if 2° holds, then, by (2.156), (8.167), and (8.28), we have Q = Rx, w) EX x WI — n(x, tv) = = Rx, w) EX x WI (Pc(x, w) =

= 1(x, w) G X x 147 1(xi x )) c (w) = 0). LI

Remark 8.27 (a) Some further characterizations of conjugations of type Lau will be given in Sections 8.8 and 9.6. (b) For any conjugation c : R x —>R w , we define the subset Q, of X x W associated to c by Q, = {(x, w) EX x WI (x {x} r(w) > —1).

(8.168)

Then, by Definition 8.2(b) of goc , for any conjugation c : R x --> R w we have Qc = Rot ,

(8.169)

where Ro, is the set (6.150) with go = go( ; or, equivalently, for any coupling function : X x W —> T? we have Qc(so)=

(8.170)

Ro•

Hence for any conjugation of type Lau L(ç9) : R x --> R w we obtain, by Corollary 8.12, (8.170), (6.150), (6.176), and (8.170), Q UO = Qc(40 1) = Rol = Q S°

Q c(49)•

(8.171)

—> Te defined by (8.164) is (c) For any set Qc Xx W, the mapping c : called the conjugation of type Lou associated to Q (Theorem 8.14 shows that this is indeed a conjugation of type Lau) and it is denoted by L (S2). Thus f1,(2)( 10

=

inf

xEx (x,w)EQ

f (x)

(f E R X , w E W).

(8.172)

(d) By Theorem 8.14 and Remark 8.26, we have one-to-one correspondences between sets Q c Xx W, conjugations of type Lau L(Q) : R x —> R w , and equivalence classes [c] of conjugations c : R x --> R w . (e) Similarly to Remark 6.14(e), the above representations c = L(p) and c = L(Q), for conjugations of type Lau c : R x —> R w , , are closely related. Indeed, for any Qc X x W and cp : X x W —> R we have, by (8.172), (8.153), and (6.151), L(Q) = L(goQ), L(q)) = L(R0 ),

(8.173)

with goQ and Ro of (6.149) and (6.150), respectively. (f) Let us denote by CL = CL(T?x , R w ) the set of all conjugations of type Lau from R x into R w . One can show that CL is a complete sublattice of C = C(R x , W w ) (of Theorem 8.3), and one can prove several algebraic properties of the mapping


281

wxxw

---> L(go) E CL (see [2721, Theorem 5.3). Also for each coupling function yo : X x W ---> 7R one can define the complementary conjugation of type Lau L(go) to L(go), by fL(ÇO)( w )

inf

f (x)

XE X

(f

E

RX , w

E

W),

(8.174)

and prove various algebraic properties, as well as the commutativity of certain related diagrams (see [2721, Definition 5.1, Theorems 5.4-5.6, and their corollaries). Let us consider now f L(0 L(0/ for a conjugation of type Lau L(p) : R x --> R w . Theorem 8.15 Let X and W be two sets and cp : X x W -> T? a coupling function. Then, for the conjugation of type Lau L(go) :R x --> Tel of (8.153), we have fl-(01,(0'

sup wEw ço(x.w),--1

(x )

inf

yEX

f (y) = fq (Np A) (x)

(f

x e X),

E

(8.175)

with A ço : 2 x ---> 2 w of (6.134). Proof Let (p i = ((p i ) koi be the unique coupling function of type {0, -co }, equivalent to go. Then, by Corollary 8.12, (8.59), (p i (X x W) c to, -oo), and (8.153), we have fL((P)L((p)'

)= sup

sup

{0 (x, w) + -f " ° ' ) (w)} =

wEw

f (y) inf VEX çoi(y,w)>-1

wEW çoi (x ,w)>-1

(f

E

RX ,

x e X),

(8.176)

whence, by go - goi and Remark 6.16, we obtain the first equality of (8.175). Finally, by (6.34) (for A = A), (6.11), (6.134) (with G = {x}), and (6.158), we obtain for any f E R x and x E X, fq (A ,,A0(x) =

f (y) = sup inf wEw\A,({x}) y Ex\Av{.})

sup wEw

inf VEX

f (y). (8.177)

(p(x,w)> —I So(Y , w)> — 1

El Remark 8.28 gL(yo)' (x)

(a) By the same argument (using now (8.52)) we obtain

sup {(p 1 (x, w) ± -g(w)} = WE

g (w ) inf wEw ço(x,w)>-1

sup

g(w)

EW

W

e(Pi)(x )

(g

E

Te,

x

E

X),

(8.178)

Conjugations

282

with yo' : W x X --> R defined by (6.138). (b) Formula (8.175) shows a relation between conjugations of type Lau L(cp) : 71--? x Te l' and dualities A 2 x —> 2 w • Some further relations between conjugations c : R x —> 7?-14/ and dualities A : 2 x --> 2 w will be given in Theorem 8.16 and Sections 8.8-8.11.

Corollary 8.14 Under the assumptions of Theorem 8.15, for f statements are equivalent:

1°. 2°.

f L(q9)L(Ço)'

f(x0)

For each d

E

E

R x and xo

X the following

E

( x) ) .

R, d < f (x0 ), there exists w = wd

E

W such that

(8.179)

— 1 < Oxo, w).

sup So(x, w) xESd(f)

Proof By (8.175) and Corollary 6.4 (for A = A ço ), we have 1° if and only if for each d E R with xo Sd(f) there exists w E W such that Sd(f) C A iço ({w}), .X0 E X \ A IÇo ({w}), which, by (2.105) and (6.158), is equivalent to 2°. El

Remark 8.29 If 2' holds, then, by Remark 4.24(c), we have f

E

Q((W , ço); x0).

Under the assumptions of Proposition 6.5, the converse is also true (see Proposition 8.12 below). For conjugations of type Lau written in the form (8.164), there holds:

Theorem 8.16 Let X and W be two sets, and let Qc X x W. Then for the conjugation of type Lau c = L(S2) : R x —> R n" of (8.172), we have f LaL(Q)'

sup

=

inf

with AQ

f (y) =

)'EX

wEw (x,w)EQ

(f

E

RX , x

E

X),

(Y,w)EQ

(8.180)

> 2 w of (6.126).

Proof If c = L(Q), then, by (8.59), (8.167), (8.164), (2.156), (6.126), (6.127), (6.34) (for A = A Q ), and (6.11), we obtain

fce i (x) = sup tve(x, w) + — RIO} = sup {—xQ(x, w) +

inf

VEX

wEw

wEw

f (y)}

(y,w)EQ

sup

inf

/DEW

VEX

f (y) =

sup

inf

f (y)

wEW\AQ({x}) yEX\A'2(fw})

(x,w)EQ (Y,w)EQ

fq(A/2AQ)(x)

(f e 7-?x , x e X).

LI

283


Remark 8.30 (a) By the same argument (using now (8.52)) we obtain g L(Q)1 (x) = sup {(p c (x, w) ± —g(w)} = sup ?Dew wEw

= — inf

wEw (x, OE Q

g(w) = gL(Q/) (x)

[—x s2(x, w) + —g(w)} (g E R X

,

w

E

W),

(8.181)

where

Q' = {(w, x) E W x Xl(x, w) e Q).

(8.182)

(b) Alternatively, Theorems 8.15 and 8.16 can be deduced from each other, using (8.173) and (6.151). Corollary 8.15 Under the assumptions of Theorem 8.16, for f e R x and x o E X the following statements are equivalent:

1°. 2°.

f (x0) = f L(Q)L(Q)/( x

For each d E R, d < f (x0), there exists w = wd E W such that sd(f)

n

Ix e X i (x, w) E Q1 = 0,

(xo , w) e Q.

(8.183)

Proof By (8.180) and Corollary 6.4 (for A = AQ), we have 1° if and only if for each d E R with xo g Sd(f) there exists w E W such that S 1 (f) c A({w}), x0 E X \ A'Q ((w)), which, by (2.105) and (6.127), is equivalent to 2'. 111

Remark 8.31 (a) One can also give the following alternative proof of Corollary 8.15: By (8.173), with (PQ of (6.149), we have 1 0 if and only if f (xo) = f I-(- x 2)L(- x 2)/ (x 0 ). By Corollary 8.14, this equality holds if and only if for each d E R, d < f (xo), there exists w = wd E W such that sup ( — n(x, w))

— 1 < — n(xo, w),

(8.184)

XESd(f)

which, by (2.156), is equivalent to 2°. Conversely, one can show that Corollary 8.15 implies Corollary 8.14 (using (8.173), with Qço of (6.150)). (b) Corollaries 8.14 and 8.15 can be also deduced from the first equalities in (8.175) and (8.180), respectively, using Lemma 8.6 below (see [258], §3, [262 ] , §3 and Martfnez-Legaz [174], thm. 4.1, respectively).

Conjugations

284

8.7 Some Particular Cases 8.7a

Conjugations of Type Lau Associated to a Family M of Subsets of a Set X

Let X be a set and M c 2 x , and let (Nat : X x JA4 --> T? be the natural coupling function (2.184) (for Jm of (2.157)), i.e., ,

(Pnat(X,

X, — Xx\m

(X E

— XX\M) — — XX\m(X)

E

J.m),

(8.185)

which is of type (0, —Do} (see also the related "normal coupling function" y9A ,, : X x M —> R of (6.184), (6.187)). Hence, by (8.153), we have, denoting L(gona) by L (Jm , (Pnat) (see Remark 8. 11 (13))

f Lu -

A4 ''")

(— xx\m) = —

inf f (x) xEx -xx \m(x) , -1 (f E RA', M c M).

— inf f (X \ M)

(8.186)

Definition 8.6 Let X be a set, and let M c 2 x . We will call the mapping Wm, (Pnat) : R X —> R J-'4 of (8.186) the conjugation of type Lau associated to the family M, and we will denote it by L (M). Thus

(f

f ") (— xx\m) = — inf f (X \ M)

E RX , M E

M).

(8.187)

Theorem 8.17 Let X be a set, and let M C 2 X . Then fL(M)L(M)'

f(04)

Proof By Theorems 8.15 (for W = Jm, q) ---fL(M )LCA/0 1 (x )

sup -xx \ mE.4, -xx\ m(x)>-1 =

(8.188)

(f e R x ). (Pnat)

and 4.3, we have

inf

f (y)

yEX

- Xx\m(Y)> -1

sup inf f (X \ M) = km ) (x) MGM xEX\M

(f

E RX , X E

X).

(8.189) 0

Remark 8.32 (a) Alternatively, one can also prove Theorem 8.17 as follows. By Theorem 8.8, applied to c((p) = L(M) = L(J.A4, (Pnat) of Definition 8.6 (hence W = J.A4, So = (Pnat of (8.185)) and by (2.193) (for W , kw), (4.124), and (4.102),

8.7 Some Particular

Cases

285

we obtain L(M)L(M)' = f c(V)c()1

= L o (jm +R) =

fq(Jm-FR)

=

fq(M)•

(b) Theorem 8.17 shows one of the main uses of conjugations of type Lau, namely, to decompose any quasi-convex hull operator q(M) : R x —> RA', expressing it as a composition of conjugations L (M )' L(A.4). A similar remark is valid for the other conjugations of type Lau occurring in the sequel (e.g., (8.204) shows a decomposition of the evenly quasi-convex hull operator as L(W , 0)' L(W , 0)), but we will not mention it again.

Similarly to the case of the Minkowski-type duality (see Theorem 6.1), given a set X and ,A4 c 2 x , it is often convenient to replace the conjugation of type Lau L(M) : Tem by a s-equivalent conjugation of type Lau L (W , 0) : R x —> R w obtained from L(M) with the aid of a parametrization (W, 0) of M (see Definition 2.8), as follows: Theorem 8.18 Let X be a set, let M c 2 x , and let (W, 0) be a parametrization of M (so W is a set

and 0 is a mapping of W onto .M). Then the mapping c = L(W , 0) : R x —> R w definby f L(14") (w) = — inf f (X \ 0(w))

(f

E R X , tV E

W)

(8.190)

is a conjugation of type Lau, and we have, in the sense of Remark 8.22(a), L(W , 0) — L(M).

(8.191)

Proof Since c = L(W , 0) satisfies (8.164), with Q=1(x,w)EXxWlx. EX\ 0(w)),

(8.192)

we have L(W , 0) = L(Q) (by Theorem 8.14 and (8.172)). Finally, by Theorem 8.16 applied to Q of (8.192) and by 0(W) = M and (8.189), we obtain fL(W,O)L(W, H)'

(x )

sup WE

X EX \

inf f (X \ 0(w))

W

( w)

sup inf f (X \ M) = f L(m)L(m)' (x) A4 cm xEx\m (f ER x , x and hence, by Proposition 5.7, L(W , 0) - 3- L (M).

E

X),

(8.193) LI

286

Conjugations

Remark 8.33 (a) By Remark 6.4(a), in general, the conjugation L(W , 0) may be more convenient for applications than L(.114). (b) Similarly to Remark 6.4(b), by abuse of language, we will call L(W , 0) : of R x ---> R w of (8.190) a parametrization of the conjugation L(M) : R x ---> R (8.187). (c) By (8.167) (for c = L(W , 0)), (8.192), and (2.203), we have (PL(w,9)(x, w) =

-

(x

Xx\e(w)(x) = (pe(x, w)

E

X,

tV E

W).

(8.194)

Also, by (8.153) (for go = goo), (2.203), and (8.190), we have

(8.195)

L(goe) = L(W , 0),

and hence, by Corollary 8.11, (8.195), L(W , 0) ^5- L(0 (W)), and Theorem 8.17 (for M = 0(W)), we obtain fc(soo)c(Soor

fL((p o )L((po Y

fl,(W,O)L(W,O)'

f Lo(W))W(W)) / = fq(ft(W))

(f

E

R X ),

which, together with Corollary 8.5, yields again the main part of Theorem 4.14(a). (d) If (W, 0) is a parametrization of M, with 0 gW c Rx , and if we extend (W, 0) to (W1, 01 ) of Theorem 6.3(a), then, by (6.113), we have X \ 0 1 (0) = 0, and hence, by (8.190) (applied to (WI , 01 )), f L( ") c R w extends to f L(w1 '9' ) e defined by fL(W 1 ,0 1 )( tv )

fL(W,8)( w )

E

fL(W1,01)(0) =

inf = -oo. (8.196)

On the other hand, if 0 E C(.114) and (W x A, 0) is a parametrization of M, with OgW C Te and A C R, and if we extend (W, 0) to (W2, 02) of Theorem 6.3(c), then we have X \ 82(0 , d) = 0 if 02(0, d) = X and = X if 92(0, d) = 0, and hence, by (8.190), f L(14 G R w extends to f L(w2 '92) G R w2 defined by f

L(w2 .62 )1 _ ft.(w,o) f 1-(w-2.02) (0 , d) iw - •

-oc

if 6+2(0, d) = X,

- inf f (X)

if 92(0, d) = 0.

(8.197) --> The usefulness of the parametrized conjugations of type Lau L(W , 0) : Tel/ of (8.190) is also shown by the fact thatfor any sets X and W every conjugation of type Lau L(go) : R x -> Tel (where go : X x W --> T?) is a parametrized conjugation of type Lou (8.190) for some suitable M c 2 x and 0 : W > M = 0(W). Indeed, we have:

Theorem 8.19 Let X and W be two sets and cp : X x W --> R a coupling function. Then there exist a family .1v1 ç, C R x and a mapping Oço of W onto M ço (so (W, 0 ) is a parametrization


287

of M0 such that

(8.198)

L(yo) = L(W , 00.

Proof Let .M„, = .A4 Aw of (6.157), with Aiço of (6.158), and define O ço : W —> M ço

v maps W onto M ço and, by (8.153), (6.159), and (8.190), we obtainby(6.159)ThenO f

inf

( w )

xEX (p(x,w)>-1

f (x)

= — inf f (X \ O ço (w)) = f L(w 'ev ) (w)

(f E R X , w E W). III

Corollary 8.16 For every conjugation c(cp) : c(go)

—> le there exists a family .11/1, C 2' such that

L(CP) L(M), namely one can take .114 to be the family of Theorem 8.19.

Proof By Remark 8.26(d), we have c(p)

8.18, we have L(yo) = L(W, 0 )

L (y9). Also, by Theorems 8.19 and

L L(M).

Remark 8.34 Corollary 8.16, Theorem 8.19, and Remark 8.32(b) show that the conjugations of type Lau L (M) associated to some family M c 2x play a similar role among all conjugations c : R x --> R w to that of the Minkowski-type dualities A m among all dualities A : 2 x —> 2 w (see Remark 6.5, Theorem 6.2, and Remark 5.11(b), respectively). For closer connections between L(A.4) and Am, see (8.272) and Proposition 8.18 below. We have seen, in Remark 2.16(b), that most of the families M c 2 x considered in Sections 2.2a—h admit natural parametrizations (W, 0). Therefore, by Remark 8.33(a), it will be useful to give now explicitly the expressions of the corresponding conjugations of type Lau L(W , 9): R x —> R w of (8.190). In the next three subsections we will arrive in this way at some of the concepts of conjugation used in the theory of quasi-convex optimization.

8.7b

Quasi-Conjugation

Let X be a locally convex space, and let M be the family consisting of X, 0, and all open half-spaces (2.53) in X parametrized by W = X* x R, B(t, d) = Ix E X (t(x) < dI

((cI), d) e W). (8.199)

Then (8.190) becomes the conjugation of type Lau L(W , 9): R x --> R x* x R defined for any f e R x by f L(w,e) (4), d)

_

inf f (x) .xE.K ca(x),d

((i) E X*,

d E R),

(8.200)

Conjugations

288

which is called the quasi-conjugate of f (in the sense of Greenberg-PierskallaCrouzeix [105], [41], [43]). Remark 8.35 (a) By Remark 6.12(a) (see also Remark 8.33(d)), we obtain the same quasi-conjugate (8.200) if we start with the family Mo of all open half-spaces (2.53) in X, parametrized by (6.90), and then extend M 0 to M = Mo U {X, 0}

and extend (6.90) to a parametrization of M, by (6.122) (which yields the above parametrization (8.199) of M). A similar remark is also valid for the parametrized conjugations of type Lau considered in the subsequent sections (but we will not mention it again), where we will assume, from the beginning, that {X, 0} c M. (b) Corresponding to Remark 8.6(c), we have the following characterization of the quasi-conjugation: If X is a locally convex space, a mapping c : Rx _>„ R x* xR is the quasi-conjugation (8.200) if and only if we have (8.1), (8.2), and (x E X, D E X*, d E R).

(X{x}) c (c13 , d) = — X{y€x I (1300,11(x)

(8.201)

Indeed, it is enough to apply Theorem 8.2 and to observe that, by (2.156), — illf

yEx

X{x)(Y) = — X{yEX

(13( y )j}

(X)

E

D

X,

E

X*, d

E R).

(c) While the (Fenchel) conjugate (8.38) is useful in convex optimization, the quasi-conjugate (8.200) is used in quasi-convex optimization. Initially in [105] there was defined for each d E R the d-quasi-conjugate of f E R x by f' (D) = — inf

xEx

f (x) ± d

(c3 E

(8.202)

X*),

which differs from the right-hand side of (8.200) only by the (inessential) additive term +d, that is, fdY 0)) = fL( ") (4), d) ± d

(q) E X* , d

(8.203)

E R);

here we use the notation fdY of [257], with y standing for "Greenberg-Pierskalla." Let us compute now the biconjugate of f G -R-x for the above conjugation L(W, 9): R x ----> R w . By (8.193), (8.199), and (4.77), we obtain f L(W ,O)L(W ( x )

= sup sup

inf

dER (1).Ex* c13(x)d

f (y) = feq (x)

y EX

(f

E

,

x

E

X).

(I)(Y)ci

(8.204) Remark 8.36 (a) Corresponding to (8.202), in [105] there have been defined for each d E R the d-quasi-conjugate of g E R X* by g' (x) = —

inf

(D EX* (1)(x) ?d

g(c1) ± d

(x

E

X),

(8.205)


and the second d-quasi-conjugate of f (fdY ),Yi (x)

— inf

(1)Ex * (13(x)d

E

289

R X by

f,,Y ez13) ± d =

sup

inf

f (y)

yEX (I) (Y)?-d

(13.X* E

(I)(x)?d

(x

E

X),

(8.206) and then the "normalized second quasi-conjugate" of f

E

R X by

f" = sup (f'), dER

(8.207)

which is nothing else than f L( " 1-( "I' 6) ' = fey of (8.204). Hence for any f E kx f" is evenly quasi-convex. Also, corresponding to Corollary 8.4, we have now, R X and d E R that fdY : (X*, weak*) is evenly quasi-convex; for any f E — indeed, this follows from (8.390) below. For further results on the d-quasi-conjugates f()1 , see, for example, [105], [41], [43]. (b) Since a function f E Te is evenly quasi-convex if and only if f E C(W), with W of (4.156), where K is the set of all nondecreasing functions k : R —> T? (see Remark 4.23(c)), one can replace the above approaches (8.204), (8.207) to the study of feg as a "second conjugate," by a different approach. Indeed, MartinezLegaz [168], [170] has introduced for any set K offunctions k : R R, closed for (pointwise) supremum, the concept of the "K-conjugate" of a function f : X —> R (where X is a locally convex space) as being the mapping fR : X* —> K defined by fA(4:13) = sup {k EKIk0 44) f}, that is, f 4 (4:I))(d) =

sup k(d)

(4) E

X* , d E R);

(8.208)

kEx ko(13f

also he has introduced the K-conjugate of a mapping g : X* —> K as being the function g ° : X —> R defined by g ° = sup {g(4)) o (NO E X*), that is, g v (x) = sup g(4))(4)(x)) E X*

(g

E

K X* , x E X),

(8.209)

and has shown that for any f E kX f 4° coincides with the K-convex hull of f (for the concept of K-convex functions, see Remark 4.23(c)), i.e., h f tiv = max h K-convex h• R, then f ILv f L(W ,O)L(W ,9) ' feci or (8.204). Also for this K we have f 4 (4:13)(d) = f L( ") (4), d) of (8.200) ([170], prop. 24'). For further results on "K- conjugation," see Martfnez-Legaz ([168], [170], [174]).

Conjugations

290

8.7c Semiconjugation Let X be a locally convex space, and let .A4 be the family consisting of X, 0, and all closed half-spaces in X parametrized by

XI

W = X* x R, 0(41), d) = Ix E

d — 1}

4) (X)

(((1), d) E W). (8.211)

Then (8.190) becomes the conjugation of type Lau L(W , 0) : R x —> R w defined for any f E R X by f

,o) (0, d)

inf

f (x)

(8.212)

((i) E X* , d E R),

X EX

c1(x)>d —

I

which is called the semiconjugate of f. W X* x R

Remark 8.37 (a) Similarly to Remark 8.35(b), a mapping c : the semiconjugation (8.212) if and only if it satisfies (8.1), (8.2), and (X{x}) c ( 0 , d) = — X{)'EX

(x E

1(1)(0>d 1)(x) -

X, (I) E X*, d

c

R). (8.213)

(b) Historically first there was defined (in [260]) for each d d-semiconjugate of f E Te by f0)= — inf X

is

E

R the

f (x) + d — 1

EX

43(x)>d--1

= f "1") (4), d) ± d —1

For the above conjugation L(W , 0) : R x (using now (4.69)), that the biconjugate is fL(w , (9 )L(w , oy (x) = sup

sup

d ER

(I) EX*

(1)(x)>d

(41) E X * ).

(8.214)

R w we obtain, similarly to (8.204)

f (y)

inf

y EX

I

= f(x)

(13( y)>d-1

(f E

RX ,

x E X).

(8.215)

Remark 8.38 (a) The above parametrization of ,A4 is slightly different from the parametrization

W = X* x R, 90(43, d) = Ix

E X 10(x)

d}

(('4), d) E W) (8.216)

(extending (2.91)), which would yield the conjugation of type Lau f 1-( "") (4), d) = —

inf

xEx

f (x)

(43 E X*, d E R),

(8.217)

(13.(x)>d f L(W,O)L(W,OY and the same biconjugate f L(W,00)L(W,00)' fzi of (8.215). The reason for introducing in [260] the d-semiconjugate (8.214), equivalent (see (b) below)


291

to L (W , 0) of (8.212), instead of using L (W , 9 0) of (8.217) was that the parametrization (W, 0) of M, given by (8.211), was more convenient for some computations (e.g., x E X \ 0(0, (I) (x)) for all x E X, (i) E X*, while (W, 90) of (8.216) does not have this property). (b) Similarly to Remark 8.36(a), let us mention that in [260] there were defined for each d E R , the d-semiconjugate ed of g E Te * and the second d-semiconjugate (fcs ) sd off E R"', and then it was shown that the normalized second semiconjugate fss = supdeR (fds) -ds of f E TV( coincides with f L(W ' °)L(W,O) ' = f of (8.215). Hence for any f E f ss is lower semicontinuous and quasi-convex. Also for any f E R X and d E R, f : X* —> R is weak* lower semicontinuous and quasi-convex; indeed, this follows from (8.393) below. (c) Similarly to Remark 8.36(b), if K is the set of all nondecreasing lower semiR, then for f 4° of (8.210) we have f = fq = continuous functions k : R fL(W,O)L(W,O)' However, for this K we have (see [170], prop. 24) inf

(0)(d) =

R) ,

E X* , d E

(8.218)

rER

doup ,z1)(S,(f))

which, in general, is different from f L( ") (43 , d) and f c'si (c13) of (8.212), (8.214). Let us also mention that Crouzeix has defined for each X() E X the conjugate off at xo (see [41], p. 33), by inf

q (x o , f; (I)) =

r

(1) E X * ),

(f E

(8.219)

rER (13 (x0)sup(1)(S,(f))

for which we have, by (8.218), Pi((10)(4)(x0)) = q(xo, f;

(f E

R X , (1)

E

X

,

X0 E X).

(8.220)

8.7d Pseudoconjugation Let X be a locally convex space, and let M be the family consisting of X, 0, and all complements of closed hyperplanes in X parametrized by

W = X* x R,

((e, d) E W). (8.221)

, d) = ( x E X I (I)(x) 0 d)

Then (8.190) becomes the conjugation of type Lau L (W , 0) : for any f E R X by f"" ) (4), d) = —

inf

xEx

f (x)

E X* , d E

—> k w defined

R) ,

(8.222)

(1)(x)=d

which is called the pseudoconjugate off. Remark 8.39 (a) A mapping c : R x —> R x* x R is the pseudoconjugation (8.222) if and only if it satisfies (8.1), (8.2), and (X{x}) c (c13 , d) = — X{yEx I (1)(y)=d)(x)

(x E X,

E X*, d E

R) .

(8.223)

292

Conjugations

(b) Historically first there was defined (in [2571) for each d d-pseudoconjugate off E R X by f7(41)) = — inf f (x) ± d = f"") (4), d) ± d xEx ck(x)=d

x*).

(l) E

E

R the

(8.224)

For the above conjugation L(W , O): R x —> R w we obtain, using (4.82), that the biconjugate is f L(w,o)L(w,o)',

, = sup sup del? (DeX* (13 (x)=d

inf

f (y) =

ycX

fqca(X)

(1) (Y)= d

(f c R x , x E X).

(8.225)

Remark 8.40 (a) Similarly to Remark 8.36(a), let us mention that in [257] there were defined for each d E R, the d-pseudoconjugate grid' of g E R X* and the second d-pseudoconjugate (jyy,r, of f E R X , and it was shown that the normalized second pseudoconjugate f" = suPdE R( )(jri of f E R X coincides with f " 1")) "") ' of the first part of (8.225). Hence, by (8.225), for any f E R X , f" is evenly quasicoaffine. Also for any f e R x and d e R, f; : X* —> R is weak* evenly quasi-coaffine (i.e., all level sets Sr (f;) are intersections of complements of weak* closed hyperplanes in X*); indeed, this follows from (8.396) below. (b) Similarly to Remark 8.36(b), if K = Ri?, the set of all functions k : R then for f : X* —> K of (8.208) we have f a (0)(d) = f L(w.9) (, d) of (8.222) (see [168], prop. 4.3 or [174], prop. 2.12) and Pi° = fqca = »,(W,O)L(W,H)' of (8.25) For other results on pseudoconjugation, and connections between pseudoconjugation and quasi-conjugation, see [2571.

8.7e

Some Extensions of the Preceding Conjugations

(1) Let X and W be two sets and ço:XxW—> —R- a coupling function, and let M = S(Vw 4 x R) of (2.191) be parametrized by (2.213). Then (8.190) becomes the conjugation of type Lau L(W , Oç') : —> R w x R defined for any f E Te by f L(V ,wP) (w d)

_

inf

f (x)

(8.226)

(w E W, d E R)

xEX

so(x.tv)>d (extending (8.217)). For this conjugation of type Lau one obtains, using also (4.159), that the biconjugate is

f r,(W,(Y)L(W.o(Py kx) =

sup sup wEw

dER

ço(x,w)>d

= fq (w.,p)(x)

inf

f (y)

VEX (y ,w)> d

(f

E

Te,

XE

X).

(8.227)


Remark 8.41

293

(a) In particular, by (8.226) for d = —1 and (8.153),

f L( 4-12 ,Ow) ( w _1)

fL(ço) ( w )

(f E R X , tV E W).

(8.228)

(b) By (8.227) and Theorem 8.15 we have

f L(f;-V,0 9')L(W,6n' (x )

sup sup

inf

dER

yEX

wEW cp(x,w)>d

sup

(19 (-Y , w) >d

inf

tvEW yo(x,w)>-1

f (y)

f (y) = f LML((p) / (x )

yEX

(PO' , w)> -1

(f E R X , X E X);

(8.229)

in Proposition 8.12 below we will see that under the assumptions of Proposition 6.5, there holds equality (instead of in (8.229). (c) We have t g L(V,O W ) I g E W}

Q(w, (p) Indeed, L(W ,

: T?x

(8.230)

Te-4-1 is a duality, for which we have

C(L(W- , O (P)'L(1712 , O (P)) = If E

= If

E RX

f

fLON,ff0n)

If=

=

Q(W, ça)

(8.231)

(by (5.32) with A = L(1;12, 9) and (8.227)). Hence, by Corollary 5.6, we obtain (8.230). Now we are able to prove the results announced in Remarks 8.41(b) and 8.29: Proposition 8.12 Under the assumptions of Proposition 6.5 we have equality in (8.229). Proof By (8.227), (4.158), (6.169), (6.33) (for A = A(p) and (8.175), we have fL(7v-,099)L(6-f,o(P) (x) = fq(w,)(x) =

= fq(A,,pA)(x)

inf d= xEc0 w w Sd(f) f L(yo)L(cp)' (x)

inf

d

xeA'9,A,(S (/(f))

(f E R X , X E X).

D

Remark 8.42 (a) Under the first assumption of Proposition 6.5, namely if ço is of type (0, —oc), one can also give the following simple direct proof. If d < 0, then (p(x, w) > d if and only if ço(x, w) > —1, while if d 0, then {w E WI (p(x, w) > d} = 0 (x E X), whence the sup VCW in (8.229) is —oc, which implies that we ot .0>d have equality in (8.229).

294

Conjugations

(b) One can give similar extensions of the quasi-conjugation (8.200) and of the pseudoconjugation (8.222). For example, using the family M = x R) of (2.218), parametrized by

fv- = w

((w, d) E

x R, 5"P (w, d) = Ad((•, w))

we arrive at the conjugation of type Lau L(i 2 , B"P) : R x f E RX by fL(w ow)(w , d) = —

fi),

(8.232)

xR defined for any

(w E W, d E R),

inf f (x) xEx (p(x,w ) d

(8.233)

which extends the quasi-conjugate (8.200) of f. For this conjugation of type Lau one obtains that the biconjugate is fL(IA:7 , 9- w)L( 14-/,FP')1 (x ) = sup

sup

inf

dER wEW

yEX

(f E R X , X E X),

f(y)

(p(x,u)d (.. 0 (Y , w)?-d

which extends the evenly quasi-convex hull (8.204) of f. If go(X x W) c similarly to (4.78), we have fL( 6>,(7w )L(I'Un'( x ) = sup wEW

inf

(f E R x , x

f (y)

y EX

(8.234) R, then,

c X). (8.235)

cp(y,w)?..cp(x,w)

For any sets X, W and any coupling function ço : X x W R U {—co } , one can express the conjugation c(ço) : R x R w with the aid of the conjugation —> Tex R of (8.233), as follows: L(W ,

Proposition 8.13 For any ço : X x W

R we have

fcM(w) = sup If L(1-4-' :(-)) (w, d) ± d}

(f E R X , W E W),

(8.236)

de R

with L(1717' ,B"P) of (8.233). Proof By (8.233) and (8.35), we have

d±f

' (7w) (w, d) = d —

inf xEx

f (x) =

sup

{c/ — f (x)}

xEX

yo(x,w)d

sup

fço(x, w) — f (x)} ( fc (`P) (w)

xEX 49 (x ,w)?-d

(f E Te, W

e

W,dE

R),


and hence the inequality we have, by (8.233), ço(x, w) — f (x)

295

in (8.236). Conversely, using again that ço(X x W) c R,

ça(x , w)

sup

(—

f y)) (

yEX (p(y,w)?..(p(x,w)

= yo(x , w) —

= ço(x, w)

inf

f (y)

yE X (P(Y. 11))%49 (x,w)

f' 0) (w, ço(x, w))

sup {d

fL

(w, c1)}

dE R

(f E Te, x whence, by (8.35), we obtain the opposite inequality

E

X,

tV E

W), (8.237)

LII

in (8.236).

Remark 8.43 Proposition 8.13 may be compared with Proposition 8.2 (for c = c(0).

M

a partially ordered set, and LI C Yx, and let (2) Let X be a set, Y = (Y, C 2x be the family consisting of X, 0, and all subsets of X of the form

M = tx E X I ti(X) (where u W=

E

(8.238)

y}

11, y E Y), parametrized by x Y, 0 (u, y) =

((u, y)

E X I u(x)

E

W).

(8.239)

Then (8.190) becomes the conjugation of type Lau L(W , 9): R x —> R w defined for any f E R X by

fL 041,0 (u, y) =

inf f (x) .KEX

(f c Rx, u

E

11, y

E

Y).

(8.240)

u(x)?-y

In other words, (8.240) is nothing else than f : X x (LI x Y) —> R defined by 40 (x, (u, Y)) = — XtzEx ticz».y1(x)

0)

= f (*P ) for the coupling function

(x EX, u E U, y

E

Y).

(8.241)

Remark 8.44 (a) The quasi-conjugate (8.240) is an extension of the quasiconjugate (8.200), which is obtained by taking Y = R and 7,1 = X* C R X in (8.240). One can give similar extensions of the semiconjugation (8.217) (rather than (8.212)) and of the pseudoconjugation (8.222). (b) In the particular case where X = R , (Y, = (R" , -L), and 14 = the relation of (8.238) is < L (since L is a total order), so (8.238), (8.239), and

296

Conjugations

(8.240) become, respectively, (2.39), (6.78), and fL(W,0)(u,

y)

inf

xe u(x)?-LY

f (x)

(f E

Ten ,

u E 11(r), y E R").

(8.242) If we define (corresponding to (8.208)) f f (u)(y) =

sup k(y)

(f E

: U(R) —> K by

Ten ,

u E 14(R"), y E R"),

(8.243)

keK

kou“

where K is the family of all lexicographically nondecreasing functions k : Rn —> (i.e., such that x, y E k(y)), then, corresponding , x Ye n by g ° = sup, Eu(R„ ) (g(v) o y) (g : 14(R") —> K), that is, g v (x) =

sup g(v)(v(x))

(g E

(8.244)

K U(R") , X E R a ),

vEu(R")

then Pi ° = f L( "1-(w '" = fq ( f E KR) ([174 1 , prop. 3.13), i.e., the "biconjugate" of any f E 7iRn is the quasi-convex hull of f. Therefore (8.242), (8.244) are called ([1711, [174]) "exact quasi-convex conjugation."

8.8

Relations Between Conjugations c : R w and Dualities A : 2x —> 2 w, Where X and W Are Two Sets

Definition 8.7 Let X and W be two sets. For any conjugation c : R x —> R w we define the duality A c : 2 x —> 2 w associated to c by

6, c (G) = {w E W I (X{g})

c (W) ‹.

(G C X).

—1 (g E G)}

(8.245)

Proposition 8.14 (a) For every conjugation c : Rx —> R w we have

(8.246)

Ac = Acp„

where ço, : X x W —> R is the coupling function (8.28) associated to c and A cp, is the duality (6.134) (with ço = (pc ) associated to ço,.. (b) For every coupling function ça : X x W —> R we have

(8.247)

Accço) = Proof (a) By (8.245), (8.28), and (6.134) with ço = (pc , we have

Ac(G) = {w E W

I (Pc(g , 11) )

‹. —1 (g E

G)} =

(p(

(G)

(G c X).

8.8 Relations Between Conjugations c :

—> Te and Dualities A : 2x —> 2 w

(b) This follows from (8.246) and (8.44).

297

El

Corollary 8.17 For two conjugations ci, c2 : R x —> R w we have c l

c2 if and only if A

Proof By Definition 8.4, we have c l c2 if and only if Go, Definition 6.3 and (8.246), is equivalent to A c , = Ac2

A (2 .

'" 49(.29 which, by

Remark 8.45 (a) By Corollary 8.17 we have a one-to-one correspondence between equivalence classes [c] of conjugations c : R x —> R w and dualities A : 2 x —> 2 w . If [c] —> A, we will denote A by A 1 .1 and [c] by H A . Thus A [c] = A c , for any C1 E [c], and [c]A = fc I A c = Al. (b) By Definition 8.7, the duality AL() 2x —> 2 w associated to a conjugation of type Lau L((p) : R x ----> R w is (G

A L( )(G) = (w E W (X{ g }) LW (W) R w there holds (8.249)

AL(w) = Aço•

Proof By Remark 8.26(d), we have L(p) formula (8.247), we obtain

c(ça), whence, by Corollary 8.17 and

AL() = A ( ((p ) — A ça ; or, alternatively, by Corollary 8.12, (8.247), and ço

, we get (8.250)

AL() — AcQ01) — A991 — A ,p; or, alternatively, by (8.248), (8.161), and (6.134), we have A 1,((p)(G) = fiv E W I — =

X{ye)( ko(y,w)>-1)(g)

E W I ça(g, w)

—1

(g c G)}

—1 (g E G)} = A(G)

(G C X).

(8.251)

In the opposite direction, let us prove: Theorem 8.20 For every duality A : 2 x —> 2 w there exists a unique coupling function (p : XxW —> R of type (0, —oo} such that A = AL(),

(8.252)

298

Conjugations

namely ça = ç oA of (6.131). Hence for every duality A: 2 x —> 2 w there exists a unique conjugation of type Lau L(A) = L(ço) : Rx —> R w for which ço : X x W —> R is of type {0, —cc} and satisfies (8.252), namely (8.253)

L(A) = with ço = ÇOA of (6.131), that is, f "A ) w) = (

inf xEx wew \A({x})

f (x)

= — inf f (X \ A'({w}))

(f E

Te,

w E W).

(8.254)

Proof By Theorem 6.5, for the given duality A there exists a unique coupling function ço of type {0, —oo}, namely ço = ÇOA of (6.131), such that A = A, p , which, together with Proposition 8.15, yields (8.252). Assume now that ço, : X xW are two coupling functions of type {0, —oo) such that A L() = A = A L(*) . Then, by Proposition 8.15, we have A AL() = AL(* ) = A v„ whence, by Definition 6.3, ( i1r. Hence, since both ço and V/ are of type {0, —co}, from Proposition 6.6 we obtain ço = r which proves the first part of Theorem 8.20. Finally, observe that, by (6.131), we have çoA (x , w) > —1 if and only if w E W \ A ({x}), or, equivalently, X E X \ A'({w}). Hence, by (8.253) and (8.153) (for ço = çoA), we obtain (8.254). CI ,

Definition 8.8 For any duality A : 2 x —> 2 w the conjugation of type Lau L(A) : R x —> R w occurring in (8.254) is called the conjugation of type Lou associated to the duality A. Remark 8.46 (a) In the definition (8.254) of L (A) no coupling function ço occurs explicitly. However, in order to obtain results on L (A), it will often be convenient to use (8.253) and apply the preceding results on L(ço) to ço = A • (b) If Qc Xx W, then for A = AQ of (6.126) we have L(A) = L(S2).

(8.255)

Indeed, by (8.254) (for A = AQ of (6.126)) and by (8.172), we obtain fL(AQ)( w )

—

inf f (x) wEw\A Q ({x}) inf

xeX (x,w)en

f (x)

= f L(Q) (w)

(f E R X

,

w E W).

(c) For any coupling function ço : X x W —> R we have L(A) = L((p).

(8.256)

8.8 Relations Between Conjugations

c : R x —> R w and Dualities A : 2 x —> 2 w

299

Indeed, by (8.254) (for A = A ( of (6.134)), (6.158), and (8.153), we obtain f L(A, o )( w )

___ — inf f (X \ A lw ({w)))

=

inf

f (x) = f "GI) (w)

xEx

(f E

R X , tt) E

W).

Proposition 8.16 For two dualities A1, A2 : 2x —> 2 w we have L(A1) -= L(A2) (if and) only if Ai = 6.2. Proof If A1, A2 : 2 X --> 2 w are two dualities such that L (A 1 ) = L(A2), then, by (8.253), we have L(ç) = L(40A2 ), whence, by Corollary 8.13, Hence, since both ÇOA, and yoA2 are of type {0, —Do), we must have çoA, = ço A2 , and thus, by (6.131) and (6.7), A1 = A2. 0 Remark 8.47 (a) Alternatively, one can give the following proof of Proposition 8.16: By (8.254) applied to f = x{ x } and by (6.69), we have for any duality A : 2x --> 2 w ,

(X{,v)) L(A) (w) = — inf x(x)(X \ A 1 ({w})) = — Xx\A , (00(x) (X E

—

X, w E W).

(8.257)

Therefore, if L(Ai) = L(A2), then W \ A1 ({x}) = W \ A2({x)) (x E X), whence, by (6.7), we obtain A1 = A2. (b) For any duality A : 2 x --> 2 w , we have the following extension of formula (8.257): (G c X).

(XG) L(A) = — XW\A(G)

(8.258)

Indeed, by (8.1) (for c = L (A)) and (8.257), (XG) L(A) = (inf X{g) gEG

= — inf gEG

)L(A) _ sup (X8})L(A) = sup (—X W\A({g ) ) ) gEG

gEG

XW\A({g}) = — XW\A(G)

(G c X).

Clearly (8.258) also follows from (8.254) applied to f = xG . Note also that for 17 G of (4.40) we have — 17 A(G) ( 170 1' 66° —

(G C X).

(8.259)

Conjugations

300

Indeed, by (8.254), (4.40), and (6.11) we obtain for any G c X and w E W,

(riG) L(A) (w) = — inf riG (X \ A'({w})) —oc

if X \ A'({w}) C X \ G

= { if X \ A'({w))

gx

if G c A'({w}) }

\ G

—oc

= { if G —oo

g A 1 ({w})

if W E A(G) 1

,

= riA(G)(w)•

if w V A (G)

{

(c) By Proposition 8.16 and Remark 8.45(a), we have one-to-one correspondences between dualities A : 2x —› 2 w , conjugations of type Lau L (A) : R x ---> R w , and equivalence classes [ c] of conjugations c : R x —> R w . For some algebraic properties of these correspondences, see [272]. (d) By (8.253) and Remark 8.26(d), for any dualities A 1 , A2 : 2 X —> 2 w we have L(A i ) — L(A 2 ) (if and) only if L(A i ) = L (A 2 ) (or, equivalently, A 1 = A2). S

Hence, in particular, L(Ai) — L(A2) implies that L(A1) — L(A2) (see Remark 8.22(a)). Proposition 8.17 For any duality Ao : 2 x —> 2 w we have (8.260)

A L (A0 ) — Ao. Proof By (8.253), (8.249), and (6.143), we obtain

A L(A 0 )

—

A L (çoA0 ) = A q)A0 = A o .

El

Remark 8.48 Alternatively, one can give the following proof of Proposition 8.17: By (8.245) (for c = L(Ao)), (8.257), and (6.7), we have AL(A 0 )(G) = tw E W I (X{g)) L(A()) (W)

= (w E W I —

— 1

XW\A o ((g))(W) -,

= gEG n Aoug» = A0(G)

(g E G)) —I

(g E G)1

(G c X).

Corollary 8.18 For any duality A : 2 x ---> 2 w we have IL(A)1 = [c] A ;

(8.261)

8.8 Relations Between Conjugations c : iix —> klY and Dualities A : 2x —> 2 w

301

in other words, the equivalence class of the conjugation of type Lau L(A) coincides with the class [c],6, of Remark 8.45(a). Proof By Remark 8.26(e), each equivalence class [c] of conjugations contains a unique conjugation of type Lau. Hence, since L(A) E [L (A)], it will be enough to show that L (A) E klp = (C I A, = A). But this holds by (8.260) (for Ao = A). 0

Let us consider now

f L(A)L()' .

Theorem 8.21 For any duality A : 2x --> 2 w we have f L(A)L(A)' ( x ) ___

sup ,Ew\A({x))

inf

Y Ex\A'

(( w))

(f E R x , x E X).

f(y) = fq (Ai 60(x)

(8.262) Proof By (8.253), (8.175), and (6.131), we have f 0,0,,g((p A )' (x) _

f L(A)L(60 1 (x )

sup weW

=

sup

inf

inf

yeX

f (y)

50A(y,w)>-1

f (y)

(f E Te, X E

wEW\A({x}) YGX\6 , '((tv))

X).

Finally, the second equality in (8.262) follows from (6.34) and (6.11).

0

Remark 8.49 (a) By the same argument (using now (8.178)) we obtain gL(61)'(x) _

inf WE

g(w) = — inf g(W \ A({x})) = g L(A ') (x)

W

xeX\A'({wD

(g E WW , x E X),

(8.263)

where the last equality holds, since, by (8.254) and (5.25), we have

g

L(Ai) (x) _ i nf ew \ A"(( x ))) = — inf g(W \ A({x)))

(g E R w , x E X).

(b) Similarly to Remark 8.41(c), from Theorem 8.21 and Corollary 5.6 one obtains that for any duality A : 2 x --> 2 w , Q(A'A) = {g L(A Y I g E

R W ).

(8.264)

(c) Combining (8.262) and (6.33), we obtain the following expressions of with the aid of Sd(f ) and Ad(f ), respectively:

f L(A)L(A)'

f L(A)L(AY z x \ ( ) =

inf xeA'A(Sd(f))

d =

inf

xcA'A(A(J(f))

d

(f E —R-X , X E X). (8.265)

Conjugations

302

For other relations between conjugate functions and level sets, see also Corollaries 8.14, 8.15, and 8.20, Proposition 8.12, Theorem 8.22, and Section 8.11. Corollary 8.19 Let X and W be two sets and A : 2 x --> 2 w a duality. (a) For any set G c X we have

(rIG) L(A)L(A)/

—

(8.266)

= 17

(8.267)

where tiG is the representation function (4.40) of G. (b) If and only if A'A (0) = 0, for any set G c X we have

(8.268)

X (XG) L(A)L(Ar — “A'A(G)•

Proof (a) By (8.262) for f =

— XX\G

and Corollary 6.2(a), we have (G c X).

(— XX\G) L(A)L(s)' — (— XX \G)q(A' A) — — XX\A' A(G)

Similarly, by (8.262) for f = riG and (6.40), ( 71G) L(AMAY

—

(r/G)q(A/A) — riA'A(G)

(G c X);

or, alternatively, by (8.259) and (8.263) for g = riA(G), we obtain (G)LL(

(G c X).

= (176,(G)) L(AY = (776,(G))" /) =

(b) If A' A (0) = 0, then, by (8.262) for f = xG and Corollary 6.2(b), (XG) L(A)L(AY

— (X ...G,) q(A/A) — X(G)

(G c X).

Conversely, (8.268), (8.262), and Corollary 6.2(b) imply that A'A (0) = 0.

D

Corollary 8.20 Let X and W be two sets and A : 2 x --> 2 w a duality. Then for f the following statements are equivalent:

E

R x and xo

E

X

10. f(x0) = f L(A)L(A)' ( xo ) .

20 .

For each d E R, d < f (x 0 ) there exists w = wd E W such that Sd (f)

ç A/ ({ wl),

xo

c X \ A t ({w}).

(8.269)

Proof For .A4 c 2x of (6.29) we have, by (8.262) and (6.30), f L(A)L(A)' _ fq(A , A) =

fq(M)

(f c R x ),

(8.270)

8.8 Relations Between Conjugations c : R x --> Rw and Dualities A : 2 x —> 2 W

303

so 1' is equivalent to f E Q(.A4 ; x0 ). Hence, by Proposition 4.4, we obtain the equivalence 10 2°.

Remark 8.50 (a) Alternatively, by Theorem 8.21, we have 1 0 if and only if f E Q(A'A; x0), which, by Corollary 6.4, is equivalent to 2°. (b) Corollary 8.20 is equivalent to each of Corollaries 8.14 and 8.15 and to Proposition 4.4 (via (8.188)). More generally, note that every result on conjugations of type Lau, say, L(yo), admits equivalent counterparts in terms of L(S2) (of (8.172)), or L (A) (of (8.254)), or L (M) (of (8.187)). In the sequel we will see that often the counterparts in terms of L (A) have more concise formulations (e.g., compare Theorems 8.26, 8.27, and 8.37). Therefore sometimes (e.g., in Section 8.10 below) we will only give the results in terms of L(A). In some other cases (e.g., in Section 9.6 below) it will be more convenient to use L(S2). In connection with Remarks 5.11(b) and 8.32(b) we can prove now a closer relation between "decompositions" of M-convex hull operators co m: 2 x --> 2 x and of M-quasi-convex hull operators q(M) : R x --> R x .

Theorem 8.22 (a) Let X, W 1 and W2 be three sets, and let A : 2 x —> 2 w' , A2 : 2 X be two dualities. The following statements are equivalent:

1 °.

A1 -"" A2.

20 .

L(A1)

L(A2).

(b) Let X and W be two sets, M C 2 X , and let A : 2 x —> 2 w be a duality. The following statements are equivalent:

com 2°. q(M) = L(A)' L(A) (i.e., fq(m) = fr,(A)L(A' ) for all f E R x ). Proof (a) 1°

2°. If 1' holds, then, by Proposition 5.7, (8.262), and (6.33),

fL(6,1)L(Al)'

inf

xe6.46.1 (Sd (f))

=

inf

d=

d

xe64 ,6,2(Sd(f))

" I., ( A2 ) 1,( 6, 2Y

(f E R X , x E X).

2° = 1°. If 2° holds, then, by (8.267),

7.746,i(G)

_

_

070 L(A2)L(A2y

=

q6,A2(G)

(G E 2 X ),

whence, by (4.40), we obtain A A1 (G) = A'2 A2(G) (G E 2x ). (b) This follows from part (a) (for W1 = M, W2 = W, A1 = AM, A2 = A), using also (6.27) and (8.275) below.

304

Conjugations

8.9 8.9a


The Conjugation of Type Lau Associated to a Minkowski-Type Duality

Let X be a set and M c 2 x , and let Am : 2 x —> 2-m be the Minkowski-type duality (6.25). Then, by (8.254) (with W = M) and (6.26), we obtain for the conjugation of type Lau L(Am) : R x --> Rm associated to the Minkowski-type duality Am, f" A (4) (M) = — inf f (X \ M)

(M E M).

(8.271)

f L(&4) (M) = f L(M) (— Xx\m)

(M E M),

(8.272)

Consequently

where L(M) : R x —> R im is the conjugation of type Lau (8.187) associated to the family M. Proposition 8.18 Let X be a set, and let M c 2 x . Then

L(Am) '- L(M).

(8.273)

q(A'm Am) = q(A/1).

(8.274)

Proof By (6.28), there holds

Hence, by (8.262) (for A = A m ), (8.274), and (8.188), we obtain fL(Am)L(Am)'

.t— Jci(A,4 A M

) = k m) =

fL(M)L(M) ,

(f E R x ).

(8.275) D

8.9b

Conjugations of Type Lau Associated to Parametrized Minkowski-Type Dualities

Proposition 8.19 Let X be a set and M c 2x , and let (W, 0) a parametrization of M (so W is a set and 0 is a mapping of W onto M). Then for the parametrization A 9 : 2x ---> 2w of the Minkowski-type duality Am, defined by (6.44), we have

L(A 9 ) = L(W, 9) -- L(M) -- L(A M ),

(8.276)

with L(W , 0) of (8.190). Consequently fL(A 0 )L(6, 0 )/

fq(A ,0,6,0) _ fq(M)

(f c R x ).

(8.277)

8.9 Some Particular

Cases

305

Proof By (8.254), (6.45), and (8.190), we have f (A (' ) (w) = — inf f (X \ _ f

,0) (w)

({w})) = — inf f (X \ (w))

(f

R - , w E W),

(8.278)

which, together with (8.191) and Proposition 8.18, proves (8.276). Finally, by (8.262), (8.276), and (8.275), we obtain (8.277). Remark 8.51 (a) By Proposition 8.19, the various parametrized conjugations of type Lau L(W , 0) : R x —> R w , , given in Section 8.7, can be also expressed as conjugations of type Lau L (A 9 ) associated to the parametrized Minkowski-type duality A o : 2 x ----> 2 w of (6.44). Also in these cases we obtain decompositions, as L (AO' L (A0), of some quasi-convex hull operators q (A4 ) of Section 4.2 and q(Ait9 A0) of (6.34), with dualities Ao of Section 6.2b. Indeed, if X is a locally convex space and W = X* x R, then, by (8.276) and (8.254), the Greenberg-Pie rskalla-Crouzeix quasi-conjugate (8.200) of f E R X can be also expressed as fL(w ,9 ) (.4 ), d) = f L(A " ) (4), d)

inf

X (41),d)E(X* x R)\A 0 ((x))

f (x)

(4) E X* , d E R),

(8.279)

VE

where, by (6.44) and (8.199), A 0 (G) = {(4), d) E X* x RIG CO (4), d)}

= {((I), d) E X * X R I (13(g) < d (g E G))

(G c X).

(8.280)

Note that A o of (8.280) is "essentially" the duality (6.91), in the sense that it is the duality obtained from (6.91) by adjoining (I) = 0 E X* to X* \ {0} as in Remark 6.12(a) (see also Remark 8.35(a)). Also, by (8.204) and (8.276), feq = f L(o)L(0)' Similarly, with the same X and W, the semiconjugate (8.212) of f E R X can be ) of (8.254), where A o : 2 x —> 2x* x R is essentialy the duality expressed as f (6.82) (with d replaced by d — 1), and, by (8.215) and (8.276), we have fEi- = f L(A)L(A0)' Also, the pseudoconjugate (8.222) of f E R X can be expressed as f "") of (8.254), where A o : 2 x ----> 2x * x R is essentially the duality (6.95), and, by (8.225) and (8.276), fq,,, = f L( ""A0)1 . Furthermore f '°) of (8.226) (where W x R) can be expressed as f' of (8.254), with A o , : 2x _÷ 2 w x R of (6.107), and fy (w) = f L(A "w)L(A°)/ . Similarly f L(11-7 :-(9-) of (8.233) coincides with f L(A) of (8.254), where 4, is the duality (6.109). Also, f "") of (8.242) can be expressed as f() of (8.254), where A o : 2x --> 2u(R " ) x R " is the duality (6.79) (since L is a total order on Ra), and we have fq = (b) One can obtain other conjugations of type Lau by considering directly L(A 0 ) Tec Rw of (8.254) for various parametrized Minkowski-type dualities Ao : 2x --> 2 w , such as those of Section 6.2b. In all cases f L(A " ) "" )/ and Q(A/0 AO can be computed from (8.277) and the results of Chapter 6, 4, or 2 on fq(64, A„) or fq(M) —›

Conjugations

306

or C(A4), respectively. Indeed, if X = R", W = 1.1(R"), and A o : the duality (6.81), then (8.254) becomes f L(A 0 ) ( u )

inf X

E Rn

f (x)

(f

u

E T?R" ,

E

(R'1 )).

> 2" (R " ) is

(8.281)

(X) L

By (8.277), (4.15), (4.2), and the remark made at the end of Section 2.2a, part (2), we have f = f L"A")' if and only if each level set Sd(f ) (d E R) is either X or a convex cone with vertex 0 g Sd (f). Furthermore, if X is a locally convex space, W = X*, and A o : 2 x —> 2 x* is the duality (6.83) (modulo the adjoining of (1). = 0 E X*), then (8.254) becomes fL(6,0) (1) ) _ _ i nf

f

xEx (1)(x)> 1

(x)

(f

E

R X , 11) E X*).

(8.282)

f L(Ao)L(Ao)' By (8.277) and a remark of Section 2.2b, part (2), we have f if and only if each Sd( f ) (d E R) is a closed convex set containing 0, or, equivalently, f is quasi-convex, lower semicontinuous, and f (0) = —oc (by (2.108)). If Ao : 2 x —> 2x* is the duality (6.85) (modulo the adjoining of (13. = 0 E X*), then (8.254) becomes

f L(A,,) (0) _ _ i nf xEx

f (x)

(f

E

R X , 14) E X*).

(8.283)

cp(x )>0 By (8.277) and a remark of Section 2.2b, part (3), we have f = f L(4OL(A0)' if and only if each Sd (f ) (d G R) is a closed convex cone with vertex 0, or, equivalently (see the Notes and Remarks), f is homogeneous (that is f (Xx) = X f (x) for all X E X,Xc R), quasi-convex, lower semicontinuous, and f (0) = —oc). If A 9 : 2 X —± 2 X* is the duality (6.89), then (8.254) becomes fL(

0

) (t) )

inf

xE X (I)(x)< 1

f (x)

(f

, (i) E X * ).

E

One can show (see the Notes and Remarks) that f = f L' ( quasi-convex, lower semicontinuous, and f (Xx)

f (x)

(x c X, X

" )L(r " )'

(8.284)

if and only if f is

1).

(8.285)

2 x* is the duality (6.92) (modulo the adjoining of (13 = 0 c X*), If A o : 2 x then (8.254) becomes f

(

)

)

inf xEx

f (x)

(f

E

RX ,

sCD E X * ).

(8.286)

By (8.277) and a remark of Section 2.2c, part (2), we have f = f)0)' if and only if each Sd(f) (d E R) is an evenly convex subset of X, containing 0, or, equivalently, f is evenly quasi-convex and f (0) = —oc.


307

If Ao : 2 x —> 2 x* x R is the duality (6.93) (modulo the adjoining of (13. = 0 E X*), then (8.254) becomes f L(A 0 ) op d)

inf

f (x)

x EX

(f E R x (13 E X* , d E R).

(8.287)

cl)(x)$d

f L(6,0)L(.Ao)' if and only if By (8.277) and a remark of Section 2.2d, part (1), f each Sd (f) (d E R) is a closed affine subset of X; such a function is, for example, f = — Xx\G + d, where G is a closed affine subset of X and d E R (by (2.161) and (2.154)). If Ao : 2 x —> 2x* is the duality (6.94) (modulo the adjoining of (I) = 0 c X*), then (8.254) becomes

f L ( A )) (1) ) =

nE xf

(f E R X , 11) E X * ).

f (x)

(8.288)

(I) (x ) 00

By (8.277) and a remark of Section 2.2d, part (2), f = f L(A0)L(A)t if and only if each Sci (f ) (d E R) is a closed linear subspace of X; such a function is, for example, d, where G is a closed linear subspace of X and d E R (by (2.161) f = —XX\G and (2.154)). _ x R , is the duality (6.96), then (8.254) If X is a metric space and A o 2x ± becomes f

(AO)

(y d)

inf

xeX p(y,x)>d

f (y)

(y E X, d > 0).

(8.289)

By (8.277) and a remark of Section 2.2f, we have f = f Lo,,g(Ao)' if and only if each Sd( f) (d E R) is spherically convex; in particular, if X is a normed linear space having Mazur's intersection property, then f = f L(AoL(Ao)' if and only if each Sd (f ) (c1 E R) is a bounded closed convex set, or, equivalently, f is lower semicontinuous, quasi-convex, and with bounded level sets Sd(f) (d E R). > 2 x is the duality (6.97), then (8.254) If X is a poset, W = X and A o : becomes f L(6, )) (y)

inf

xEx

f (x) = — inf f (x) xEX Y

yEX\A 0 ({x})

(y E X),

(8.290)

whence f

)L(A°)' = f< of (4.88), the nondecreasing hull of f. Hence we have if and only if f is nondecreasing (or, equivalently, each Sd( ), d E R, is an order ideal in X). If A o : 2x —> 2 x is the duality (6.99), then (8.254) becomes f

f L(A0)L(A0)1

fL(A0) (y ) _ _ inf f(x)

x_r E<Xy

(y E X) ,

(8.291)

and, symmetrically to the above, f " A " ) ""' = f>., the "nonincreasing hull" of f.

308

Conjugations

If A o : 2x _>. 2xx(zDu({z}xx) is the duality (6.100), then (8.254) becomes (since b} for each x} U {(z, b) E {z} x Xix W \ A9({xp = Va, z) E X x {z} I a X E X)

Xy % f (x)

f L() (Y, z)

f

(A

)

yl) =

f (x ) xEx

le , (y,

X {Z)),

(8.292)

(f E R X , (Z, y') E (z) X X).

(8.293)

(f E

z) E

X

By (8.277) and Proposition 2.5(a), we have f = f l-(° () )1-( "/ if and only if each x x2, d E Sd(f) (d E R) is order convex (i.e., the relations x 1 , x2, x E X, x 1 R, and f (xi), f (x2) d imply that f (x) d). Similar remarks can be made for f' with the dualities A o of (6.102)—(6.106). For example, for A o of (6.102) (resp. (6.103)), by (8.277) and Proposition 2.7 we have f = f l-(° " ) "°" )/ if and only if each Sd(f) (d E R) is of the form {x E XIx a f,d} (resp. {x E X taf.d x}) for some a fd E X. Also for A o of (6.102) (resp. (6.103)), f _ f L(6,0)L(A0)' if and only if for every family {x i },,/ c X we have f (supiEi xi ) = El f (x,)); this has been shown by Volte ([302 1, = suP,,/ f (xi) (resP. f (infi El thm. 11.2.1).

8.10

The Conjugate of Type Lau of max {J , —h}, Where f, —h E R x

In the present section we will assume that X and W are two sets and A : 2 x --> 2 w a duality, satisfying the following condition, corresponding to (8.141): There exists is a binary operation on W, say, A:Wx W --> W, such that X \ 6. / ({WI A W2}) = (X \

Yow,D) n

(X \ A'({w2}))

(W1, w2 E

(8.294) with A' : 2 w --> 2x of (6.10); or, equivalently, Y({wi

A W21) = A l ({W1)) U A / ({W21)

(W1, W2 E W).

(8.295)

Condition (8.295) is satisfied, for example, when W = (W, 4) is a complete lattice, A = inf (with respect to =--, A' ({w }) is an inf-homomorphism from (W, -,, R w is the conjugation (8.254), with the aid of f and h") . Remark 8.52 Alternatively, one could also use any other equivalent form of conjugation of type Lau, such as, L(yo) of (8.153) or L(Q) of (8.172), but L (A) of (8.254) has the advantage that it satisfies the concise relation (8.267), which will be used in the sequel (the corresponding formulas for (rIG) L W L( `PY , (rIG)L(Q)L(Ç2Y

8.10 The Conjugate of Type Lau of max

(f, —h),

Where f,—h E R"

are longer). Note that in the case of L((p) (where yo : X x W L(yo) = L(A) (by Remark 8.46(c)), and condition (8.294) for A = to E X ça(x , w1 A W2) >

—1} =

E X

I min {(p(x,

309

R) we have A 99 is equivalent

Ox, w2)) > — 1); (8.296)

indeed, by (6.158) we have X\

A W2) =

(X \ No (Iw i D) n (X \ A({w2}))

—1),

E X I (p(x , w 1 A W2) >

= tx E X IÇO(X, WI) > —11

n

tx E X I (P(X, W2) > -11

= tx E X I

min {(,o(x, wi), 40 (x, w2)} > —11.

Clearly (8.296) is satisfied when W), (8.297) which holds, for example, when W c Te, = çonat, and A = min. Similarly in the case of L(Q) (where Qc Xx W) we have L(Q) = L(A 2 ) (by Remark 8.46(b)), and condition (8.294) for A = AQ is equivalent to (P(X,

WI

A W2) = min Iço(x, WI), 40(x, w2))

E X I (X,

(X E X,

WI A W2) E Q1 = tx E X I (X, WI) E Q}

WI, W2 E

n Ix E x (x, (WI, W2

E

W2) E

W);

Q)

(8.298)

indeed, this follows from (6.127) (with P = (w)). The results of this section on (max { f, —h }) L(A) will parallel those of Section 8.5 on (f —h)*. Thus, while condition (8.294) corresponds to (8.141), the following result corresponds to Lemma 8.4: Lemma 8.5 Let X and W be two sets and A : 2x —* 2 w a duality, and let A:Wx W --> W satisfy (8.294). Then for any f, h E 'Fe( and w, y E W we have

(max {f, —h})" °) (w) = — inf max {max { f (x), rIxw({w})(x)), — h(x)}, xeX

(8.299)

(max { f, nxvvowol) L(A) (Y) = f L(A) (W A V),

(8.300)

where ria is the representation function (4.40) of G c X. Proof By (8.254) and (4.40), we have for any f, h E

re and w

(max ( f, —hp L(A) (w) = — inf max If (x), —h(x)j, xex\ A'aw))

E

W, (8.301)

310

Conjugations

f (x) max f (x), x\A , {

if x

E

X \ .6.' ({w}), (8.302)

(x)) =

I +oo if x

E

whence (8.299) follows. Also, by (8.254), (8.302), and (8.294), we have for any f, h E R X and w, y E W, (max If,

\A , ((.1))) L(A) (v) =

inf

max If (x), 77X\A , ({0)(x))

x Ex \ A , ({0)

inf

xea\ A'((w)))(1(X \ A'av)))

inf xeX\A'

(J WA1)))

f (x)

f (x) = f L(A)(W

A y).

Remark 8.53 (a) Formula (8.299) reduces the study of (max (f, —hp L(A) to the study of the infimum on X of the maximum of two functions, and formula (8.300) expresses (max (f, 71,X\CV({w})D L(A) for any w E W, with the aid of f L(A) . Therefore we will first give some duality results for the optimization problem occurring in the right-hand side of (8.299) (even without assuming (8.294)), and then we will combine them with Lemma 8.5 to obtain the desired results on (max { f, —h }) L(A) . (b) If there exists an element +00 E W such that A'({d-oo}) == 0,

(8.303)

then formulas (8.299) (for w = +oo), (8.303), and (4.40) (for G = X) yield (max {f, —h}) L(A) (+Do) = — inf max (max If (x), 17x \A' ({+00))(x)1, — h(x)) xex

= — inf max { f (x), —h(x)}.

(8.304)

xex

Hence in this case one can also go in the converse direction; that is, applying the results on (max { f, —h}) L(A) (w) to w = +oo, one can obtain results on infx E X max If (x), —h(x)}. Furthermore, by (8.294) and (8.303), we have

X \ A'({+oo A y}) = (X \ A'({±oo})) n (X \ AZ({y})) = X \ A l ({y})

(1)

E W),

whence, by (8.254), fL(A)( ±co A u) =

inf f (X \ .6:({+co A v))) inf f (X \ A'({v})) = f L(A) (V)

(y E W),

(8.305)

which is useful, for instance, when applying (8.320) or (8.322) below for w = +00. Note also that if (8.303) holds, then, by (6.7), we have Ai( W) = nwEw Ai ({ w } ) = 0, that is, (6.38), or, equivalently, 0 E C(A'A) (see Remark 6.3(a)).

8.10 The Conjugate of Type Lau of max (f, —h), Where f,—h

E

RX

311

Corresponding to Proposition 8.7, we have now: Proposition 8.20 Let X and W be two sets and A : 2x ---> 2"7 a duality. Then

inf max { f (x), —h'' (x)} =

inf max {h

xeX

oo, fL(A)(10}

wEW

(f, h

(8.306)

Ttx. ).

E

Proof By (8.263) (with g = hL(A) ), (6.11), and (8.254), we have inf max { f (X), — h L(A)L(AY GO} xEx inf xEx

inf

weW\A({x})

= inf

inf

weW xeX\A'({w})

=

inf max If (x), inf h L(A) (w)) xEx wew\A({x})

max { f (x), h L(A) (w)} max If (x), h L(A) (w))

= inf max {11'(w),

inf.

wEw

= inf max {h L(A) (w), —f L(A) wEW

f (x))

(01

(f, h

E

El

R X ).

Corollary 8.21 Let X and W be two sets and A : 2 x ----›- 2 w a duality. Then

inf max If (x), — 11 1-(A)L(A)' (X)} xEx

—

1, ( inf max If L(A) L(Ar (x), —n.(A)L(AY xEx

(f, h

E

R X ).

x ))

(8.307)

Proof Applying Proposition 8.20 to f replaced by fL(A)L(A)' and using the fact that (f L(A)L(A) ') L(A) = f L(A) , we obtain inf max If L(A)L(A) '(x), —h L(A)L(A Y(x)} = inf max {h L(A) (w), —f L(A)

xeX

weW

(f, h

E

R x. ),

(W))

(8.308)


0

Corresponding to Proposition 8.8, we have now: Proposition 8.21 Let X and W be two sets and A : 2x ---> 2 w a duality. For a function h following statements are equivalent: 10 .

h = h L(A M A Y .

E

R x the

312

Conjugations

2°.

We have

inf max { f (x), —h(x)} = inf max { f (x),

xeX

_hL(A)L(A)' ( x ))

(f

xEX

E

R X ). (8.309)

3°.

We have inf max { f (x), —h(x)} =

xEX

inf max th LUX)

(w) ,

f

( w ))

wEW

(f G R x ). 4°.

We have inf max { f (x), —h(x)} = inf max

xEX

L(A)L(AY

(X), — h L(A)L(AY (X)}

xEX

(f 5°.

There exists a function (ph : R x

There exists a function ilt h :

E

R x ).

(8.311)

> R such that

—

inf max { f (x), —h(x)} = (p h ( f L(A)L(A)) _vex 6°.

(8.310)

(f

E

R x. ).

(8.312)

—> R such that

inf max {f (x), — h(x)} = 1Ifh(f L(A) )

(f

xeX

E

R x ).

(8.313)

7°. We have inf max { f (x), —h(x)} = inf max If L (AWAY (x), —h(x)} xEx xEx (f ER x. ).

(8.314)

Proof Similarly to Proposition 8.8, each of the implications 1° = . . . = 7° is either obvious or an immediate consequence of Proposition 8.20 or (8.308). 7' = 1°. Let f = risd (h), where d E R. Then, by (4.40), the left-hand side of 7° is inf max Nsd (h)(x), —h(x)} =

xEX

inf {—h (x)} = —

xeSd(h)

sup h(x), xeSd(h)

and, by (8.267) and (4.40), the right-hand side of 7' is inf max {(77,5(, (h ) )"°)L(A Y (x), —h(x)} =

xEX

inf max1,71 A, A(Sd(h))(X),

xeX

inf

xEA'A(Sd (h))

—

{—h (x)}

sup xEA'A(S(j(h))

h(x).

— h(x))

8.10 The Conjugate of Type Lau of max V, —h}, Where f, —h c R x

313

Therefore, if 7° holds, then sup

h(x) = sup h(x)

xeA' A(Sd(h))

d,

x€S,i(h)

whence h(x) d for all x E A'A(S(j(h)), that is, A'A(Sd(h)) C Sd(h), which, since A'A is a hull operator, yields A'A(Sd (h)) = S 1 (h) for any d c R. Hence, by (8.265), =

inf

xeA' A(Sd(h))

d =

(x

inf d = h(x)

x€S,i(h)

E

X).

hL(A) , the implication 1° Remark 8.54 Since h L(A)L(AY L(A) Proposition 8.21 is equivalent to Proposition 8.20.

El 3' of

Corollary 8.22 Let X and W be two sets and A : 2 x —> 2 w a duality. (a) For a function h E Te we have h E Q(A' A) if and only if

sup h(G) = sup h(A' z\(G))

(G C X).

(8.315)

(b) For a subset G of X the following statements are equivalent: 1°. 2°.

G E C(A'A).

We have inf f (X \ G) = inf f (X \ A'A(G))

3".

E

R X ).

(8.316)

R X ).

(8.317)

We have inf f (X \ G) = inf fq() (X \ G)

4".

(f

(f

E

We have inf f (X \ G) = inf fq(A , A) (X \ A'A(G))

(f

E

R x ).

(8.318)

Proof (a) If h E Q(A'A), then, by Theorem 8.21, we have h = hy (A , A) = h L(A)L(A Y whence, by Proposition 8.21, implication 1" = 7° (applied to f = r)G of (4.40)) and by (8.267), we obtain for any G C X, sup h(G) = — inf (—h )(G) = — inf max {nG(x), — h(x)) xEx — inf max {776, , A(G)(x), —h(x)} = sup h(A'A(G)). (8.319) xEx

Conversely, if (8.315) holds, then, by (8.315) for G = Sd (h), and part of the above proof of Proposition 8.21, implication 7° = 1°, we obtain h E Q (A' A).

314

Conjugations

(b) 1° = 2 0 . If 1° holds, that is, if G = A'A(G), then, by (4.40), we have inf f (X \ G) = inf max If (x), ',Ex

-- 71G(x))

= inf max If (x), — rIA'A(G)(x)) xex = inf f (X \ A'A(G))

(f

E R x ).

2° = 1°. If 2° holds, then, by (4.40) and (8.267), we have inf max If (x), —1 7G(x)) = inf f (X \ G) = inf f (X \ A'A(G)) xEX xEX

xeX

= inf max If (x),

— 11A'A(G)(x)}

= inf max If (x),

— (17G) L(A)L(AY }

xEx

xEx

(f

E

RA').

Hence, by Proposition 8.21, implication 2° = 1°, and (8.267), we obtain IIG = ( 71G) L(A)LGAY = 71A' A(G), and thus, by (4.40), G = A'A(G). 1° 3° and 1° 4°. The proofs are similar, using now (8.262), (8.267), and Proposition 8.21, equivalences 1° 7' and I° 4', respectively (with h 0

Let us pass now to the study of (max If, —hp L(A) . We have the following result corresponding to Proposition 8.9:

Proposition 8.22 Let X and W be two sets and /\ : 2'— 2 14' a duality, and let A:Wx W —* W satisfy (8.294). Then for any f, h E Te and W E W we have (max f f, _ h L(A)L(60) ) L(A) (w ) = sup min {f L(A) (w vEw

A 0,

—h L(A) (v)}.

(8.320)

Proof By Lemma 8.5 and Proposition 8.20, we have (max (f, _hLo)L(AyDL(A)(w) = — inf max (max If (x), 71x \ A' qw))(x)} ,

—h L

(A)L(Ar (x))

xEX

= — inf

vEW

max {h L(A) (v), —(max If, 17X\A'({w}))) /,(A) ( 1) )}

= — inf max fh L(A) (v), —f L(A) (w

A 1.)))

VEW

= sup min { f L(A) (w

A U),

—h L(A) (V)}

vEW

(f, h

E

T?X , WE W). 0

8.10 The Conjugate of Type Lau of max (f, —h), Where f, —h E R x

315

Corollary 8.23 Under the assumptions of Proposition 8.22, we have (max If,

= (max

{ f L(A)L(A)'

(f, h E R X ).

(8.321)

Proof Applying Proposition 8.22 to f replaced by f L(A)L(A)' and using the fact that f L(A)L(AY L(A) _ f L(A) , we obtain (max if L(A)L(A)' , _hl.(A)L(A)))1,(64) (W)= sup min { f L(A) (W VE

A U),

—h L(A) (V)}

W

(f, h E R X ),

(8.322)


D

Finally, we have the following result corresponding to Theorem 8.12:

Theorem 8.23 Let X and W be two sets, A : 2 x —> 2 w a duality satisying (6.38), and A : W x W —> W a binary operation satisfying (8.294). Then for a function h e Te the following statements are equivalent:

1 0. h _ 2". 30.

hL(A)L(AY .

(max If, —hD L(A) = (max { f, _ h L(A)L(Ay DL(A) We have

( max { f, — h}) L(A) ( w) = sup min (f L(A) (w

(f E RX ).

A y),

—h L(A) (V))

vEW

(f E R x , 1.1) E W). (8.323)

4". 5'.

(max (f, —hD L(A) = (max if L(A)L(A)' , _fri L(A)L(A)' DL(A) ( f There exists a mapping ah : Te —> Tel such that

( max (f, —hD L(A) 6°.

=

There exists a mapping

a h ( f L(A)L(A))

I3h : R W --->

(max If, —hD L(A) _ max If L(A)L(AY , —h)L(A)

Proof Each of the implications 1 0 = . . .

R X ).

(8.324)

(f E R X ).

R w such that

( max (f, —hi) L(A) _ /3h(f) 7°.

E

(8.325)

(f E R X ). (f E

R x ).

7° is either obvious or an immediate consequence of Proposition 8.22 or (8.322), even without assuming (6.38).

316

Conjugations

70 = 1 0 . If 7' holds, then, by (8.254), we have inf

xExwv({10)

max If (x), —h(x)) =

inf

xexvv((w))

max IfL(A)L(Ar (x), —h(x)) (f E R X , w E W). (8.326)

But, by (6.38) and (6.7) (for A'), there holds X = X \ A l (W) = X \

n

A'({w}) = U (X \ A'({w})). wE w wE w

(8.327)

Hence, taking inf we w in both sides of (8.326), we obtain (8.314), whence, by Proposition 8.21, implication 7 0 = 1°, h = h L(A)L(AY (a) The main part of Theorem 8.23 is the implication 1 0 30, which holds even without assuming (6.38). (b) By Theorem 8.21, one can replace in the above results f L(A)L(A)' and h L(A)L(A) ' by fq(A, A) and h q(A, A) , respectively. For example, formula (8.307) can be written in the equivalent form Remark 8.55

inf max { f (x), —h q(A , A) (x)) = inf max { fq(A'A) (X),

xEX

xEX

- hq(A'A) (X)}

(f E R x ).

(8.328)

Also condition h = h L(A)L(AY in 1' of Proposition 8.21 and Theorem 8.23 is equivalent to h E Q(A'A). Thus the equivalence 1' . 7' of Proposition 8.21 means that we have h E Q(A' A) if and only if inf max { f (x), —h(x)}

inf max { fq(A'A)(x), — h(x))

xEx

xEx

(f E R x ).

(8.329)

Since for any M c 2' the duality Am : 2 x —> 21" of (6.25) satisfies (6.28) and (6.27), applying the above results (and Remark 8.55(b)) to A = A m , one can obtain some results on Q(M) = Q(A'm Am ) and C(M) = C(A'.A4 Am) (and conversely, these results on Q(M) and C(M) imply again the preceding results on Q(A' A) and C(A' A), where A : 2 x —> 2 w is a duality, using the standard generating class (6.29), but we will not pursue this converse direction). Note that, by (6.26), condition (8.295) for W = M C 2 x and A = Am becomes MI A M2 =

U M2

(M1, M2 E M),

(8.330)

that is, A = U; also, by (6.26), the condition of Remark 8.53(b) becomes 0 --=A({H-oo}) = +cc E W = M, whence 0 E M.

(8.331)

8.10 The Conjugate of Type Lau of max {f, —h}, Where f, —h E R x

317

and conversely, if (8.331) holds, then taking +Do = 0 E M = W, we obtain A lm ({±oo}) = 0. Let us mention now some results on X and A4 c 2' obtained in the above way (i.e., taking W = M c 2x , A = Am in the preceding results) in which Am does not occur explicitly. From Corollary 8.21, Proposition 8.21, and Corollary 8.22 we obtain, respectively:

Corollary 8.24 Let X be a set, and let M c 2 x . Then

inf max {f (x), —17, 1 04 ) (x)) = inf max { fq(m)(x), —h y (m)(x)}

xEx

xEx

(f, h E R X ).

(8.332)

Proposition 8.23 Let X be a set, and let M C 2 x . For a function h equivalent:

F. 20 .

E

R x the following statements are

h E Q(M). We have

inf max {f(x), —h(x)} = inf max If (x), —12, 104) (x))

xEx

xEx

(f E R X ).

30.

(8.333)

We have

inf max {f (x), —h(x)) = inf max { fq(m ) (x), —17, 1 04 ) (x))

xEx

xEx

(f E R X ).

40 .

(8.334)

We have

inf max { f (x), —h(x)) = inf max {fq(m ) (x), —h(x))

xEx

xex

(f E R X ).

(8.335)

Corollary 8.25 Let X be a set, and let M C 2x . (a) For a function h E T?" we have h

E

Q(M) if and only if

sup h(G) = sup h(comG)

(G

C

X).

(b) For a subset G of X, the following statements are equivalent:

(8.336)

318

Conjugations

P.

G E C(M).

2°.

We have inf f (X \ G) = inf f (X \ com G)

3°.

(f E R x ).

We have inf f (X \ G) =- inf fq(m ) (X \ G)

4°.

(8.337)

(f E R X ).

(8.338)

We have (f E R X ) •

inf f (X \ G) = inf fq(m ) (X \ comG)

(8.339)

Also, by (8.271) and (8.330), the implication 1' = 3' of Theorem 8.23 yields (for W = M c 2 x. and A = Am) that if h E Q(M), then — inf max { f (x), —h(x)) =- (max { f, —h }) L( " ) (M) xEx\m

sup min { —

=

inf

f (x),

x eX \ (MU/171 )

flieM

inf h(x)) xeX\fi

(f E R X , M E ./14).

8.11

(8.340)

Conjugate Functions and Level Sets

Due to the importance of level sets in the theory of quasi-convexity, we will show now that the conjugates (and hence the biconjugates too) of a function f E R X can be also expressed with the aid of the level sets S,( f) or Ad( f) of f. Also we will compute the level sets of conjugate and biconjugate functions. Theorem 8.24

Let X and W be two sets and yo : X x W --> Ra coupling function. Then (f E R X , w E W).

f(w) = sup { sup cp(x, w) — d} dER xEs d (f) Proof Let f E Te and w —oo), we have

E

W. Since X0 E Spxo)(f) (X0 E dom f, f(x 0) >

sup ( sup yo(x, w) — d) deR xeSd(f)

(8.341)

sup

{yo(x , w) ± — f (x))

xeS H , 0) (f)

ço(xo, w) ± — f (x0) (X0 E

dom f, f(x0) > —cc).

(8.342)

8.11 Conjugate Functions and Level Sets

On the other hand, if f(x0) = —oc, then xo yo(xo, w) (d E R), and hence sup yo(x , w)

319

Sd(f) (d

E

E

R), whence

xES,i(f)

supi sup (p(x, w) — di ?. sup {ço(xo, w) — d } = +oo dER

del? xeSd(f)

= go(xo, w) ± — f (x0)

(x 0 E dom (—yo • , w)), f(x0) =

—

oc). (8.343)

From (8.342), (8.343), and (8.32) (where c = c(yo)), we obtain the inequality in (8.341). Assume now, a contrario, that this inequality is strict. Then there exists do E R such that f c(99) (w)
—oc) such that

f (w) < ço (x0, w) — do

ço (x0, w) ± — f (xo),

in contradiction with (8.35).

0

Theorem 8.25 We have fc(w)( w )

inf {d c RIA-d(f

w)) = 0)

= inf {d e R I S-d(f

w)) = 0)

= sup {d

RI A-d(f

w)) 0 0)

= sup td E R I S-d(f

w)) 0 0)

E

(f E R X ,

Proof Let f

E

E

W). (8.344)

Te and w c W. We claim that

Ad(f)

C

lw

E

{w E W I A-d(f . C

w)) = 0)

W I S-d(f

Sd(f c4) )

— (p(• , w)) = 0) (d E R).

Indeed, for any d E R we have, by (8.35) and (8.25), E Ad(f) R, and X c R. (a) We have inf f (S) X if and only if E n Ax (f) = 0. (b) If inf f (s) > )1/4., then E n Sx(f) = 0. (c) We have sup (X ERIEn A x (f) = 0) = inf ERIEn Ax(f) 01

inf f(E)

= sup (X E RI:En sx(f) = 0) = inf P1/4. ERIEn Sx(f )

0). (8.348)

f(x 0 ) < A. Conversely, if Proof (a) If x0 E E n Ax(f), then inf f (Ed') inf f (S) < A, then there exists .X0 E E such that f (x0 ) < A, so xo E E n Ax(f)• f(x 0 ) A. (b) If x0 E E n Sa.(f), then inf f (s) (c) By (a) and A x ( f) C Sx(f), we obtain inf f(E) = sup (AERI inf PE)

X)

= sup (A E RIEn AX(f) = 0)

sup (A ERIEnsA.(f)=0).

On the other hand, by (b), inf f(E) = sup (À E R I inf f(E) > X)

sup (A ERIEn S(f) = 0),

whence inf f(E) = sup P. E RI -En Ax(f) = = sup (A ERIEn sA.(f) = 0). These equalities also imply the other equalities of (8.348). Indeed, the sets (X E RISC) AX(f) = 0) and (X E RIEn Ax(f ) (4) are complementary half-lines in R, and for any X1, X2 E R with E n AA(f) 0, say, xo E E n A x , (f), and E n AX2 (f) = 0, we have À1 > f (xo) X2, which, together with (1.1), proves the E second equality in (8.348). The proof for A x (f ) replaced by S(f) is similar. Remark 8.57

One can also write (8.348) in the form

inf f(E) = sup {X E R A x ( f) X \ El = inf = sup {X E R I Sx (f) c X \ E)

R I Ax(f )

X \ E)

inf (AERI SÀ (f) g X \ E). (8.349)

Theorem 8.26

Let X and W be two sets and yo X x W —> R a coupling function. Then we have f"P) (w) = inf fy E R I yo(x, w)

—1 (x E A,(f)))

322

Conjugations

= sup {v E RI]x E A,(f), (p(x,w) > —1} (f

Te,

E

E

W), (8.350)

w e W),

(8.351)

and similar equalities with A_(f) replaced by S,(f). Proof By (8.153), we have (f

f L( ') (w) = - inf f (SD)

Te,

E

where {x

E

X I yo(x, w) > —1}

W).

E

(8.352)

Hence, by Lemma 8.6(c) applied to E = E„„ we obtain f

(çO)

— sup {X

( w )

= inf {—X

E

E

R I (p(x , w)

—1

R I (p(x , w)

A x (f))}

(X E

—1 (x

E

A x (f))},

whence (putting y = —X) the first equality in (8.350). The proofs of the other equalities are similar. Remark 8.58 Alternatively, one can also deduce Theorem 8.26 from Theorems 8.25 and 8.13, as follows: By Theorem 8.13 (see also Remark 8.24), for any coupling function (p : X x W —> R we have L((p) = c((p i ), where (p i = ((pi)[(p i is the coupling function (6.176). Hence, by Theorem 8.25, we obtain

f " (P ) (w) = f

(w ) = inf {d

E

w)) = 0}

RI A-d(f

= inf {d

E

R I f (x)

xf,TEw 1 99(x 07,)>-t)(w)

= inf {d

E

RI f (x)

—d (x

= inf {d

E

R yo(x , w) ( —1 (x

E

—d

(x

E

X))

E

W).

X, (p(x, w) > —1)) E

A-d(f))} (f

E

RX , w

Theorem 8.27 We have f000pY (x)

sup

sup {X E R l(p(y , w)

—1 (y E A x (f)))

wEw (p(x

sup

inf

E

R Iy

E

Ax(f), (P(Y , w) >

—11

weW

(f

and similar equalities with A x (f) replaced by Sx ( f).

E RA',

x

E

X),

(8.353)


323

Proof This result follows from Theorem 8.15 and Lemma 8.6(c) applied to 3 =-

Fl u, of (8.352). The above method of applying Lemma 8.6(c) to suitable subsets 3 of X permits us to express also the other conjugates and second conjugates of type Lau of the preceding sections, with the aid of the level sets A x (f) or Sx (f). For example, for el ' of (8.226), applying Lemma the conjugation of type Lau L(W ,B(P) : Rx 8.6(c) to w ,d

=

EX

I yo(x, w) >

d}

(8.354)

(w E W, d E R),

and using also (8.227), we obtain: Theorem 8.28

T? a coupling function. Then for

Let X and W be two sets and go : X x W L(f , (V) of (8.226) we have

f L( 'V ' 9') (w, d) = inf {v E RI ç.o(x , w)

= sup ly

E

d (x E

A—v(f)))

R I]x E A,(f), yo(x,w) > d} (f E RA', 21) E W,d E R),

fL(17V- ,9`P) L (VV,Ow)'( x )

sup sup sup (X E R I yo(y, w) dER wew 90(x,w)>d

(8.355)

d (y E &(0)}

= sup sup inf (X E RI3y E Ax(f), go(Y, w) > d} dER wEw (x,w)>d (f E R X x E X),

(8.356)

and similar equalities with A,(f), A x (f) replaced by S_(f) and Sx(f), respectively. Remark 8.59 Combining (8.227) and (4.158), we obtain other expressions of f i--(1-''"w)L(V •9°)/ with the aid of Scl(f), Ad(f), namely

f r.,(W,oço)LoV,oy (x) =

inf

xeco w. ,Sd (f)

d =

inf

d

xecowAd(f)

(f E

Te,

x E X),

(8.357) which we have used in the proof of Proposition 8.12. Let us consider now the conjugation of type Lau L(W , 9) : R A' (8.233). Applying Lemma 8.6(c) to F.; =

(

I

= x E X ÇO(X, W)

and using also (8.234), we obtain:

d)

(//) E

W, de R)

R WxR

of

(8.358)

324

Conjugations

Theorem 8.29

Let X and W be two sets and yo :XxW—>Ra coupling function. Then for L(1 i ,'-6 2 ) of (8.233) we have f

- , t-i‘P) (w, d)

inf (I)

E

R I ya(x, w) EX*

ci)(x)?-d

sup sup inf dER q)Ex* (Do )?-c/

E

(f

A x (f), CD(y)

x

E

E

d}

X),

(8.364)

and similar equalities with A,(f), A x (f) replaced by S_(f) and Sx (f), respectively. (b) One can apply these methods to obtain, for the semiconjugation L (X* x R, 0) of (8.212), the formulas f L(x* xR,9)

d) = inf

(1) E

sup

(1) E

RI sup (I)(A,(f)) RIx

E

A --,(f), CD(x) > d — 1) (f

f(x)

d — 1)

E

R X , CD

E

X* , d

E

R),

(8.365)

f L(X* x R,O)L(X* x R,H )' (x )

= sup dER

sup (D EX*

sup P.

d — 1)

E R I sup (1)(Ax(f))

4:13(x)>d-1

= sup dER

inf sup 43Ex* 43(x)>d-1

(X E

RI]y

Ax(f), (I) (Y) > d — 11

E

(f

E

,

x

E

X),

(8.366)

and similar equalities with A,(f), A x (f) replaced by S_(f) and Sx (f), respectively. (c) The case of the pseudoconjugation L(X* x R, 0) of (8.222) can be treated similarly, starting with Lemma 8.6(c) applied to =

=

tx E

X I CD(x) = d}

E

X* , d

E

R).

(8.367)

Conjugations

326

Let us give some formulas for the level sets of conjugate and biconjugate functions.

Theorem 8.30 Let X and W be two sets and ça : X x 4 sd(fc('9)) =

R a coupling function. Then

n

fl Sd((p(x,•— f (x))

X EX

sf(x)±d((p(x,.»

xeX

(f E X , d E R).

(8.368)

Proof Let f E R x , d E R. For any w E W we have, by (8.35) and (8.8), the equivalences f e(ÇD) (w)

d yo(x, w) . yo(x, w)

— f (x)

d

(x E X) (x E X),

f (x) ± d

whence (8.368) follows.

LI

Corollary 8.27 We have fc (') (w) = inf (c/ E R1w En sd((09(x,•) ± — f (x))1 xEx

(f E

W E W). (8.369)

Proof This formula follows from (8.368) and f'°(w) = inf Id E Rlw E Li sd(fc (99) )1-

Theorem 8.31 Let X and W be two sets and yo : X x 4 sd(fL(0)

=n

R a coupling function. Then

s_,((p(x,.))

(f E

, d E R).

(8.370)

xEA_d(f)

Proof Let f E R X and d E R. For any w E W we have, by (8.153), the equivalences WE

Sd(f L(99) ) . —

inf f (x) xEx 99(x,w)>-1

f (x)

—d

d

(x E X, (p(x, w) > —1) (x E X, f (X) < —d)

w E

n

xeA_d(f)

LI


327

Remark 8.62 (a) Alternatively, one can also deduce Theorem 8.31 from Theorem 8.26 and Lemma 4.1. Indeed, given f E R X and d E R, let

((p(x , -))

U, =fi

(t E R).

(8.371)

xEA_,(f)

Then, since r < t implies that A_,(f) D A -t(f ), we have (4.20). Also, by Theorem 8.26, we have f L (0 (w) = inf (r E Rlw E Ur ) = Mud (w)

(w E W).

(8.372)

Hence, by Lemma 4.1, we obtain Sd (f L(Ç°) ) =

n =n n

n

s_,((p(x,.))=

fit

t>d

t>d xEA_ 1 (f)

xe/i_d(f)

where the last equality follows from the fact that Ad(f) = Ut>d A—t(f)• (b) From Theorem 8.31 and f(w) = inf Id E Rlw E S1(f L(99) )) there follows again the first equality of Theorem 8.26. (c) Theorem 8.31 shows that for any f : X - -> R and d E R, S d (f L(99) ) is an intersection of level sets of functions ICÇÛ (X) G R W , with IC- 0 of (8.89), which means that each Sd (f L( g)) ) is K 0 (X) -convex or, in other words, that f L(99) is K- 0 (X)-quasiconvex. In particular, if X is a locally convex space, W = X* and go (Pnat, then each S_I (cPnat(x, .)) = { (1) E X * I 4)(X) —11 (x E X), and hence, by (8.370), also each Sd (f L( "wi ) ), is convex and weak* closed, so f "wt.° is quasi-convex and weak* lower semicontinuous. Corollary 8.28 We have, for any f E Te and d E R,

sd (f LL( ') i

)

s_1(0, ID))

—

weA _d(fl-W)

= tx E X I yo(x, w)

inf

(I)

—1 OD E W,

sup

E RI

(p(x, w)

—1 ) < — d)).

(8.373)

xEA_„(f)

Proof By (8.178), Theorem 8.31, and (6.138), we have

sd (g 1-((p)'

=

sci (8,0,)))

n WEA d(g)

n

wEA_,,(g)

(g E R w , d E R),

(8.374)

and hence (for g = f"P) ) the first equality of (8.373). This, in turn, together with Theorem 8.26, implies the second equality of (8.373).

Conjugations

328

Remark 8.63

Corollary 8.28 shows that for any f : X --> R and d E R, Sd(f L( W ) "99Y) is an intersection of level sets of functions v„,, ço = (-, w). Thus each Sd (f L( Ç°)L( ÇD)j ) is Vw,,p -convex, that is (by (2.189)), (W, 0-convex or, in other words, fL () L (Ç°Y is (W, yo)-quasi-convex. Note that this fact also follows from the inequalities f f q(w ,,p) f"(19) "PY (which hold by (8.227) and (8.229)), applied to f replaced by f"P ) ' , and from (5.31). The above method can be used to obtain the level sets of the other conjugates and second conjugates of type Lau of the preceding sections. For example, let us prove:

Theorem 8.32

Let X and W be two sets and yo : X x W —> "ix T?WxR of (8.226), we have L(I ,

2-

sx(f 1( 'TI w°) ) =((w,d)Ewx,,we

a coupling function. Then, for

n

sd((p(x,.)))

xEA x (f)

Sx( f

L(f4-1(fi-/ oçoy '

TO,A

R),

(8.375)

(f E Tec , X E R).

(8.376)

(f E

E

sd ( (p (., w))

= dER wEA -,,(f Lffi `o(-,d))

Proof Let us consider, for any f G , the "partial functions" fdL f L(W ,ew) (., d) : W ---> R (d E R) of f l-(1T1 '9) , that is, fdt, (w)

f L(kow) (w d)

inf

E RA', W E

Then for any f WE

Sjfdll

(w EW, d E R).

f (x)

xEX

(8.377)

W, and X, d E R, we have —

inf

f (x)

xEX

(x E X, yo(x, w) > d)

- f (x) ?, yo(x, w) WE

d

(x E X, f (x)
d

(x EX, d E R).

(8.379)


329

Then, by (8.378) applied to this case, we have S(g)

But, by (8.227), (8.226), (8.377), and (8.379) (for g = fdL f L(V,9P)L(ffi,v)/(x) = sup(— dER

(8.380)

(g E R W , À, d E R).

= n sd(,(.,w» WEA x(g)

inf

W (p(x,w)>d E

fl-

(

E R W ),

we have

w ) )

(f

= SUp (kid 14" Id \L kA"I dER " \

E RX ,

x

E

X),

(8.381)

whence, by (4.9) and (8.380) (for g = fdL ), we obtain

Sx(f L0-4-1 '

oço)Loil- oPy ' ) = n s„.((fcf , )/J) dER

= dER n

(f

n

E

re, À E

R),

IDEA

which, by (8.377), proves (8.376).

LII

Remark 8.64 Combining (8.227) and (4.160), we obtain other expressions of Sx (f L(1311-.6n L(IV '(P)/ ) with the aid of St (f) , Al (f) , namely =

fl cow, v,St(f) = n cowAr(f

r>x

(f

)

E RX , ÀE

R).

i>x

(8.382) Naturally, in the converse direction, combining (8.227) with Theorems 8.28 and 8.32, one obtains new expressions for fy(w.,p) and Sx (fq.(w, ) ). With a similar method to that of Theorem 8.32, we obtain:

Theorem 8.33 For L(Ç17- ,

éço)

RW

Rx

.(

S;t(f 1

xR

of (8.233) we have

)) ={(w,d)EW.R,wE n xcA

Ad(ox,.»}

'JP

(f c R x , A

sx(f

L(14-/ FP)L(1-4>

FPY )

=n dR

E

R),

(8.383)

R).

(8.384)

n wEA_ À (f 1- 01-1 •Fiw )(•,d))

(f

E RX ,

A

E

Conjugations

330

Remark 8.65 (a) In Theorems 8.32 and 8.33 we have obtained only level sets of type Sd of conjugate and biconjugate functions. Using the expressions of the conjugates of f as an inf, one can give various expressions for their level sets of type Ad. For example, from Theorems 8.25 and 8.26 one obtains, respectively, that, for any f E R X and d E R, A d (f

)

A d(f “") )

= 1w E WI = 1w

E

= tw

EW

inf E R S_x(f

w)) = 0}
d

A(St(f)) =

n

A'A(At(f ))

(f

E

Rx , d

E

R).

t>d

(8.406)

334

Conjugations

This formula can be also deduced from (8.404), as follows: By (8.404), (8.401), and (6.6) (applied to the duality A'), we have Sd(f L(A)L(Ar ) = A'({w E W I inf NERiwe A(A_v(f ))) < — d})

= Al (fw E W 13 V G R, y < —d, w E -=-- A l (fw E Wt > d, w E A(At(f))1) = A' (Li A(A,(f))) = f>d

n

A I A(A,(f))

t> d

(f c R x , ci E R) , and the first equality in (8.406) follows similarly, using (8.401) with A, (f) replaced by S, (f). 0

Chapter Nine

v-Dualities and _L-Dualities In the present chapter we will first consider "v-dualities," that is, dualities A : R x —> kw defined by a condition on (f v d) A (instead of condition (8.2)), namely condition (9.22) below, where v and A stand for (pointwise) sup and inf, in R x and R w , respectively. In contrast with the case of conjugations (see Theorem 8.1), the dual A' : R w —> R x of a v-duality need not be a v-duality. To determine the duals A' : R w —> R x of v -dualities, we will introduce two new binary operations on R, denoted 1 and T, respectively, and then, extending these operations (pointwise) to kx, we will introduce and study "1-dualities" A : R x —> R w defined by a condition on (f Id) ° , namely condition (9.44) below. It will turn out that v-dualities and 1dualities are dual to each other. We will then show that the conjugations of type Lau are simultaneously conjugations, v -dualities and 1-dualities, and that, conversely, they are the only dualities having any two (and hence all three) of the above properties. Let us also mention that v-dualities are a useful tool for the study of certain dual optimization problems (e.g., see Flachs and Pollatschek [88]; Flachs [86], 1 87]).

—

9.1

The Binary Operations _I_ and T

Definition 9.1 For any a, b

E

T? let a

if a < b,

-Hoc

if a

a

if a > b,

—oc

if a

alb =

(9.1)

b,

aTb = f

(9.2)

b.

Remark 9.1

The binary operations 1 and T on R are noncommutative and nonassociative. Indeed, for example, we have (OTO)TO = — ocTO = —oc, 335

(9.3)

v-Dualities and I-Dualities

336

OT(OTO) = OT(—oo) = 0.

(9.4)

Proposition 9.1

For any a, b, c E — R we have a .b.vc aTb

aTc

(9.5) (9.6)

Proof If a > b, then for any c E T? we have the equivalences

bvc.-#.cic.aTbc,

a

while if a b, then for anycET?we have a b V c and aTb = —oc c. The second equivalence in (9.5) follows from the first one, since bvc = cv b. The proof of (9.6) is similar. Remark 9.2 (a) By Proposition 9.1, the operation T (resp. 1) is, in a certain sense, the " inverse" of v (resp. A). This property shows that the binary operations 1 and T may be useful for other applications too. (b) Proposition 9.1 is similar to Lemma 8.2. Corollary 9.1

For any b, c E R we have b V c = max a = max a, aE-RaTc.1)

b

A

c = min a = min a, aER alc?-13

(9.7)

aER aTb.‹,c,

(9.8)

aER albc

bic = max a,

(9.9)

aET? aAch

bT c = min a.

(9.10)

aEW avoh

Proof We have b V c = max {a E Ria b v c}, whence, by (9.5), we obtain

(9.7). The proofs of (9.8)—(9.10) are similar.

LI

A connection between I and T is given by Proposition 9.2 For any a, b E R we have alb = —(—aT — b), aTb = —(—al — b).

(9.11)

9.1 The Binary Operations 1 and T

337

Proof This follows from (9.1) and (9.2), using that a < b if and only if —a > —b. 0 Remark 9.3 (a) By Proposition 9.2, each result on 1 is equivalent to a result on T (e.g., (9.6) and (9.5) are equivalent). (b) Proposition 9.2 is similar to (8.25). Proposition 9.3 We have aT

oc = —oc = (—oc)Ta

(a

E

R),

(9.12)

aT — oc = a = a1(±oo)

(a

G

R),

(9.13)

al — oc = +oo = +cola

(a

E

R),

(9.14)

+ ooTa = +oo

(a

E

R U {—oc}),

(9.15)

(—oc)la = —oc

(a

E

R U (±oo)),

(9.16)

(aT b)T c = aT (b V c) = (aT c)T b

(a, b, c

E

T?).

(9.17)

Proof Formulas (9.12)—(9.16) are particular cases of (9.1) and (9.2), but we have mentioned them separately because they will be used in the sequel. Finally, by (9.2) and (9.5), we have for any a, b, c E T?, IaTb

if aTb>c

aTb=a

ifa>bvc

—oc

ifa“vc

=

(aTb)T c = —oc

if aTb c

= aT (b v c) = aT (c v b) = (aT c)Tb.

0

Proposition 9.4 For any set I we have (sup ai )Tb = sup (ai Tb) iEi

(inf ai )_Lb = inf (ai _Lb) jEt iet Proof If /

({a i } i , i

C

R, b

E

R),

(9.18)

iEi

({a,),E1 g R, b E R).

(9.19)

0 and supia a, > b, then fi c / jai > b} 0 0 and, by (9.2),

(sup ai )Tb = sup a• = sup ai = sup (ai Tb) = sup (ai Tb). il iEt iCI, a,>b iEl, a, >b iEl b (i E I), If sup / ai b, then (supie/ ai )Tb = —oc. On the other hand, ai whence a•Tb = —oc (i c I), so sup, Ei (ai Tb) = —oc. This, together with (1.1) and (9.2) for a = —oc, proves (9.18).

v-Dualities and 1-Dualities

338

Finally, (9.19) follows from (9.18) and (9.11).

El

The binary operations 1 and T can be extended to R x , where X is any set, as follows: Definition 9.2 For any f, h

E

Te

let

(f Ih)(x) = f(x)1h(x)

(x

X),

(9.20)

( fT h)(x) = f (x)T h(x)

(x e X).

(9.21)

E

9.2 V-Dualities Definition 9.3 Let X and W be two sets. A duality A : R x —> R w is called a v-duality (or, a max-duality) if (f v d) A = f A A —d

(f

E

Rx, d

E

R).

(9.22)

Remark 9.4 It is enough to assume (9.22) for all d E R, since for d = -koo it reduces to (7.12) and for d = —oc it reduces to f ° = f ° (f E R x ).

Let us consider the problem of the representation of v-dualities with the aid of coupling functions. To this end, we will proceed similarly to the case of conjugations, using now the representation functions 77G of (4.40) instead of the indicator functions x G . Corresponding to Lemma 3.1, we have now: Lemma 9.1 For any set X and any function f : X —> R we have f = inf {q {x} v f (x)}. xex

(9.23)

Proof This formula follows from Lemma 3.1, observing that

XG d = riG v d

(G c X, d

E

T?) .

(9.24) 0

We have the following representation theorem: Theorem 9.1 Let X and W be two sets. For a mapping A : R x —> R w the following statements are equivalent: 10.

A is a v-duality.

9.2 v-Dualities

2°.

339

There exists a coupling function * : X x 4 /

f ( w ) = sup (*(x, w) xEx

A

—.12- such that

(f E R X , w E W).

—f (x)}

(9.25)

Moreover, * of 2' is uniquely determined by A, namely

(x, w Proof

)

(X E X, w E W).

= Oixe (w)

1 0 = 2'. If 1° holds, then, by (9.23), (7.1) (with

d -= f (x) E

T?), we obtain for any f

I

(9.26)

= X), and (9.22) (with

E Rx,

f °= (inf {n{ x} V f( x )})A = sup(r){ x } V f (x)) ° = sup {( 77{x}) A A - f (x)} • xeX xeX

xEX

Thus for V/ of (9.26) we have (9.25). 1°. If 2° holds, then, by (9.25), we have for any I 0 0,

(inf iel

fi ) ° (w)

= sup {*(x, w) A — inf = sup xEx

(x)}

iel

xeX

{*(x,

w)

sup (— fi (x))) = sup {*(x, w) iEi xEx,iEl

A

= sup fi A(w )

({fi } iEI C R x ,

,

A

—fi(x)}

W E W),

iel

(f v d) A (w) = sup xEx

(*(x,

w)

= sup (*(x, w) A xEx = f A (w)

A

—(f v d)(x)}

A

—

—d

f (x) A —d} (f c x , d E R, w E W).

Also, by (9.25), (±oo) ° = —oc. Finally, if 2° holds, then, by (9.25) for f = nIx) and (4.40), we obtain (T/{x}) A (W) =

sup

{*(Y , w) A —77{x}(Y)} = *(X,

111)

(X E X, w E W),

yeX

which proves (9.26) and the uniqueness of *. Remark

9.5 (a) The implication 1° = 2' also follows from Theorem 7.3 or 7.4,

since by (7.64), (9.24), and (9.22), we have e(x, w, a) = (x{x)

a) ° (w) = (11(x) V a) A (w) = [(1{x}) ° A —a] (w)

= (n{x }) A (W)

A

—a

(x E X, wEW,aE R).

(9.27)

Note also that, by (9.27) and (9.26), we have e(x, w, a) = *(x, w)

A

—a

(x

EX, WE

W, a E

(9.28)

340


whence, in particular,

*(x, w) = e(x, w, —oc)

E

X, w

E

(b) Since the elements xo E X with *(xo, w) A disregarded in the supxEx of (9.25), we have f (w) =

(1/ (x, w) A

sup XEdOM (-1,11(•,10)ndOM

—f

W).

(9.29)

(f

(x)}

E

may be

— 00

(X0) =

Te,

w

E

W).

f

(9.30) (c) Using (3.23), (9.24), (7.1) (with I = Epi f), (9.22), and (9.26), we obtain

sup

f ° (w) =

{07 fx)) ° (w) A —d} = sup

(x.d)eEpi f

{ilf (x, w)

A —d}

(x,d)eEpi f

(f

w E

E

W). (9.31)

Definition 9.4 Let X and W be two sets. (a) For any coupling function * : X x W —> R the v-duality A : -/7x —> R w defined by (9.25) is called the v -duality associated to *, and it is denoted by A (ik, V). Thus f A(k.v) (w ‘) = sup (*(x, w) A — f (x)}

(f

w

E

W).

(9.32)

xEX

(b) For any v-duality A : X --> Te the unique coupling function * : X x W --> R satisfying (9.25), that is, the function (9.26), is called the coupling function associated to A, and it is denoted by *A , \/ . Thus f

= sup Ilk ,v(x,

tV) A

— f (x)}

(f

E

Rx,

WE

W).

(9.33)

xeX

çOnat (of (2.184)) Remark 9.6 (a) The case where X is a set, W C R i" and * is important for applications. The general case can be reduced to this particular case using V = Vw, * c R x defined by (2.188) (similarly to Remark 8.6(e)). (b) By Theorem 9.1, we have a one-to-one correspondence between v-dualities A : R x ----> R w and coupling functions V/ : X x W --> R. Note that for any v-duality Ao : R x R w and any coupling function *0 R, we have

A(1/fp 0 ,v, y) =- AO,

*LS,(*(),v),v = *0,

(9.34)

and therefore each result on a pair (ip. , A ( p, V)) can be also formulated, equivalently, as a result on a pair (A, *A , v ). In the sequel we will state the results only in one of the two forms. On the other hand (see Remark 8.7(a)), we have a one-to-one correspondence between conjugations c : --> Rw and coupling functions ço : X x W --> R. Hence we obtain a one-to-one correspondence between v -dualities A : R x --> R w and conjugations c : Rx namely A —> c(*A , v ) (with

341

9.2 v-Dualities

A ((pc , v)), satisfying

inverse c

(x

(r) {x}) ° = (X{x}) c(*"

E

X).

(c) The conjugations c((,o) : R x --> Te (of (8.35)) can be expressed with the aid Te, as follows: If : X x 147 --> T? is any of v-dualities A(*, TexR coupling function, then f c(ço)

FAf,v)

(f

R X ),

E

(9.35)

where çlr((x, r), w) = 2yo(x, w) — r

(x EX, rE R, w

-

F(x, r) = 2f (x) — r

(x

E

W),

E

(9.36)

X, r c R).

(9.37)

Indeed, by the identity sup (a — r)

A

(r — b) =

(a ± —b)

(a, b

E

R),

(9.38)

rE R

due, essentially, to Flachs and Pollatschek ([88 1 , lmm. 1; see also Voile [303], p. 126, lmm.), we obtain

I* ((x , r), w)

sup

FA(* , v)(w)

A

— F (x, r)}

(x,r)EX xR

= sup sup 1[4(x, w) — ri

A

[r — 2f (x)1}

xEX rER

[4(x, w) ± —2f (x)1 = f`W (w)

= sup

(W E

W).

XEX

Let us give now some corollaries of Theorem 9.1.

Corollary 9.2

For A : R x --> R w and *A , v =

as in Theorem 9.1, we have

(f

min d dET? v(•,11))-f

f (w)

Proof By (9.33), (9.6), and (9.11), for any f f A (w) =

min dER f ° (11))

One can express f

d =

E

Te

d =

min dER

11/A,v(• , w)A — f

E

Te,

w

and w

E

E

W).

W we have

min dER

d

(9.39)

—dT-11/ A . v (-,w)f

with the aid of the level sets of f, as follows:

d.

(9.40)

342


Corollary 9.3 For A and11 fA, v = fA

as in Theorem 9.1, we have min

( w )

d

dER

sup E-

E

E

RX , w

W).

E

(9.41)

VIA V (X

A_d(J)

Proof Since lifA,v(x, w) A — f (x) —j(x) > d (x E A—d(f)), we have fd

(f

(

d (x

— f (x)

R I *A.v(x , w) A — f (x) d (x = Id

E

R

d (x

VfA,v (X, w)

E

X

E

\

A-d(f)) and

X)}

A—d(f))},

E

whence, by (9.40), we obtain (9.41). Remark 9.7

Formula (9.41) is equivalent to

P(w) = min {d E

inf

f (x) —d}

xe X

(f

E

RX ,

W E W).

(9.42) Finally, let us mention another expression for f ° (*.v ) . Proposition 9.5 For any v -duality A(1r, v) we have = sup xEx

min

d

(f

E

E

W).

(9.43)

dE -12d

(x

— f (x)

LI

Proof This formula follows from (9.32) and (9.8).

9.3 i-Dualities Definition 9.5 Let X and W be two sets. A duality A : R x —> R w is called a I-duality if (f _Ld) A = f A T — d

(f

E

,d

E

T?).

(9.44)

Remark 9.8 It is enough to assume (9.44) for all d E R, since for d = —00 it reduces (by (9.14) and (9.12)) to (7.12) and for d = -Foo it reduces (by (9.13)) to f ° = f ° (f E R X ). We have the following representation theorem:

9.3 I-Dualities

343

Theorem 9.2 Let X and W be two sets. For a mapping A : R x —> R w the following statements are equivalent:

1°. 2°.

A is a _L-duality. There exists a coupling function v :X x W—> R such that f ° (w) = sup (— f (x)T — v(x , w)) xeX

=

sup

—f(x)

xeX f(x) —a,

(9.49)

whence e(x, w, a) -oc, then, by the above, it follows that e(x, w, —a) = —a. Thus e(x, w, a)

E (—DO,

—a}

(a

E

R).

(9.50)


344

Hence, since e(x,w,.) : R —> R is nonincreasing and lower semicontinuous (by Theorem 7.4), there exists v(x, w) e R such that

I—a

if a < v(x, w)

e(x, w, a) =

= —aT — v(x , w).

if a

(9.51)

v(x, w)

Consequently, by (7.63) and (9.51), we obtain (9.45). 2° = F. If 2° holds, then, by (9.45), (9.18), (9.11), and (9.17), we obtain for any I

0 0,

(inf fi) A (w) = sup {[— inf fi(x)]T — v(x, w)) ier iEr xEx

sup {—fi (x)T — v(x, w))

= sup {[sup (—f i (x))]T — v(x, w)} = lei

xeX

xeX,iel

= sup fi° (w) iEr

({f i }, Ei c R x , w c W),

(f Id) ° (w) = sup {—(fid)(x)T — v(x, w)) xeX

= sup {(— f (x)T — d)T — v(x, w)} xeX

= f° (w)T — d

(f

E

R x , d c R, w G W).

Also, by (9.45), (-Foo) A = —oc. Finally, according to Theorem 7.4, e of (7.63) is uniquely determined by A. Hence, since (by (9.51)) v(x, w) =

sup

a =

aER e(x,w,a)=--a

a,

min

(9.52)

aeT? e(x,w,a)= oo

v is also uniquely determined by A and, by (9.52) and (9.47), we have (9.46).

El

Remark 9.9 (a) For Theorem 9.2 there is no proof similar to the above proof of Theorem 9.1, since the only result (corresponding to Lemmas 3.1 and 9.1) expressing f G R x with the aid of the operation 1 is the formula

f = inf If (x)1 — 77 1,0 ) xeX

(f

E

whence, by (7.1) (with I = X), f ° = sup XEX

If (x)1 — 77{X}}

(f

R X ),

(9.53)

9.3 I-Dualities

345

to which we cannot apply (9.44) (since 1 is noncommutative). To show (9.53), it is enough to observe that for any x, y E X we have, by (4.40), (9.13), and (9.14),

f (x)± - 17{x)(y) =

i

f (y)

if x = y,

+00

if x 0 y.

(b) By Theorem 9.2, we have a one-to-one correspondence between 1-dualities A : R x —> R w and coupling functions v:Xx W ---> R. Similarly to Definition 9.4, the 1-duality A : R x ---> R w defined by (9.45) will be denoted by A (v, 1) and called the 1-duality associated to y, and the coupling function v :X x W —> T? defined by (9.46) will be denoted by vA.± and called the coupling function associated to the _L-duality A. (c) The conjugations c(go) :R X —> R w can be expressed with the aid of Idualities A(v, I): R x x R —> R w , as follows: If : X x W —> R is any coupling function, then fc(so) = FA(v,I)

(f c R x ),

(9.54)

where y((x , r), w) = (p(x, w) — 2r

(x c X, r

F(x, r) = f (x) — r

E

R, w c W),

(x c X, wc W).

Indeed, we have

sup

F(w) =

—F(x,r) =

(x ,r)e X x R F(x,r) R x is a I-duality, and for their associated coupling functions we have

x) = *A.v (x, w)

(9.62)

E W, X E X).

Proof By Theorem 9.2, we have to prove that for any v-duality A : there holds g '

sup {—g(w)T — buov

(g E R w ).

w)}

—> T?w ,

(9.63)

By (7.13), (9.33), (9.6), and (9.11), we have g A' =

inf h =

-11. R x is the 1-duality A(Afr , 1) (in the sense of (9.45) and Remark 9.9(b)), where v* : W x X --- > .T? is the coupling function (w, x) = * (x , w)

E W, X E

X).

(9.69)

Corollary 9.6 R x ---> R w we have For any v -duality A : — (771,0) ° ( w )

sup

=

b=

min

(x G X, w

b

bER

12€7-2

000 -i-12) ° ' (x)=-b

(xi +b)' (x)=-oo

E W).

(9.70) T?X ),

Proof By (9.26), (9.69), and (9.46) (applied to the 1-duality A' : R w we obtain (9.70). Proposition 9.7

Let X and W be two sets and * : X x W R a coupling function. For two functions f E R x and g E R w the following statements are equivalent:

1°. f g A( C v)i . 2°. We have g(w)± — f (x) 30

g

f

(x , w)

(x G X, w

E

W).

(9.71)

(1 ,V)

Proof 2° - 3°. By (9.6) and (9.32), we have the equivalences 20 . g(w)

*(x, w) A —f(x)

(x

E

X, w

E W)

. g (w) ?. sup (*(x, w) A — f (x)} = f A( C v) (w) X€X

(w

E W).

9.4 The Duals of v-Dualities

349

Finally, the equivalence 1° -. 3° holds by (7.15) (with A = A(*, V)). Proposition 9.8

R, f E R x , x E X, and w E W, we have the following For any Vf :Xxl inequalities that are equivalent to each other: — f A( C v) (w)T — *(x, w) f (x) f A( "(w)1 — f (x) *(x, w) A

—

f (x)

(x, w)

f A( *' v) (w). (9.72)

Proof The last inequality of (9.72) holds by (9.32) and the inequalities of (9.72)

are equivalent to each other by (9.11) and (9.6). Remark 9.12 One can make similar observations to those of Remark 8.9(a), (b), using now e and e' of (9.67), (9.68). In particular, (only) the first one of the inequalities (9.72) is the Fenchel-Young inequality (of Definition 7.2, for A = A (*, V)).

Let us consider now, for a v-duality A : R x --> R w , the second dual f °°' of a function f E R x . Theorem 9.4

For any v-duality A(*,

>

:

f A(iii.v)Accvr (

x) ,

, we have

sup { — JçA(Cv)( w )T — *(x, w)}

weW

f6,(0. , V)( w

inf

)

well/ f A ( 'fr , v ) (w)—*(x,w)} ±

b

(w E W, bc R)

correspond to the "(W, 0-elementary functions" of Remark 3.25(d).

(9.84)

9.5 The Duals of 1-Dualities

351

Corollary 9.8

> R w we have

For any v-duality A(*, v) : (

1,1f,v)6. 1/1,vy

sup

(

ibT —

w)}

(f E R X , w E W).

wE1V,bEW

sup

1//(x,w).. —b, w E W).

LI

9.5

The Duals of _L-Dualities

Theorem 9.6 If A :7 X ---> R w is a _L-duality, then its dual A' : R w ---> Te is a v-duality, and for their associated coupling functions we have

x) =

w)

E W, X E

X).

(9.86)

Proof By Theorem 9.1, we have to prove that for any I-duality A : R x there holds

g ° = sup {vA,±e, w) A —g(w))

(9.87)

(g E R w ).

wew

Rw

By (7.13), (9.45), and (9.5), we have

g

6.1 =

inf h=

inf

hERx

hERx

h=

h°. Tel is a I-duality, then A (v, I)' is the v-duality A (*v , y), where i lî, : W x X ---> T? is the coupling function ,

(w, x) = v(x, w)

(tV E

W, x

E

(9.92)

X).

Corollary 9.9 For any _L-duality A : R x ----> R w we have

( 77{.}) °' (x = )

sup aER (X(Ai-i-a) A (w)=—a

a =

min

a

(x

E

X,

WE

W).

aeT?

(9.93) Proof By (9.26) (applied to the v-duality A'), (9.86), and (9.46), we obtain (9.93). E1

9.5 The Duals of I-Dualities

353

Corollary 9.10 (a) Every v-duality is the dual of a I-duality. (b) Every I-duality is the dual of a v-duality. Proof If A is a v-duality (resp. a I-duality), then A = (A')' (by Theorem 5.3), where A' is a I-duality (resp. a v-duality). 0 Corollary 9.11 (a) A duality A : R x (b) A duality A : R x

R w is a v -duality if and only if A' is a I-duality. R w is a I-duality if and only if A' is a v -duality.

Proposition 9.9 Let X and W be two sets and v:XxW R a coupling function. For two functions f E R x and g E R w the following statements are equivalent: 10 .

f

2°.

We have f(x)_L — g(w)

30

v(x, w)

(x E X, wE W).

(9.94)

g

Proof 2° 3°. By (9.6), (9.11), and (9.45), we have the equivalences

2° f (x)Iv(x, w) —g(w)

g (w)

sup

{—

(x E X, w E W)

f(x)T — v(x, w)} = f A( 'I) (w)

(w E W).

xeX

Finally, the equivalence 1° 3° holds by (7.15) (with A -= A (v,

0

Proposition 9.10 For any v:Xx W —> R, f E R x , x E X, and w E W, we have the following inequalities that are equivalent to each other: V(x, w) A —f A(v ' 1) (W)

f (x) f(x)_L — f ' 64 '1) (w) — f (x)T — v(x, w)

v(x, w) f

' ±) (w). (9.95)

Proof The last inequality of (9.95) holds by (9.45) and Remark 9.9(b), and the inequalities of (9.95) are equivalent to each other by (9.6) and (9.11). Remark 9.16 One can make similar observations to those of Remark 8.9(a), (b), using now e and e' of (9.51), (9.91). In particular, (only) the first one of the inequalities (9.95) is the Fenchel-Young inequality (of Definition 7.2, for A = A (v,


354

Let us consider now, for a I-duality A : R x function f E R X .

R w , the second dual f

' of a

Theorem 9.7 For any _L-duality A(v,

: R x ---> R w we have (f E R X , X E X).

f A(v ' j-)A(v-LY (X) = sup 11)(X, W) A —f A(v ' 1) (W)} wEW

(9.96) Proof This formula follows from (9.87) applied to A = A(v, I) and g = p(v,1) E Te.

ii

Remark 9.17 Remark 9.9(c), which expresses fc ((P) as F--L) (with F : X x R of (9.56) and v : (X x R) x W —> R of (9.55)) cannot be used to express R c f c ((,) (01 with the aid of F °( '-" ( ''' )/ . Nevertheless, this aim can be also achieved with a different method, as follows: If ço : X x W —> --E? is any coupling function and f E Rx, g E R w , define v : (X x R) x (W x R) F:XxR--->R andG:WxR—>Rby (x E X, W E W, r, s E R), (9.97)

v((x , r), (w, s)) = 2(p(x, w) — 2r — s F (x, r) = f (x) — r G(w, s) = 2g(w) — s

(x E X, r E R),

(9.98)

(w E W, s E R).

(9.99)

Then, one can show (see [182], example 5.1) that F" / (x, r) = fc(`P)c(`P)/ (x) — r GA(v , i-) 1

( w , s ) — 2g

/

(x E X, r E R),

(w) — s

(9.100)

(w E W, s E R).

(9.101)

We have the following corollary of Theorem 9.7:

Corollary 9.12 For any 1-duality A(v, _L) : Ti x —> R w , we have f")-1-)A( ' ±)/ (x) = sup wEw

d

min

(f E

Te , .3c

E

X).

d ET?

(9.102) Proof By (9.96) and (9.8), we have

f' ±)/ (x) = sup

min

well/

deT?

d

(f E R X , w E W),

— f A ( v , ±)(w)d1v(x ,w)

whence, using (9.11), we obtain (9.102).

Ii

9.6 Characterizations of Conjugations of Type Lau

355

Theorem 9.8 For any I-duality A(v,

R w we have

: Rx

fA (v,_L)A( v,_L Y (x )

sup

Ive, w)

A

b)

(f E R X ).

(9.103)

wEW, bel? 1,4x,w)A1).R w the following statements are equivalent:

1 0 . A is a conjugation of type Lau L(Q) for some (unique) Qc X x 2°. A is both a conjugation and a v -duality. 3 0• A is both a conjugation and a 1-duality. 4 0• A is both a v-duality and a I-duality. 5°. Both A and A' are v -dualities. 6°. Both A and A' are I-dualities.

W.

Moreover, in these cases, we have cp A j(x ,

=

(x

,4•(w x) = — xQ(x ,

*A,v(x, w) = vA,±(x, w) =

x) =

E

X,

WE

x) = —rm(x, w) (x EX, w

Proof The implications 1 0

W), (9.114)

E

W). (9.115)

2°, 3°, 4°, and the equalities

(PA,j- = — XQ,

VfA,v =

= — r/Q,

(9.116)


358

follow from Theorems 8.14 and 9.9 and from the uniqueness of yoA ,4_, VfA,v and VA ,1 (see the uniqueness parts of Theorems 8.2, 9.1 and 9.2, respectively), using formulas (8.167), (9.111), and (9.112). Furthermore the equivalences 40 5° . 6° and the other equalities of (9.114) and (9.115) follow from Corollary 9.11 and (8.51), (9.86), and (9.62), respectively. 2' = 1°. If 2° holds, then for *A,\/ = * of (9.26) we have, by (4.40) and (8.5) (for c = A),

*A,v(x,

=

(r/{x) A (w)

= 07{x}

ViA,v(X , tV) — I

which proves that IfrA, v (x, qQ, with

1 ) A (w) = (r/I x 0 A (w) — 1 (X E

1.1)) E { - 00, +00)

((x, ID)

X, w E W), E

X x W), whence *A,v =

—

Q = {(x, w) E X x 147 HfrA,v(x, w) = +Do).

Hence, by Theorem 9.9, implication 2° = 1°, we obtain A = L(Q). 3° = V. If 3° holds, then, by Theorems 8.1 and 9.6, A': R w ---> R x is both a conjugation and a v-duality. Hence, by the implication 2° = 1', proved above, and = by Theorem 8.14, there exists a (unique) set Q' Ç WxX such that Thus, by (8.51) (with c = A), we obtain ço6.,+(x, w) =

X)

= — XS-21 (W, X)

= — XQ(X,

(X E X,

we W), (9.117)

where Q is the (uniqely determined) set

Q = {(x, w) EX x WI (w, x) E Q');

(9.118)

therefore A = c(—n) = L(S2) of (8.164), (8.172). 1'. Assume 4', and let (x, w) EXx W be such that *A, v (x, w) > Then, since for any d E R we have rifx ) = riix iid (by (4.40), (9.14), and (9.16)), we obtain, by (9.26) and (9.44), —00
—d (d E R), that is, *A (x, w) = +oo. This proves that *A , v (x, w) E {-00, +00} for all (x, w) E X x W, whence, as in the above proof of the implication 2° = 1°, we obtain A = L(S-2).

Chapter Ten

Abstract Subdtfferentials 10.1 Subdtfferentials with Respect to a Duality A Where X and W Are Two Sets

:

Throughout this section, we will assume that X and W are two (nonempty) sets, Rw > — A : Rx R w is a duality, with dual A' : — R x , and e = e : XxW are the mappings corresponding to them by R, e' = eA , : Wx X x Theorems 7.3-7.5. Definition 10.1 Let f E R X and X0 E X. The subset

a° f (x0) = two

a° f (x0)

of W, defined by

E W I e t (w 0 , xo, ° (wo)) = (x0)1,

(10.1)

is called the subdifferential off at xo with respect to the duality A, or, briefly, the Af(x0) 0 0, then f is said to be A-subdifferentiable subdifferential off at xo. If at xo, and each element w o E a° f (x 0 ) is called a A-subgradient of f at xo . If aApxo) o 0 for all X0 E X, then f is said to be A-subdifferentiable.

a°

Remark 10.1 (a) No assumptions of finiteness (of e, f(x 0 ) or f ° (w 0 )) are made in Definition 10.1. (b) We will see below that Definition 10.1 is a natural one, since the Asubdifferential a° f (x0) of Definition 10.1 generalizes the usual subdifferential and preserves some of its main properties. (c) By (10.1) and the generalized Fenchel-Young inequality (7.93), we have (no, X0,

f(x 0 ) = {WO E WIf

J'A (w0))).

(10.2)

Hence, if f(X 0 ) = —Do, then a° f (xo) = W. (d) Combining (10.1) and (7.83), we obtain

a° f (x0) = two which expresses

E W I min {a E

a° f (x0)

I e(X0, ivo, a)

P(w0)1 = f(xo)1, (10.3)

with the aid of e (instead of e'). 359

360

Abstract Subdifferentials

(e) For f

±oo and any

a ° (+ 00)(xo) = {w0

E

X0 E

X we have, by (10.1), (7.12), and (10.3),

W I e (w o ,x 0 , —00 ) = +cc}

= {Wo E W

'

I min {a

E

R I e(x o , w o , a) = —oo) =

Hence, if z\ is strict at (xo, wo) (see Definition 7.4(a)), then wo In the sequel we will assume that f

E

E

a° (+00)(4).

R X and X0 E X.

Theorem 10.1

For an element 1°. 2°.

WO E

w0 E W, the following statements

are equivalent:

a° f (x0).

There exists b E R such that

(wo, • , b)

(10.4)

f,

(10.5)

(wo, xo, b) = f (xo). Moreover in this case we have

f ° (wo) = min {b E R I e f (wo, •, b) Proof 1°

f, e/ (wo, xo, b) = f (xo))•

(10.6)

2°. If 1° holds, then, by (7.93) and (10.1), b = f ° (w() satisfies

(10.4) and (10.5). 2° 1° and (10.6). Let b E R satisfy (10.4) and (10.5). Then, by (10.4) and Proposition 7.2, we have e(x, w o , f (x)) b for all x E X, whence, by (7.63), f ° (wo) = sup e(x, wo, f (x)) xEx

(10.7)

b.

Hence, since e' (w o , x o , .) is nonincreasing, we obtain, using also (10.5), (wo, x0, fA(wo))

(wo, xo,

= ,Rxo),

a°

and therefore, by (10.2), WO E f (4). Thus 2° = 1 0 . Finally, by (10.7) (which holds for each b E R satisfying (10.4), (10.5)), we have the inequality in (10.6); hence, by 1° and the above proof of the implication 1° = 2°, we obtain (10.6). D

a°

Remark 10.2 (a) In other words, Theorem 10.1 says that w o E f (x0) if and only if there exists an elementary function (in the sense of Remark 7.10(b)) with "slope" w o, which is a minorant of f and coincides with f at xo (for the term "slope," see Remark 10.6(b) below). (b) In the case where wo E f (x0), formula (10.6) is a sharpening of (7.89) (with A = B = R).

10.1 Subdifferentials with Respect to a Duality A : R x ---> R w

361

Proposition 10.1

If a' 1 (x0) 0 0,

then (10.8)

f(x 0 ) =- .f' (x0).

Proof If wo E

a° f (x 0 ), then, by (7.96), (10.1),

f A A ' (x0)

and f'

f, we have

f AA ' (x0)•

e/ (wo, x0, f Nw0)) = f(x 0 )

Remark 10.3 (a) By (7.94), Proposition 10.1 says that if f is A-subdifferentiable at xo, then f E C(.F; x0), where = fe' (w , • , b) Iw E W, b E

(C

R X ).

(10.9)

Hence, if f is A-subdifferentiable, then f E C(.F). (b) Remarks 10.2 and 10.3(a) above substantiate the statement of Remark 10.1(b), f (x o ) preserves some of the main properties of the usual subdifferential of f that at xo (see Remark 10.6(b) below).

a°

Proposition 10.2 If (10.8) holds (in particular, if a° f (x0) 0 0), then

0 A f(x0)

8 A f AA , (x0).

Proof By (10.1) (applied to f °A ' ) and f AA'

a° f °Ai (x0) =

two E W I

=

(10.10)

f ° , we have

(wo, x0, f Nw0)) = f' (x0)},

(10.11)

whence, by (10.8) and (10.1), we obtain (10.10). Theorem 10.2 For any element W o E W we have the implication e(xo, wo, f (x0)) = f

A

(wo)= X0 E aA" f (wo).

(10.12)

Moreover, if (10.8) holds (in particular if a° f (x 0 ) 0 0), then we have the equivalence e(xo, wo, f (xo)) =

(wo) .#> .)co E 8A 7 A (W0)-

(10.13)

Proof By (7.87) and Definition 10.1 (applied to the duality A' : R w ---> a A' g(W0) = (x0 E

X I e(xo, wo, g Ai (xo)) = g(wo))

(g E W W ),

(10.14)


362

whence, in particular, Of A (WO) = {X0 E X e(xo, wo,

f

AA'

00)

=

f

A

(W0)).

(10.15)

Observe now that we have the inequalities e(x o , w o , f (x 0 ))

e(x o , w o , f

AA‘ 'A(wo) = f A (W0);

(X0))

(10.16)

indeed, the first inequality follows from f °°' f, since e(xo, wo, •) : R —> —. nonincreasing, while the second inequality follows from (7.92) (applied to R f A A ' ). Thus, if e(xo, /Do, f (x0)) = f ° (wo), then e(xo, 11)0, f ' (x0)) = f (wo), which, by (10.15), proves the implication (10.12). Moreover, if (10.8) holds and X0 E a'f (WO) , then, by (10.15), we obtain e(xo, wo, f (xo)) = f (WO), which proves the equivalence (10.13). Remark 10.4 Formulas (10.12) and (10.13) can be also written in the form of inclusion and, respectively, equality, between the sets {w0 G W I e(x o , w o , f (x 0 )) = f

(wo ))

(10.17)

and {WO E W I X0 E eifA(W0)}-

(10.18)

Note that (10.18) is nothing else than (8A'f°) -1 (x0 ), where (8' f')' :X -÷ 2w is the inverse of the multifunction a AV A : w E W —> 8 A V A (tv) E 2x (of (10.15)). On the other hand, (10.17) is the set of all wo E W for which the equality sign holds in (7.92). We will give below some relations between these sets and a A f (x0). R x --> We recall (see Definition 7.4(a), with A = B = T?) that a duality A : — is said to be strict at (x o , wo) E X x W if the function e(xo, w o , •) :R— >R is strictly decreasing (where e = eA :XxWxR —> R is the mapping corresponding to A, by Theorem 7.3). Theorem 10.3

If A' is a strict duality at (wo, X0) E W x X, then we have the implication WO E

Proof If wo E

aAf (x0) = x 0 E a'f°(v0).

a° f (x 0 ), then, by (7.83), f

'

f and (10.1), we have

(wo, x o , e(xo, wo, f °°' (x0)))

= min fa E R I e(xo, too, a) f AA' (x 0 )

e(xo, wo, f AA ' (x0)1

f(x 0 ) = e' (w0 , xo, f A (wo))•

(10.19)

10.1 Subdifferentials with Respect to a Duality A : R x ---> R w

363

Hence, since e'(w o , xo , •) is strictly decreasing, we obtain e(xo, w0, f

f (wo) ,

(xo))

which, together with (10.16), yields e(xo, w o , f' (xo)) = f (w0). Thus, by (10.15), x0 E aA'fA(w0 ). Corollary 10.1 If A is a strict duality at (x0, WO) e X x W, and A' is a strict duality at (wo, xo), then we have the implications wo E a A f(x0)

xo c a A, p (wo)

wo E a A f AA, (x0).

(10.20)

If in addition, (10.8) holds (in particular; if a° f (x 0 ) 0 0), then we have the equivalences WO E

8f(X0)

X0 E

a°1 ,0(wo ) f(x 0 ) — E

Also, by (7.94) and Definition 10.2, for each e 0 sA f(x0)

0O

f AA (xo) '

3ff(x0) o 0.

(10.23)

0 we have the implication f (x0) —

E.

(10.24)

364


(d) If f (xo) = f AA ' (xo) < +cc, then, by (10.23), we have f(x0) o 0

(Es >

0).

Conversely, if (10.25) holds, then, by (10.24) and f °° '

(10.25) f , we have f (xo)

(x0)•

(e) Some of the preceding results on A-subdifferentials can be generalized to A-e-subdifferentials, in the usual way.

10.2 Subdtfferentials with Respect to a Conjugation c : Rx—> Rw, Where X and W Are Two Sets Throughout this section, we will assume that X and W are two (nonempty) sets, C: R x —> R w is a conjugation with dual c' : Rw --> R x , and ço = (pc : X x W R, (pi = (pc, W x X —> 1?- are the coupling functions associated to them by Theorems 8.2 and 8.4, so c = c((p) and c' = c((pi). Then, by Definition 10.1, applied toA= c((p) and e' = ec, :WxXxR—>Rof(8.53), the c((p)-subdifferential of f E R x at xo E X is the set = 1/1)0 E W I

CP(X0,

w0) ± — fc (') (wo) = f (x0)}.

(10.26)

Remark 10.6 (a) The set ac(99) f(X 0 ) of (10.26), obtained by requiring equality in the Fenchel-Young inequality (7.93) (with e' = cc, of (8.53)), that is, in the first inequality of (8.58), is different from (10.17), which is now (by (8.30)) the set {wo E W I ço(xo, wo) ± — f (xo) = fc (') (wo)),

(10.27)

obtained by requiring equality in the third inequality of (8.58); indeed, for example, if cp(xo, wo) = —00, fc ((°) (wo) = +oo and f(X 0 ) = —oc, then wo E ac(9') .1' (x0), but wo does not belong to the set (10.27), nor to the set obtained by requiring equality in the second inequality of (8.58). However, if f(X 0 ) E R, then all three sets (obtained by requiring equality in the inequalities (8.58)) coincide (by Theorem 10.4 below). (b) For e' = ec, :Wx Xx R —>R of (8.53), Theorem 10.1 says that given a coupling function (p:Xx14/.--›- T?, we have wo E 3f (X0 ) if and only if there exists b E T? such that 49 (•, 11)0) + — b

f,

Oxo, wo) + — 12 = f (xo);

(10.28) (10.29)

in particular, if W c Rx and (i9 = çonat , then conditions (10.28) and (10.29) become f and W0 (X0) — b = f (x0), which motivates the term "slope" used in —b Remark 10.2(a) above (take X = R and W = R*). Also, for e' = e( , of (8.53), Remark 10.3(a) shows that if acw f (x 0 ) o 0, then f E C(Vw, ço ± R).

10.2 Subdifferentials with Respect to a Conjugation c R x ----> R w

365

Theorem 10.4 and x0 E X, such that Let X and W be two sets, ço : X x W —> 172, f e f (X0) E R. For an element Wo E W the following statements are equivalent: 1 0. 2°.

3°.

WO E 0c(0) f(X0).

We have ÇO (x0 , WO) E

R,

(xo) +

(wo) = (x0, wo).

(10.30)

49 (x0 , WO) E

R,

(xo, wo) — (xo) = fe m (wo)

(10.31)

We have

(i.e., ço(x0, w0) E R and the supx , x in the definition (8.35) of fc (0 (wo) is attained for x0). 4°. We have

ço(xo , wo)

E

R, (p(x, wo) — (P(xo, wo)

.f (x) — f (xo)

(x E X).

(10.32)

2°. If 1° holds, that is, if go(xo, wo) + — fc ('°) (wo) = f (x0), Proof 1 0 then, since (by our assumption) f(x 0 ) E R, we must have cp(xo, WO) E R and fc ('') (wo) E R, whence we obtain 2°. The implication 2° = 1' and the equivalences 2° 3° 4' follow obviously from (10.26) and (8.35). Remark 10.7 (a) Condition 4° of Theorem 10.4 is convenient for aplications, since it does not contain explicitly fc ((P) (as do conditions 1°, 2°, and 3°). Following Balder [19] (who has considered the particular case go (X x W) c R), some authors use the term "(p-subdifferential of f at x0 ," and the notation aço px0 ), for the set of all wo E W satisfying condition 4° of Theorem 10.4 (with f(x 0 ) E R). However, R we will not use them here, since for a given coupling function go : X x 14/ Rx —> Rw associated we have not only the (Fenchel-Moreau) conjugation c(y9) : — R w (of (8.153)) associated to go but also the conjugation of type Lau L(p) : R x --> — to ço, which will lead to a different subdifferential aL(o f (x 0 ) (see (10.56) below). Moreover the concept of "subdifferential with respect to a conjugation" encompasses also other particular cases, for example, that of a L(°) f GO, where L (A) : R x —> R w is the conjugation of type Lau (8.254) associated to a duality A : 2 x —> 2 w ; note that one can also express a L(6 f (x o ) as (10.59), in which no coupling function occurs explicitly. (b) The assumption f(x 0 ) E R of Theorem 10.4 excludes only two easy cases. Indeed, if f(x 0 ) = —oc, then, by Remark 10.1(c), ' )

ac((P) f (xo) = v v .

(10.33)

On the other hand, if f(x 0 ) =- +0o, then, by (10.26), =

{WO E W I 49 (X0, WO) = + 00 ,

fc ('') (wo)
— 00, r (c) (wo) = —oc).

(10.34)

Hence, in particular, if f(x 0 ) = +Do and çoi is of type {0, —Do), then = ( WO E W I 401(xo, wo) = 0, r (" ) (wo) = —oc).

(10.35)

(c) Condition 2°, with ± replaced by :k in (10.30), implies that f(x0) E R. However, such a remark is no longer true for 1', 3°, and 4', even with — replaced by — in (10.32). (d) If X is a set, W c R x , and (p = (pnat , the natural coupling function (see (2.184)), then ac4n1t ) f(x0) is called the (W -)subdifferential off E Rx at xo, and it is denoted by af (x0); that is, (10.36)

af (x0) = {wo E W I wo(xo) ± — f* (wo) =

with f* of (8.37). By Theorem 10.4, when f(x0) E R, we have

af (x0) = {wo

E W I WO(xO) E

R, wo(x) — wo(xo)

f (x) — f(x 0 ) (x E X));

(10.37)

in particular, if f (x0) E R and w (X) c R (w E W), then, by (10.37), we have

af (x0) = {wo

E W I w0(x) — w0(x0)

f (x) — f(x 0 ) (x E X)).

(10.38)

0, then aec (cP"- ) f (x0) More generally, if X is a set, W c R x , = (Pnat, and s (of (10.22), for A = c((Pnat)) is called the (W-)s-subdifferential of f E RV at xo, and it is denoted by as f (4); that is, 8 Ef(X0)

= (IV E W I W(XO) ± — P(W)

f(x0) — s).

(10.39)

(e) Historically af (x0) (which has appeared before acm f (x0)) was first defined by condition (10.38) for f(x0) E R and some particular X and W c R x (see the Notes and Remarks). Finally, let us mention that if g E R W and wo E W, then, by (10.14) applied to A = c((p) and e = ec ((p) :XxWxR —> R of (8.30) (or, by (10.26) applied to the triple (W, X, (p'), with (p' of (6.138), and using (8.54)), we have ac(9'Y g(Wo) = {xo E X I (p(x o , wo) ±—g c(99)/ (x0) = g(wo)}.

(10.40)

Remark 10.8 (a) For 0`.(0 g(wo ) there hold results corresponding to those on acm f (x0). For example (see Theorem 10.4, equivalence 1' 4"), when g (wo) E R, we have xo E ac((P)/ g(wo) if and only if P(X0 , WO) E

(

R, Oxo, w) - 49(xo, wo) g(w) g(wo)

(w E W).

MAO

(b) By (8.30) (resp. (8.53)), c((p) (resp. c((,o)') is a strict duality at (xo, WO) E X X W ((Wo, X0) E W x X) if and only if ço(xo, Wo) E R, which is certainly the case


367

if Wo E acm f (x 0 ) and f (x0 ) E R (by Theorem 10.4). Hence, by Corollary 10.1, if f (xo) E R, then we have the implications (10.20) for A = c(49 ); also, if (p(xo, tV0) E R and (10.8) holds, then we have the equivalences (10.21) for A = c(ço)). 1 g(wo ) = ac (c° iat ) g(wo) (see (10.40) and (c) If W c R x and yo = (Nat , then ac (8.54)) is called the (X-)subdifferential of g E R w at wo, and it is denoted by ag(wo); that is,

0 g(WO) =

PC0 E

with g* of (8.55). When g(wo)

E

X I wo(xo) ± — g * (x0) = g(wo)1,

(10.42)

R, we have

ag(w o ) = Ixo E X I wo(xo) E R, w(x0) — wo(xo) g(w) — g(wo) (w E W)).

(10.43)

Note that if the canonical mapping K : X --> W (of (2.140)) is one-to-one, then (10.42) is consistent with (10.36) applied to W and X = K (X) C R w .

10.3


In this section we will give, for some particular cases, descriptions of the sets f(x 0 ) (and, sometimes, 3c(9')" g(wo)), and conditions in order that ac@P) f(4)

(resp.

ac(`Pr g(wo) 0 0).

10.3a

The Case Where X = {0, 1}", W = (R n ) * Ix, and ço = çonat

If X = {0, 1)°, W c (R n ) * x (C R X ) and ç = (Pnat, then, by the above, for any f E R X and xo E X we have ac(c9,-) f (x 0 ) = af (x 0 ) (of (10.36)). Moreover, if f (x0) E R, then, since w(X) c R (w E W), we have (10.38). Theorem 10.5 Let X = {0, lr and W = (R")*I x . For f following statements are equivalent:

1°. 2'.

af (xo) A

E

R x and xo E X with f (x0) E R, the

0.

We have f (x) E R U

(x E

X \ {x0}).

(10.44)

Proof 10 2°. If 1° holds, say, Wo E af (x0), then, by wo (X) C R, f (x0) E R and (10.38), we have (10.44). 2° = 1°. If 2° holds, then, by Theorem 8.9 and formula (10.36) we have Wo E El af (x0), for each Wo E (R")* x = W for which the max in (8.92) is attained. Corollary

10.2

Let X and W be as in Theorem 10.5.

368


(a) For f E le and x o E X with f (x0) E R, the following statements are

equivalent: 1°. 8f (x) 0. 0 2 . f = f**. 3° .

f(x0) = f** (x0).

(b) For f E R x , the following statements are equivalent:

There exists xo E X such that f (x0) 2 0 . f =- f** and f 7L +00. 10 .

E

R,

af ( 4 )

0.

Proof (a) The implication 1° = 2° follows from Theorems 10.5 and 8.9. The implication 2° 3° is obvious. Finally, by Theorem 8.6, Definition 3.5, and Corollary 3.2, we have the equivalences f (x0) = f** (x0) N),

whence, by Theorem 10.7, implication 3' = 1', we obtain (N 1 > N).

ac((P.

f (x 0 ) o

10.3c The Case Where X Is a Metric Space, W = X X (R + \ {0}), and cp = cpc, Let X = (X, p) be a metric space, W = X x (R ± \ (0)), 0 < a (pa : X x W ---> R be the coupling function (8.111).

1, and let

Remark 10.9 For f E R X , xo, y E X, and 0 < N < -i-oo we have, by (10.26) and Remark 8.18(a), (c), the equivalence (Y, N) E

f (x0) Y E

ac' ° a N )

f (x0).

Theorem 10.8 For f

E

R x and X0 E X the following statements are equivalent:

(10.49)

370


10.

20 . 3°.

ac(c)f(x)

0. There exists N > 0 such that (x0, N) E ac("' ) f (x0). There exists N > 0 such that we have (10.47).

Proof This result follows from Remark 10.9 and Theorem 10.7.

10.4

LI

The Subdifferential of f -i- —h at xo, Where f,h E Rx and xo E X

Proposition 10.3 Let X be a set, let W C R x satisfy (8.117), and let ço = yoflat (so acm f (x 0 ) af (4)). Then for any f, h E R x and xo E X with f(4), h(x0) E R, we have

af (x0) + a (—h)(x0) g Proof Let w 1

E af (x 0 ), w 2 E

(10.50)

3(f + —h)(x0)•

a (—h)(x 0 ). Then, by Remark 10.7(d), we have f (x)

(x E X),

—h(x)

(x E X),

w (x) — wi (xo) Pxo) w2(x) — w2(x0) — h(x0)

whence (v) + w2)(x) — (u) + w2)(4) + f(x 0 ) — h(x0)

(f 4- —h)(x)

Therefore, by (8.117) and Remark 10.7(d), w1

E

8(f

+ —h)(x0)•

Since 8f(x 0 ) has been defined with the aid of f* (see (10.36)), it is natural to —h)*, such as those of Section 8.5, may yield expect that a suitable formula for (f a formula for a ( f —h)(x 0 ), where f, h E Ti." and xo E X. Indeed, for example, there holds: Proposition 10.4 Let X be a set, and let W C R x satis W = —W and (8.117). If f, h (8.143), then for any xo E X with f (x 0 ), h(x0) E R, we have

8(f

+ —h)(x0) = af (x0) +

a (-17)(x0)•

E

k" satisfy

(10.52)

Proof By Proposition 10.3, we have to prove only the inclusion c in (10.52). Let w0 E 0(f + —h)(x0). Then, by W c Rx , f (x0), h(x0) E R and Theorem 10.4, implication 1° = 2' (with yo = yonat ), we have f(x 0 ) — h(x0) + (f

—h(w 0 ) = wo(x0);

(10.53)

10.5 Subdifferentials with Respect to Conjugations of Type Lau

371

also, by (8.143), there exists Vo E W such that

(10.54)

( f + —h)* (wo) = f*(wo — vo) - .F- (—h)*(v0)• Combining (10.53) and (10.54), we obtain f(x 0 ) — h(xo)

f*(wo — vo) - .F- (—h)*(vo) = wo(x0)

(10.55)

= (wo — vo)(4) + vo(x0),

whence, by the second inequality of (8.58) (with ço = çonat ),

f(x0) + f* (wo — vo) = (wo — vo)(xo),

— h(xo) + ( — h) * (vo) = vaxo)•

Therefore, by Theorem 10.4, implication 2' 1', we have WO — VO E af (x 0 ) and o VO E a ( — h)(x0), and hence wo = (wo — yo) + VO E 8f(x 0 ) + a (—h)(x0)•

10.5 Subdifferentials with Respect to Conjugations of Type Lau Let us consider the conjugation of type Lau c = L((P) of (8.153).

Theorem 10.9 Let X and W be two sets, ço : X x VV ---> ii a coupling function, f E W x and x0 E X with f(4) > —oc. Then a L() f (x0) = {wo E W I ça(xo, wo) > —1 , f (x0) = — f L( `P) (14)1 = {WO E W I 40 (X0, WO) >

—1, f(x0) =

min

xEx (p(x,w())>- I

f (x)}. (10.56)

Proof Let 0 = ((p i ) kid, the unique coupling function of type (0, —Doh equivalent to (p. Then, by (6.176) and (2.156), we have the equivalences SOI(X0 , WO) E

R goi(xo, wo) = 0 yo(xo, wo) > —1.

(10.57)

Hence, if f(x0) E R, then, by Theorems 8.13 and 10.4, we obtain = ac("' ) f (x0) = {wo E W I 0 (X0, Wo) E R, f (xo) + f c*P1) (wo)=(Pi(xo, wo))

= {WO

E W I 40 (X0, WO) >

—1, f(x0) + f L( '') (wo) = 0),

which, together with (8.153), yields (10.56). On the other hand, if f(x 0 ) = +oo, then, by Theorem 8.13, (10.35), and (10.57), we get a L( ") f (x0) = 0c (991) f (xo) = {WO E W I çoi (xo, wo) = 0, fc (° ) (wo) = —oc)

= {wo E W I ÇO(X0, WO > — 1, f"" ) (wo) = — f (x0))•

Ill

372


For conjugations of type Lau written in the form L(Q) (of (8.172)), from Theorem 10.9 we obtain:

Corollary 10.4 Let X and W be two sets, QcXx W, f c R x and x0 e X with f(x0) > —oc. Then aL(Q) f (x0) = {WO G W I (X0, WO) G Q,

= {WO e W I (X0, WO) G Q,

f (x()) = — f L(Q) (wo))

f (xo) =

min f(x)). (10.58) xcx (x,w0ES2

Proof By (8.173) and (6.149), we have L(Q) = L(40s2), with çoQ = —n, so ços2(x, wo) > —1 if and only if (x, wo) E Q. Hence, by (10.56), we get (10.58). D For conjugations of type Lau written in the form L (A) (of (8.254)), we obtain

Corollary 10.5 Let X and W be two sets, A : 2 x ---> 2 w a duality, f E -12-x and xo E X with f(x0) > —oc. Then aL(A)f( x0 ) = {Wo e W I XO

E X \ 6,' ((w0)), f(x0) = — f L(A) (wo))

= {WO G W I X0 E X \ A i (tWO}),

f (xo) = min f (X \ A l (two)))}. (10.59)

Proof By (8.253), we have L(A) = L(4 0A), with ÇOA of (6.131), so cpA(x , wo) > —1 if and only if x E X \ A'((w 0 }). Hence, by (10.56), we get (10.59). D Remark 10.10

(a) One can also write (10.59) in the form

aL(A)

f (x 0 ) = tWo E W 1 X0 E X \ 6,' ({w 0 }), f (x)

f(x0) (x E X \ 6/({w0}))).

(10.60)

(b) In particular, from Corollary 10.5 it follows that if f(x 0 ) > — 00 and = 0, (whence X \ Al ( { wo)) = X), then wo E aL(A) f (x0) if and only

if

f (xo) = min f (X).

(10.61)

Let us observe now that if f (x0 ) > —oc, then, by (10.59), we have wo E aL(A) f (xo) if and only if xo is an optimal solution of the constrained infimization problem

a = inf f (X \ A l ({wo}))•

(10.62)

10.5 Subdifferentials with Respect to Conjugations of Type Lau

373

Hence, using any characterization of optimal solutions of infimization problems, one obtains necessary and sufficient conditions in order that wo E aL(A) f (x0 ). For example, one can use the characterization of optimal solutions, with the aid of level sets, given in the following simple consequence of Lemma 8.6(a):

Lemma 10.1 Let X be a set, 2 c X, f :X —> R and xo c E. The following statements are equivalent: f (x 0) = min f (2). For each À E R, À < f(x 0 ) we have 2 3° For each À c R, À < f(x 0 ) we have E

1'. 2'.

n A x ( f) = 0. n S),( f) = 0.

Proof By (4.26) and Lemma 8.6(a), we have the equivalences

2° oc for all X0 E X), (10.66), and (6.8), we obtain (10.65). Finally, if G = A'A(G) and xo V G = A'A(G), then A((x01) 2 A(G) (since otherwise we would have X0 E A'A(G)), so (W \ A({xo})) n A(G) 0 0, whence, by (10.65), aL(A) xG(xo) 0 O. —

375


10.6


10.6a L() -Subdifferentials, for Minkowski-Type and Parametrized Minkowski-Type Set-Dualities A Proposition 10.6 Let X be a set, .A4 C 2, AM : 2 X —> 2m the Minkowski-type duality (6.25), L(Am) : R x R -A4 the conjugation of type Lau (8.271) associated to Am, f E R x and x0 E X with f (xo) > —oc. Then

aL(A m )

r (x0

) = {M0 E M I X0 E X

Mo, f(x0) =

f

(A )}

= (MO E M I X0 E X \ M0 , f(x 0) = min f (X \ Mo)).

(10.67)

Proof By (8.253), we have L(Am) = L(ÇoA m ), with ÇOp m : X x A / 1 —> R of (6.184), so (pA m (x, Mo) > —1 if and only if x e X \ Mo. Hence, by (10.56), we obtain (10.67). Lii Remark 10.12 In particular, from Proposition 10.6 it follows that if f (xo) > —oc and 0 E M, then 0 E a L(Am) f(xo) if and only if we have (10.61). Proposition 10.7 Let X be a set, »I c 2 x , (W, 9) a parametrization of M, Ao : 2x ---> 2 w the parametrization (6.44) of the Minkowski-type duality Am, L(A0) = L(W, 9) : Ttx —> R w the conjugation of type Lau (8.254), (8.276) associated to A 0 , f E R x E X with f(x 0 ) > —oc. Then andx0

aL(A 0 ) f ( xo )

aL(W,O) f ( xo )

= {WO E W I xo E X \e(w 0 ), f (x0 ) = — f L(A " ) (w0)} =

two E W I Xo

E X \ 9 (W0),

f(x 0 ) = min f (X \ 0 (wo))1. (10.68)

Proof By (8.253), we have L(A0) = L((l)A,), with (PA0 = (PL(w,o) of (8.194) (by (6.131) and (6.45)), so çoAo (x, wo ) > —1 if and only if x E X \ 9(w 0 ). Hence, by (10.56) and (8.276), we obtain (10.68). LI Remark 10.13 If f(x0 ) > —oc, then, by Propositions 10.6 and 10.7, for any parametrization (W, 9) of M (so M = 9(W)) we have aL(A o )f (x0 )

t wo E

9 (w0 ) E 3L(A .A,Of (x0 ))

9-1 ( 3L(A .A4 )

f (xo))•

(10.69)


376

10.6b Subdifferentials with Respect to Quasi-Conjugations; Quasi-Subdifferentials Proposition 10.8 Rx* xR Let X be a locally convex space, (W, 0) the pair (8.199), L(W, 9) : RA' the Greenberg-Pierskalla-Crouzeix quasi-conjugation (8.200), f 1-7x and xo E X with f(x0) > —oc. Then

a L(w ' 9) f (xo) =

I

( (Do, do) G X* x R do

n

G

(.0(x), .0(x0)] ,

-vGAr(A0)(f)

(10.70) where, A+00(f) = dom f (by (2.109)) and 04)(0), '2130(x0)] = {d E R I (1)0(x) < d (13 0(xo)}.

Proof By (10.68) applied to (W, 0) of (8.199), we have a L(w ' 9) f (xo) = {( 431 o, do) E X* x R I Oo(xo)

do, f (xo) =

min

xex cr) 0(x)

f (x)}

= t(cDo, do) E X* x R I (13 0(x) el)

(10.75)

of X* x R onto X*.

Corollary 10.8 Under the assumptions of Proposition 10.9 we have the equivalence

aL(wm f (x 0 )

o 2x of Definition 2.3 have been introduced, with a different terminology, by Moore [202]. Without aiming to give a history, let us only mention here that the theories of such families B and operators u have developed in several directions. For example, one direction has been to regard the families B C 2x satisfying (2.26) and the hull operators u : 2x —> 2x of Definition 2.3 as generalizations of the family of all closed subsets of a topological space X and of the usual (Kuratowski) closure operator u, respectively (indeed, adding to (2.26) the condition that the union of any two sets in B should belong to B, one obtains the usual axioms for the closed subsets of a topological space, and, adding to (2.13)—(2.15) the condition u(G 1 U G2) -= U(Gi) U U(G2) for any GI, G2 C X, one obtains the axioms of Kuratowski for the closure operator in a topological space), and later to regard the hull operators u : >2 x as simultaneous generalizations of the topological closure and of the convex hull operators; for example, see a series of papers by Hammer ([1 l 0]4112], etc.). In another direction, the families B c 2x satisfying (2.26) have been regarded as generalizations of the family of all R-closed subsets of a set X, where R. is a set of finitary operations in X, in the sense of "general algebra" (of [26]), and the members B 0 of B (and, in some cases, B = 0, too) have been called "ideals"; for example, see the papers of Schmidt, who has mentioned the family B of all convex subsets of the plane X too as an example of an "ideal" ([245], p. 80, example 2; [246], p. 38, example 9). Apparently the first paper where for a family B c 2x satisfying (2.26) the pair (X, B) has been called a "convexity space" and the sets B E B "convex sets" was that of Levi [157]. In this direction there have been many contributions; for example, see Jamison ([127], [133], etc.), de Smet [48], Sierskma [252], van Maaren [295], Soltan [279], Evers and van Maaren [80], van de Vel [293], and the references therein. In some of these works there have been given interpretations and applications of the theory of abstract convexity to various domains, such as modules, graphs, matroids, ordered sets, number theory, ordered fields, approximation theory, mathematical logic, projective geometry, mathematical linguistics, and mathematical systems theory. For some contributions to abstract convexity theory, obtained in Romania, see Ghika ([95]—[97]), Voiculescu ([298], [299]), Precup [228], and Cipu [39], among others. In the particular case when X is a topological space (see Remark 2.3(c)), a family B C 2" of closed subsets of X that is stable under arbitrary intersections, is called by Wieczorek [312] (see also Keimel and Wieczorek [138]) a convexity on X, and the sets B E B are called closed convex sets. A convexity B C 2 X is said to be regular

390

Notes and Remarks

[138] if for each G E B and x V G there exists B E B such that G c Int B, x X \ B. Van de Vel ([292], [293]) gave conditions of compatibility between a topology and a convexity system B on a set X. Following van de Vel [292], a pair (X, B) is called a topological convexity structure if X is a topological space, (X, B) is an alignment (see

Definition 2.18 below) and if the (B-)convex hull of every finite set is closed (this is a slight generalization of a concept of topological convexity structure introduced by Jamison [127]). However, we do not consider here convexity theories for topological spaces X; for convexity concepts in topological spaces, see Fuchssteiner [91], Komiya [146], among others. Now we will mention some complements to Section 2.1, describing briefly some further developments related to hull operators and convexity systems in the theory of abstract convexity (we will not consider here "extended topologies," "algebraic closure systems," etc.). Definition 2.17 Let X be a set. (a) A hull operator u : 2x 2x is said to be algebraic (e.g., see Schmidt [245], Evers and van Maaren [801) if u(G) = U{u(F)I F

C G.

IFI 2)( and every convexity system B C 2 X is algebraic. Theorem 2.10 Let X be a set. (a) A hull operator u : 2x —> 2 x is algebraic if and only if for any nonempty chain (i.e., any nonempty family, totally ordered by inclusion C) of subsets {G of X we have

u G i) = U u(G i ) * u( iEr iEr

(2.220)

Notes and Remarks

391

(b) A convexity system 13 C 2 X is algebraic if and only if it is "inductive" (i.e., the union of any nonempty chain of subsets of 13 belongs to 8). Proofs of Theorem 2.10 and further characterizations of algebraicity, as well as other related results, can be found in Schmidt [245], Hammer [112], Jamison [127], Soltan [279], Evers and van Maaren [80], among others. For example, for every convexity system 8 in X there exists a smallest (with respect to inclusion) algebraic convexity system in X, containing 8, that admits some simple descriptions (see Sierksma [252], p. 13, or Soltan [279], p. 19, and the references therein). On the other hand, a convexity system 8 contains a maximal algebraic convexity (sub)system if and only if 8 itself is algebraic ([279], p. 19). Definition 2.18 An algebraic convexity system 8 c 2x satisfying (2.221)

coB = 0,

is called an alignment on X and any such pair (X, 8) is called an aligned space (Jamison-Waldner [133]). In [133] aligned spaces are considered as the natural framework for the theory of abstract convexity. The following strengthening of the concept of an algebraic hull operator has been also studied: A hull operator u : —> 2 x is said to be domain bounded, with domain bound n (Hammer [1121), if there exists a smallest integer n such that u(G) =U{u(F)I F C G, I Fi

nJ

(G

C

X).

(2.222)

For example, the (usual) convex hull operator in a locally convex space X is domain bounded, with domain bound 2. Generalizing the theory of hull operators, Wieczorek [311] has developed the beginnings of a theory of "spot operators" (which he called, in [311], "spot functions"). Given a set X, a spot operator in X is any operator u : dom u —> 2 x , where dom u (the "domain" of u) is a family of subsets of X (e.g., it may be the family of all finite subsets of X, or of all two-element subsets of X). A set G C X is said to be (a) u-convex if u(F) c G for every F E dom u with F C G; (b) u-convexoidal if there exists F E dom u such that G = u(F). If F C u (F) for each F E dom u, then every u -convex set in dom u is u-convexoidal 13111 On the other hand, every u-convexoidal set is u-convex if and only if the relations F, G E dom u, F C u(G) imply that u(F) C u(G), and this holds whenever we have the implications F, G e dom u, FcGu (F) c u(G) and F dom u u(F) dom u, u(u(F)) = u(F) [311]. Also ([311], prop. 5) the family of all u-convex sets is stable under arbitrary intersections, and if every element of dom u is a finite set, then the union of every upward directed (by inclusion) family of u -convex sets is also u -convex. For further results on spot operators and applications to Krein-Milman type theorems, see [311].

392

Notes and Remarks

Given a set X, a convexity system B C 2 X is said to be n-ary (Soltan [279], p. 20), if

B = {B c X I coB F

B (F

B, F n)}. n)}.

(2.223)

(G C X),

(2.224)

Defining, for any convexity system B C 2 X ,

u n (G) = U{coB F I F C G, I Fi n} u(G) = G, u`n+1 (G) = un (u n (G))

(G C X, i = 0, 1, 2, . . .), (2.225)

we have ([279], thm. 2.8) that a convexity system B C 2X is n-ary if and only if CKD

coB G =

(G C X).

u 1 (G)

(2.226)

j=0 Using this result, it is shown in [279] that for each n 1 there exists an n-ary convexity system which is not (n — 1)-aty, and that every n-ary convexity system is algebraic. A convexity system B C 2 X is said to be (finitely) join-hull commutative (see Kay and Womble [137] and Ellis [71], respectively), if for each (finite) subset G of X and each X E X, we have

coB({x} U G) =

coBlx, y);

(2.227)

yEcoBG

here the right-hand side is called the (8-)join of x and G (in particular, coB{x , y) is the join of x and y, or the "segment" (x, y)). Clearly join-hull commutativity implies finitely join-hull commutativity. One can show that if B C 2 X is an algebraic convexity system, then the converse is also true [137]. Also, if B C 2X is a join-hull commutative algebraic convexity system, then a subset G of X is convex if and only if coB[x, yl C G for all x,y E G (i.e., if and only if G is "segmentally convex") [137]. For a convexity system B C 2x , the convex hull of a finite subset of X is called a B-polytope. A convexity system B C 2X is said to satisfy the cone-union condition (Sierksma [252]) if for each X E X and each finite collection of B-polytopes {B, B 1 , . . . , Bn with B C LL 1 B i , we have

COB({X}

U B)

C

COB({X} U

B i ).

(2.228)

One can show [252] that every finitely join-hull commutative convexity system B g_ 2' satisfies the cone-union condition, but the converse is not true [252]. For a convexity system B C 2, a subset G of X is said to be (a) star-shaped with respect to go E G if coB lgo , g) C G for all g G G; (b) star-shaped if there exists such a go E G (e.g., see Soltan [279], pp. 38-40, and the references therein).

Notes and Remarks

393

Extending the terminology of algebra to a convexity system B c 2x (where X is an arbitrary set), an element X E X is said to be (e.g., see Schmidt [247]) dependent (resp. independent) on a subset G of X, if x e coB G (resp. x ■;Z coB G). A subset G of X is said to be independent if no element g e G is dependent on the set of the other elements of G, that is, if g V coB(G \ {g})

(g E G);

(2.229)

otherwise, G is said to be dependent [247]. Clearly the property of independence is "hereditary"; that is, if G is independent, then so is every subset of G. One can show (see [245] or [279], thm. 3.15) that for an algebraic convexity system B C 2x , a subset G of X is independent if and only if so is every finite subset of G. Starting with van der Waerden [294] (see also Marczewski [167]), various axiomatic theories of independence in abstract algebras, and other concepts of independence in convexity systems, have been also studied (see e.g. [247], [279], and the references therein). As can be seen from the above, for some time the theory of general convexity systems B C 2x has developed independently on that of the convexity systems B = C(M), where M c 2 x . The latter systems have been chosen as the main objects of study of Section 2.1 because of their versatility, due to the possible choices of M (see the great variety of particular cases mentioned in Section 2.2), and because of their usefulness in optimization theory, where one of the main tools is separation. Let us also recall that, by Remark 1.12(c), the theory of general convexity systems B C 2 X is equivalent to that of the convexity systems B = C(M), where M c 2 x . For a convexity system B C 2x bases M of B (see Definition 2.5) and B-semispaces (see Definition 2.6) have been considered by several authors. A Bsemispace at Z E X is also called (Jamison [130], [133]) a copoint attached at z. B-semispaces are also called "absolutely irreducibles" [246], or "copoints" [133]. Theorems 2.4, 2.5, and Corollary 2.1 are due to Soltan (see [279] and the references therein). The following result of Hammer (1112], thm. 7 or [111], cor. 2.5) shows another reason of the importance of algebraic convexity systems, in the sense of Definition 2.17(b) above. Theorem 2.11 Let X be a set and B C 2 x an algebraic convexity system. Then B has a (unique) smallest base consisting of all B-semispaces. For a related version of Theorem 2.11, see also Schmidt ([245], thm. 7). Hammer [112], [111] has observed that there are many natural examples of algebraic convexity systems, especially in algebra (the subgroups of a group, the subrings of a ring, etc.) for which it is useful to apply Theorem 2.11. Let us also mention the following result (Soltan [279], Imm. 3.2): Proposition 2.11 Let X be a set. A convexity system B C 2x coincides with its smallest base if and only i f B is well ordered with respect to inclusion C (i.e., B is totally ordered, and every nonempty subfamily g of B has a smallest element).

Notes and Remarks

394

Extending the usual linear space terminology to a convexity system B C 2. a point g e G satisfying (2.229) is called (e.g., in [279 ] ) a (B-)extreme point of G (hence G is independent if and only if every point of G is an extreme point of G). However, although this is an extension of the concept of an extreme point of a set G in a linear space X (taking as B the family of all usual convex subsets of X), it is not suitable for convexity systems B which are not algebraic. Indeed, if X is the plane R2 and B is the family of all closed convex subsets of X, then coB G = E5 G, and hence, for example, the closed unit disk G in X has no extreme points in the above sense (i.e., no points g E G satisfying (2.229)), although every point of its boundary (the unit circumference) is an extreme point of G in the usual linear space sense. Extending some linear space results of Hammer [111], it has been shown in Soltan ([279], thm. 3.5), that if X is a set, B C 2 X is an algebraic convexity system, and G C X, then (1) g E G is an extreme point of G if and only if there exists a B-semispace Bg at g such that G \ {g} C Bg ; 2) denoting by ExtBG the set of all extreme points of G, we have G = coB(ExtB G) if and only if for each g E G and each B-semispace Bg at g there holds (X \ B g ) n ExtB G 0 0. Furthermore ([279], thm. 3.6) we have (

ExtB G =

n IF c

GI coB F = coB GI

(G

C

X).

(2.230)

For a general convexity system B C 2x , various separation properties have been studied. A convexity system B C 2 X is said (Jamison-Waldner 11331) to satisfy axiom To (resp. TO if for any x,y e X with x 0 y we have coB Ix ) n coBly ) = 0 (respectively, if for each x e X, the singleton {x} is convex); thus Ti = To. The following statements are known to be equivalent (see Pereira [222] or Soltan [279], p. 33): 1 0 . B satisfies axiom T1 . 2'. For any x,y E X with x 0 y there exists B E B containing only one of x, y, and for any G E B and x g G there exists B G B such that GcX\B,xVX\B

(2.231)

(i.e., G can be separated from x by the complement of a convex set). A subset B of X is called a B-hemispace (e.g., see Jamison [131], [133] and the references therein) if both B E B and X \ B E B. Both B-semispaces (e.g., see Theorem 2.5 and Remark 2.2) and B-hemispaces are useful objects for separation. A convexity system B C 2 X is said [133] to satisfy axiom T4 if any two nonempty disjoint convex sets GI, G2 can be separated by a B-hemispace. One can show (e.g., see Soltan [2791) that if I XI > 1 and if a convexity system B C 2x satisfies axioms T1 and T4, then every B-semispace is a B-hemispace; it is known (see [222] or [2791) that there exist convexity systems satisfying T1 , but not T4. Let us also mention that a convexity system B C 2 X is said [133] to satisfy axiom T2 (resp. T3 ) if any two distinct points x,y E X can be separated by a B-hemispace (resp., if for any G E B and x g G there exists a B-hemispace B satisfying (2.231)). However, note that similarly to the case of B-semispaces, for some natural convexity systems B C 2"

Notes and Remarks

395

there exist no nontrivial B-hemispaces; indeed, for example, if B is the family of all closed convex subsets of a locally convex space X, then there exists no closed convex set B with closed convex complement X \ B, except B = X and B = 0. Some important classes of convexity systems B C 2x that have been thoroughly studied are those satisfying the "exchange law" or the "antiexchange law." An aligned space (X, B) (see Definition 2.18) is called a matroid (or a combinatorial geometry), if it satisfies the following exchange law: For any B E B and x, y g B, we have the implication y

X E COB(B U {y}) =

E COB(B U

{x }).

(2.232)

An aligned space (X, B) is called an antimatroid [133] (or, a convex geometry [671) if it satisfies the following antiexchange law: For any B E B and x,y g B with x 0 y, we have the implication x E coB(B U {y }) = y g coB(B U {x}).

(2.233)

The convexity systems B C 2x satisfying (2.233) are said to be extremally detachable (Jamison-Waldner [133]). For example, a linear space X, together with the family B of all affine subsets of X, is a matroid, while (X, B), where B is the family of all usual convex subsets of X, is an antimatroid. For the theory of matroids, see e.g. Welsh [307]. For some results of the theory of antimatroids, see Jamison 1130] and Edelman and Jamison [67]. Let us mention here that for any aligned space (X, B), the following statements are equivalent: 1°. (X, B) is an antimatroid. 2'. For any z E X and any B-semispace B, at z, we have B. u {z} E B. 3°. For any B-semispace B in X there exists at most one point Z E X such that B is the B-semispace attached at z ([130], thm. 1). On the other extreme, in any matroid (X, B), each B-semispace B is attached at all points Z E X\B [130]. An aligned space (X, B) is simultaneously a matroid and an antimatroid if and only if B = 2x , "the free alignment," that is, if and only if every subset of X is convex ((130], thm. 2). Various numerical parameters associated to convexity systems B C 2x have been also much studied, such as the "Helly number," the "Radon number," the "Carathéodory number," and the "exchange number," (e.g., see [252], [133], [279], and the references therein). Jamison-Waldner11331 introduced and studied "varieties of alignments." If (X, B) is an aligned space and Y is a subset of X, then the "restriction of B to Y," defined by Bly , {BnYIB

E B } (c 2 Y ),

(2.234)

is an alignment on Y, with associated hull operator coB ly (G) = (coB G)

n

Y

(G c Y),

(2.235)

396

Notes and Remarks

and the aligned space (Y, Bly) is called a subspace of the aligned space (X, 8) [133]. An isomorphism between two aligned spaces (X ] , Bp and (X2, 82), is a one-to-one correspondence between X ! and X2, preserving convexity in both ways. A variety of alignments is defined [133] as a class V of aligned spaces with the following three properties: (V!) Any aligned space isomorphic to a space in V is also in V. (V2) Any subspace of a space in V is also in V. (V3) If every finite subspace of an aligned space (X, 8) is in V, then (X, 8) E V. For example, the class of all matroids, and the class of all antimatroids are varieties of alignments [133]. By axiom (V3), if V is a variety of alignments and (X, 8) is an alignment, (X, 8) g V, then there exists a finite subspace (F, OF) of (X , 8 ) such that (F, O r ) g V, and hence there exists a "minimal forbidden subspace" (Y, y) of (X, 8) i.e., such that (Y, Bl y ) SZ V but every proper subspace of (Y, Bly) is in V. Hence, if one knows all minimal forbidden subspaces for V, then one has a characterization of V (namely, (X, 8) e V if and only if (X, 13) has no subspace isomorphic to one of the minimal forbidden alignments for (X, 8)). In [133] there are given such "minimal forbidden structure" characterizations for various concrete alignments (X, B). 2.2 Semispaces in linear spaces have been defined by Hammer (see [109], [111])

and, independently, in the finite-dimensional case, by Motzkin [209]; in the twodimensional case they have appeared also in Schmidt ([245], p. 180, example 2; [246], p. 38, example 9) as an example of "irreducible ideals" in the "hull system" B of all convex subsets of the plane X = R 2 . The semispaces in linear spaces are also called "hypercones," in K6the [148]. The structure of semispaces in linear spaces has been investigated by Hammer [109], [111], Klee [140], Moore [201], Jamison [127], [131], and others. Hammer showed that the family M of all semispaces in a linear space X is the smallest (intersectional) base of the family 8 of all convex subsets of X ([109], cor. 4). Actually, as noted by Danzer, Griinbaum, and Klee 1451, it was this result, together with the known consequence (discovered much earlier) of the usual separation theorem (or, equivalently, of the Hahn-Banach theorem), that the family M of all closed half-spaces in a locally convex space X is a base of the family B of all closed convex subsets of X, that led them to Definition 2.1 of M-convex subsets of a set X, where M is an arbitrary family of subsets of X. Hemispaces (i.e., convex sets with convex complements) in linear spaces, which are the particular case of the 8-hemispaces (mentioned after (2.231) above) in which X is a linear space and B is the family of all (usual) convex subsets of X, have been studied by Hammer [113], who called them "demispaces." He has shown that they are generalizations of semispaces, namely maximal convex sets disjoint from a given affine set V (see [113], thm. 3.2); semispaces are the particular case in which V consists of one point. The structure of hemispaces in I?' has been studied in [180], where several geometric characterizations of hemispaces and several ways of representing them with the aid of linear operators and lexicographical order (various extensions of (2.39)) are given, as well as a metric-affine classification of hemispaces

Notes and Remarks

397

in terms of their "rank" and "type." Hemispaces in linear spaces have been studied, with other methods, in Lassak [153 ] and Lassak and Pr6szyfiski [154], where they are called "convex half-spaces;" see also Dupin and Coquet [66]. Evenly convex subsets of R" were introduced by Fenchel [84] and rediscovered independently by Martfnez-Legaz [168], [170], who called them "normal convex sets." For applications of evenly convex sets in optimization theory, see e.g., Passy and Prisman [213]. Coaffine subsets (i.e., intersections of complements of hyperplanes) of linear spaces over an arbitrary field have been studied by Jamison [129], but in Section 2.2e we have considered only evenly coaffine subsets (defined as intersections of complements of closed hyperplanes) in real locally convex spaces. Jamison showed ([129], thm. 3) that every connected coaffine set in a Euclidean space is convex, but Proposition 2.2(b) and Corollary 2.2 state slightly more (even in a Euclidean space). The idea of regarding the families of all order convex sets in a poset (which is a "segmental" convexity concept) and of all order ideals in a poset, as alignments, can be found, for example, in Jamison-Waldner [132], [130], where they are called the order alignment and the downset alignment, respectively. Propositions 2.4 and 2.6 are stated, without proof, in Jamison [130], pp. 538 and 539, respectively. The intersectional base (2.77) for the order convex alignment (i.e., Proposition 2.5(a)) was given in [183], prop. 4.2. In the converse direction, Jamison-Waldner [132] has considered the problem of characterizing the order convex subsets of posets in terms of aligned spaces. An aligned space (X, B) (see Definition 2.18) is said to be determined by an order [132] if there exists an order ( on X such that B coincides with the family of all order convex In [132] Jamison-Waldner gave some characterizations of aligned subsets of (X, spaces (X, B) determined by an order. For example ([132], thm. A), an alignment B on a set X is determined by a total order if and only if (a) each independent subset G of X (in the sense (2.229)) has cardinality 2; (b) (X, B) satisfies the antiexchange law (2.233); (c) (X, B) satisfies the separation axiom T2 (mentioned after formula (2.231)). Also ([132], thm. B) an alignment B on a set X is determined by a partial (resp. a total) order on X, if and only if for any finite subset Y of X, the alignment B1 (of (2.234)) is determined by a partial (resp. a total) order on Y; moreover in the case of total orders, one can replace here "finite subset Y of X" by "subset Y of X with at most 4 points" ([132], thm. C). Edelman and Jamison have given the following characterization of order ideals of finite posets in terms of alignments ([671, thm. 3.2): For a finite convex geometry (X, B) (see (2.233)) there exists a partial order on X such that B is the downset alignment on X under this partial ordering if and only if co8(G1 U G2) = cos G1 U cos G2

(G1, G2 C

X).

(2.236)

Let us mention that results of the above type have been given not only for order structures on X but, for example, also for linear structures. Indeed, the problem of obtaining necessary and sufficient conditions on a pair (X, B), where X is a set and B C 2 )( is a convexity system, for X to be a linear space for which the members of 13 are the (usual) convex sets has been called, by Kay and Womble [137], the linearization problem. A solution to this problem has been given by Mah, Naimpally, and Whitfield [165], who have shown that for a convexity system (X, B) that is domain

Notes and Remarks

398

finite, join-hull commutative (see (2.227)) and such that

.x, y, z e

x,

coBtx, yI = coB{z, y}

x --- z,

(2.237)

there exists a (real) linear space structure on X such that B is the family of all convex sets generated by this structure if and only if there exists a family of functions X* C R x (called, in [165], a linearization family for X) with the following properties: (a) each w E X* is "convexity preserving" (i.e., such that w(B) is a convex subset of R, for each B E B); (b) we have {x} = n w _i w(x) w Ex. and there exists a distinguished element w (x0) = 0 (c) the restriction of each w

E

.X0 E

( Y E X),

(2.238)

X such that

(w

E

(2.239)

X*);

X * to any "line," that is, to any subset of X of the form

C(x, y) = (z e X I either z e coB{x, y} or x E coB(z, YI or y

E COBIZ,

xll

(2.240)

(where x, y e X) is either a bijection or a constant; (d) the relations x, y c X, w, w' E X*, w(x) $ w(y), w' (x) 0 w'(y) imply the existence of scalars X, p. G R such that w l (z) = Xw(z) ± p, for all z G C(x, y) (of (2.240)). For some other solutions of the linearization problem, see Meyer and Kay [197], Hammer [114], Guay and Naimpally [108], Szafron and Weston [282], and Whitfield and Yong [309], among others. Concerning the concept of order convexity in posets, in [183], prop. 2.1., it has been shown that there exists no order relation on R" (n ?., 2) for which the order convex subsets of (Rn, ) are the usual (vector) convex subsets of Rn (however, as a complement to the proof given in [183], let us note that it should start with the following observation: If for some order on Rn the order convex sets are the usual convex sets, then must be a total order; indeed, if there existed x 0 y in Rn that are not comparable for .., then the two-element set (x, y} would be order convex and hence convex, which is impossible.) Therefore, in [183], instead of posets, the natural framework of multi-ordered sets has been considered (by [1831, def. 0.1, a multi-ordered set is an ordered pair (X, 0), where X is a set and 0 is a nonempty family of partial orders on X), and two concepts of convexity, segmental and separational convexity, of subsets of multi-ordered sets have been introduced and studied. Among other results, in [183] it has been shown that in (Rn , T), where T is the family of all total orders on Rn compatible with the linear space structure of Rn , both of these multi-order convexities coincide with the usual convexity in R'. Some applications of these multi-order convexities to discrete convexity in Zn , where Z = {. . . , —2, —1, 0, 1, 2, . . .}, and to order convexity in posets, have been also given in [183]. Let us mention that, as noted in [183], remarks 4.1 and 4.2, the

Notes and Remarks

399

problems of extending some results on order convexity to multi-order convexity (e.g., finding the semispaces or an intersectional basis, or characterizations of multi-order convex sets in terms of alignments) are unsolved. Parametrizations of families M c 2 x , where X is a set, with the aim of applications to parametrizations of the Minkowski-type dualities Am : 2 x —> 2•A4 (see Chapter 6), have been introduced by Evers and van Maaren ([80], def. 2.12), in a slightly more restrictive way; namely they have required the mapping 9 : W M (see Definition 2.8) to be bijective. Some of the examples of parametrizations of families A4 C 2, mentioned in Remark 2.16(b), have been given in [266]; for Remark 2.16(c), see [261]. Voile ([303], p. 208) has defined a parametrization of a base A4 of C(u), where u : 2x —> 2 x is a hull operator, as a pair (W, 0), where W is a set and 0 is a mapping of W with values in C(u) and such that u(G) =

fl

0(w)

(G c X),

(2.241)

WEW

GC0(w)

and has given some examples of such pairs (W, 0) (where the base .A4 of C(u) did not appear explicitly, but apparently one should take M = 0(W) = (0(w) IwEW}). There exist also other concepts of parametrizations of families M c 2 x (e.g., see Lojasziewicz [1611). Finally, as a general remark, let us note that the reader may find it useful to summarize the various examples of Section 2.2 in a concise table, as follows: X

M c 2x

G E C(M) .

(I) linear space

All semispaces M in X

G is convex

COm

G

co G

Parametrization(s) of M (6.79), in X

= R"

Similar tables can be also made for the subsequent examples of Chapter 2 on W c R x , 1C(W), co w G, B C 2X , cos G, u : 2x 2x , C(u), etc., and for the examples of Chapters 3 and 4 on C( W), Q(M), Q(W), Q(13), Q(u), etc., of Chapter 6 on A : 2 x —> 2 w , Chapter 7 on A : R x R w , Chapter 8 on etc.

2.3 For W c R x , where X is a set and R = (—oc, -Foc), the concept of Wconvex subsets of X has been introduced by Fan [82] (actually in [82], a subset G of X is called W-convex if it is an intersection of sets of the form Ix E X I w(x) d} =- Ix E X I (—w)(x) —d}, where w E W, d E R, i.e., if G is (—W)-convex in the sense of Definition 2.9, where —W = I—w Iw E WI). In order to generalize the classical Krein-Milman theorem on extreme points of compact convex subsets of locally convex spaces (e.g., see [148], p. 335, thm. 4), Fan has also introduced [82] the following concept: If W c Rx, a point go of a set G c X is called a W -extreme point of G if there exists no pair of distinct points g2, g2 E G such that go is "W-between"

400

Notes and Remarks

g i and g2 , i.e., such that the relations w E W, w(go) ( min(w (g ), w(g2)) imply that w(gi) = w (g2) = w(go). Remark 2.18(b), (c) (including (2.116), (2.117)) and Proposition 2.9 have been given in [266], remark 1.7, and prop. 1.9(a), respectively. In [266] only level sets Sd(f) and Ad(f) f )with d E R have been used, but here we need the cases d = ±oo, too (e.g., see (2.131), (2.137), etc.). For W c R x , the W-convex hull (2.119) of a set G c X has been defined by Fan [82]; for the general case W c R x , see [266]. Note that it is important to consider C R X (instead of W c R x ), since this permits us to use, for example, W = of (2.157). Theorem 2.6 was given, essentially, in [266], prop. 1.10, equivalence 3'. Fan has shown the first three equalities of (2.131), assuming that W c R x and sup w(G) < -koo for all ZU E W (see [82], lmm. 5); in the general case, they have been given in [266], remark 1.10. The last equality of (2.131) has been shown in [266], prop. 1.11. For W c R x , the "separational" characterization (2.132) of W-convex subsets of X has been given by Danzer, Griinbaum, and Klee ([45], p. 156), and for W c Rx in [266], prop. 1.8. Corollary 2.3, equivalence 1' 3 0 , is due essentially to Klee and Olech [143]. Corollary 2.5 was given in Kutateladze and Rubinov [152], prop. 1.4 (for W C R X , see also [63], thm. 2.2). For W c R x , Corollary 2.6 was given by Dolecki and Kurcyusz ([63], cor. 2.4); however, note that in [63], the claim that the canonical mapping K : X ---> R W of (2.140) is always one-to-one should be replaced by the assumption that it is one-to-one (or, equivalently, that W separates the points of X). For finite-dimensional linear spaces X, the correspondence between convex subsets of X and their support functions has been studied by Minkowski ([199 1, [200]) and Fenchel ([83], [85]), and for infinite-dimensional locally convex spaces X and W = X* by 1-16rmander [121]. For further historical comments on the theory of convex sets and functions in locally convex spaces, see e.g. the section "Comments and References" in Rockafellar [235]. For an arbitrary set X, and W C R X , Definition 2.11(a) of the W-support function (2.146) of a subset G of X has been given by Dolecki and Kurcyusz ([63], §2). In the particular case where X is a locally convex space and W = X*, the W-support functions and W-barrier cones (called simply support functions and barrier cones) are useful tools of convex analysis, optimization theory, computational methods, and the like. Proposition 2.10 has been observed in [266], prop. 1.7. The set Jm of (2.157) has been introduced, in the particular cases of Remark 2.16(c), in [261], and in the general case, in [266], formulas (1.55), (1.56). Theorem 2.8 and Remarks 2.28(a)—(c) were given in [266], thm. 1.1 and remarks 1.11, 1.12; also, remark 2.28(e), on the equivalence of the theories of M-convex sets, where M c 2 x , and W-convex sets, where W c R x , was made in [266] (before thm. 1.1 of [2661). 2.4 The observation that a subset G of a locally convex space X is X*-convex, respectively, (X* ± R)-convex, if and only if it is closed and convex, can be found in Schrader ([248], thm. 2.8) and Dolecki and Kurcyusz ([63 1, §1). In the particular case when X is the conjugate space B* of a locally convex space B, endowed with

Notes and Remarks

401

the weak* topology, X* can be identified with B, and the X*-convex subsets of X are nothing else than the weak* closed convex subsets of B*, called also "regularly convex subsets" of B*, which are used in functional analysis, game theory (e.g., see Berge [251), and so on.

2.5 Part (1) of Section 2.5 is a correction to [266], remark 1.8, where it has been claimed that if we replace M = S(W x R) of (2.111) by Mw = A(W x R) of (2.178), in Definition 2.9, then we arrive at the same family 1C( W) of W-convex sets; the correct statement is that we arrive at C(M w ) of (2.179), which contains (possibly strictly) 1C( W). W-convexlike subsets of a set X, where 0 W c R x (see Definition 2.12) have been introduced in [277], §4, where they have been called "W-regular" subsets. Lemma 2.1 can be found in [259], lmm. 2.1. Remarks 2.30 and 2.31(a) have been made in [277], §4. In the particular case where X is a topological space and W c R x (see Remark 2.31(b)), a subset G of X is called W -separated (see Dolecki [56]), if for each x V Int G there exists w E W satisfying (2.182). Note that if W contains a constant function w, then this function satisfies (2.182), so every subset G of X is W-separated. For a locally convex space X and W = X*, Klee and Olech [143] have studied the class of proper subsets G of X such that G is W-convex for every dense subset W of X* (relative to a given admissible topology for X*). 2.6 The term "coupling function" (see Definition 2.13(a)) has been introduced by Moreau ([205], [206]); the term "pairing function" is also used (see e.g., Evers and van Maaren [801). The term "natural coupling function" (see Definition 2.13(b)) has appeared in [267]. (W, (p)-convex subsets of a set X, where W is a set and (p : X x W —> R is a coupling function, have been introduced by Schrader (see [248], def. 2.5, where they are called "ça-convex" subsets). Schrader [248] has called the set G (pA = { (w , d) E W x RI sup ço(g, w) ( d},

(2.242)

gEG

occurring in (2.185) the "ço-conjugate set" to G, and for any set Nc WxR he has introduced the "(p-conjugate set" to N by N go^ -= Ix E X I cp(x , w)

d ((w, d) E N)1;

(2.243)

then, he has observed (12481, thm. 2.18) that G is (W, yo)-convex if and only if G = (G ) ^p (this follows from (2.185)). The set Vw,(p of (2.188) has been introduced in [267], where Remark 2.33(d) has been also made (see [267], remark 2.2). The (W, 0-support function (2.198) of G has been considered, for example, in [267], formula (3.9). Coupling functions of type {0, —Do} (see Definition 2.16) have been used in [267], but the term "of type {0, —oo}" has been introduced only in [270]. Coupling functions ço : X x W ---> R that can assume only the values 0 or

402

Notes and Remarks

+oo (which may be called "of type {0, + }") have been used by Lindberg [160] and Martfnez-Legaz [169].

Chapter 3 3.1. In the particular case where X is a compact topological space, W is a cone of continuous functions on X, and f is continuous, the concept of W-convexity of f is due to Kutateladze and Rubinov [151], [152]. W-convex functions f : X ---> R, where X is a set and W c R x (see Definition 3.1) have been introduced and studied by Dolecki and Kurcyusz [62], [63], who have also given applications to duality in optimization theory. However, most of their results (see also [571) have been given for W c R x satisfying

W + R = W,

(3.162)

where R is identified with the set of all (real-valued) constant functions on X; this condition is rather restrictive, since it is not satisfied, for example, when X is a locally convex space and W = X* (but it is satisfied for W = X* + R). Moreover, soon after the appearence of [63], [57], Dolecki has replaced the terminology of these papers by the following one: The (W + R)-convex functions (resp. the W-convex functions), in the sense of Definition 3.1, have been called, in [58], §2, "W-convex" (resp. "Wsublinear") functions, motivated by the particular case where W = X* (see Theorems 3.7 and 3.8). The functions that are (W + R)-convex, in the sense of Definition 3.1, have been called "W-convex" by some other authors, too (e.g., see Rolewicz [240]). However, we prefer the terminology of Definition 3.1 (i.e., of [151], [152], [63]), since it permits us to deduce the results on W-convexity in (R x where W c --k x is arbitrary (it need not satisfy (3.162)), as particular cases of results on M -convexity and infimal generators in complete lattices E = (E, this has not been achieved in [63], [57], [58], but only in [272], since Lemma 3.1, and hence the infimal generators Y and Yo of R x given by (3.18) and (3.25), have been discovered only in [267], [272]. Note also that, as in the case of W-convexity of subsets of X (see Notes and Remarks to Section 2.3), for W-convexity of functions f : X —> R too, the extension from W c R x to W c R x is important. Some concepts of convexity of functions f : X ---> T? , defined with the aid of a set W and a coupling function cp :XxW —> R, have been introduced independently by several authors (See Notes and Remarks to Section 3.3); in the particular case where Ç Rx and ço = (Pnat (of (2.184)), these yield concepts of "(W R)-convexity." The W-convex hull (3.7) of a function f : X —› R is a particular case of the "hull" concept of Moreau [204]; namely, according to [204], p. 149, by a hull of f is meant the function—if it exists—which, among a given set of functions, is the greatest minorant of f. In the case (3.162), formula (3.11) has been given in [631, [57], and, in the general case, in [266], thm. 3.3 (however, let us mention that in [266], W-convexity of functions f : X ---> R has been defined by (3.37), and the equivalence of (3.37) with (3.5), i.e., Proposition 3.2, has been observed in [2661, prop. 3.3). The upper addition and the lower addition + on R were introduced and studied by Moreau (e.g., see [205], [206]).

Notes and Remarks

403

Formulas (3.17) and (3.19) were given in [267], formulas (3.1) and (3.2). Remarks 3.2(a)—(d) were made in [272], example 1.6 and remark 1.4b (see also [1811). Formula (3.30), with --fi instead of R, was given in [272], formula (5.21); note that this result, as well as some of the further formulas of Chapter 3 involving x ix) ± d, were not obtained in [267], [266] but only in [272]. For W C R X , formula (3.31) of Theorem 3.1 was given by Dolecki and Kurcyusz ([63], prop. 1.6i). The family E(W) C 2Xx R of (3.36) has been considered, essentially, by Evers and van Maaren in [80], §8.3. As mentioned above, Proposition 3.2 was given in [266], prop. 3.3. Theorem 3.2 was given in [266], cor. 3.2. The function A m of (3.40), used in the proof of Theorem 3.2 and also later (e.g., in Proposition 3.4), can be found, for example, in Kutateladze and Rubinov ([152], p. 9) and Crouzeix ([41], p. 10); formulas (3.41) and A m c C(W) (for M E C(E(W))), of the proof of Theorem 3.2, were given in [266], lmm. 3.1. Theorem 3.3, Remark 3.6(b), and Theorem 3.4 were given in [266], cor. 3.3, thm. 3.4, remark 3.9, and thm. 3.2, respectively. Proposition 3.3 and formula (3.52) are due to Moreau ([206], p. 140). Remark 3.8(a) can be found, for example, in Kutateladze and Rubinov ([1521, p. 10). The epigraphic subsets M of X x R were characterized by Moreau ([206], p. 141) as those having the property that for each xo E X, the intersection M n (x0, R) is either empty, or R, or the half-line [f (xo), -Foo). Proposition 3.4 was given, essentially, in Kutateladze and Rubinov ([152], p. 9), where the sets Mc XxR with property (3.54) are called "R + -stable." In the case where X is a compact topological space, W is a cone of continuous functions on X, and f is continuous, the concept of W-convexity of f at xo is due to Kutateladze and Rubinov [151], [152]. Definition 3.5 of W-convexity of f : X --> T? at xo can be found, for an arbitrary set X and W = W-ERCR x , in Dolecki and Kurcyusz ([63], def. 1.21) and, in the general case W C R X , in [266], def. 3.3. Theorem 3.5 was given in [266], prop. 3.5: for a locally convex space X, and W = X* ± R, it was observed earlier, in [262], remark 3.4d. In the particular case where X is a topological space, some mild conditions on W C WX , under which C( W; x0) is easy to identify, have been given, independently, by Dolecki and Kurcyusz [63], Dolecki [58] and others (see also the Notes and Remarks to Section 3.3). Let us give here an example of such a result. To this end, we need: Definition 3.5 ([58]; see also Balder [19], p. 338) Let X be a topological space and xo E X. A set W c R x , satisfying (3.162), is said to be flexible at x0 if for every WO G W,r E R, and E > 0 there exist ti) E W and a neighborhood V(xo) of xo such

that

(3.163)

(x E X \V(x(J)),

(3.164)

(X G V(X0)).

(3.165)

404

Notes and Remarks

Theorem 3.18 (Dolecki [58], thm. 2.9) Let X be a topological space, xo E X and W C R x , satisfying (3.162) and flexible at xo . If a function f : X —> R is minorized by an element of W and if f is lower semicontinuous at xo, then f E C(W; X0). Proof Let w0

E W,

wo f, and E > O. Put r = f (xo ) — wo(x0) — 2e.

(3.166)

Then, since W is flexible at xo , there exist IV E W and a neighborhood V(x 0 ) of xo satisfying (3.163)—(3.165). We may assume, without loss of generality, that equality holds in (3.163); indeed, by (3.162), one can replace w c W by w — w(x0 ) + wo(xo) + ✓ E W, which has all the above properties of w (since — w(xo) + wo(xo) + r 0), with equality in (3.163). Then, by (3.163) (with equality) and (3.166), we have w(x0) = w o (x0) r = f (x o ) — 2E.

(3.167)

We claim that w f. Indeed, if x G X \ V(xo), then, by (3.164) and wo f, hand, if On the other we have w(x) ( w0 (x) f (x). X E V(X0), then, by (3.165), (3.167), f(x 0 ) — E < f (x 0 ), and the lower semicontinuity of f, we obtain (replacing w(x0 ) + E = f(x0) V(x0) by a smaller neighborhood of xo , if necessary) w(x) • < f (x), which proves the claim. Thus for each E > 0 there exists W E, = w E W such that w(xo) = Pxo) — 2E and w, f. Hence f(x0) = suPE>o ws(xo), so f E C(W; X0)• 3.2 The concept of W-convexity is a generalization of (usual) convexity of functions f : Rn —> R. Indeed, we have: Theorem 3.19 [192] For any integer r with 0 r ( n, any u E L(Rn , Rr) (the set of all linear mappings of R" into Rr, where R ° = {0}), with rank u = r, and any z E Rr, D E (R)*, and d E R define w = Wu,z,c1).d R n —> R by —00

w (x)

=

(x)

=

)(x) ± d

if u(x) < L z, if u(x) if u(x)

z,

(3.168)

>L Z.

Let W

Then for a function f :

{Wu , z , (13,d }u , z , (13.d •

--> R the following statements are equivalent:

10 . f E C(W). 2°. f is convex (in the usual sense).

(3.169)

Notes and Remarks

Moreover, if f is convex, then for each xo w = wu.z,cD,d E W such that

w

f,

E

405

X with f(x 0 ) c R there exists a function

(3.170)

w(x0) = f (x0).

Note that the set W of Theorem 3.19 contains all affine functions (I) ± d (indeed, if r = 0, u = 0, z = 0, then Ix E R I u(x) = z} = Rn). Among other results, in [192] it has been shown that a function f : R" —> T? is simultaneously convex and concave if and only if f E W Utw i I u, z as above, i E {—oc, +a }} , where

—oo Wu , z ,

— oo

=

if u(x) (3.171)

+oo

if u(x) >L z,

00

if u(x) R U I+ oo } admits affine minorants. However for an infinite-dimensional locally convex space X, Moreau ([205], p. 29) has observed that if G c X is a closed convex set with Int G 0 0 and if f = h + XG, where h : X ---> R is a discontinuous linear function, then the convex function f admits no continuous affine minorant (since for any open subset F of G, h takes arbitrary negative values on F). Some further results, showing the role of lower semicontinuity of f for the existence of continuous affine minorants (and for Hahn-Banach type theorems), can be found, for example, in Anger and Lembcke [2]. Lemma 3.3 can be found, for example, in [118], ch. V, proof of prop. 1.1.3. Theorem 3.8, in the equivalent form of Theorem 3.9, was given by 1-16rmander ([121], thm. 5), with a different proof. The proof of Theorem 3.8, given here, follows the lines of [118], Ch. V, proof of thm. 3.1.1.

406

Notes and Remarks

Theorem 3.10 has been obtained in [181], thm. 3.8 and cor. 3.1a. Theorem 3.11 was given in [181], cor. 3.1b. Theorem 3.12 and Corollary 3.2 have been given in [181], thm. 5.3 and cors. 5.3a and 5.1, respectively, but the proofs in [181] have used conjugates of functions, Corollary 8.2, and formulas (8.70), (8.71). Theorem 3.13 has been obtained in [269], thm. 4.1 (see also [269], remark 4.2b); in [269] some applications of polyhedral convex extensions of functions f : {0, 1}" R to combinatorial optimization, have been also given. Remarks 3.20(a) and (c) have been made in [269], remark 4.2a and p. 52, respectively. In a survey paper, Bachem ([18 1, p. 10) has observed that the problem of finding "the appropriate convexity concepts in discrete structures" is still open. In this connection the following strengthening of Definition 3.1 has been proposed in [269]: Definition 3.10 ([269], def. 4.1) Let X be a set, and let W C R X . A function f : X —> R is said to be finitely W -convex if there exists a finite subset Wf of W, such that f (x) = max {w(x) Iw

E

Wf}

(X E

X).

(3.173)

Clearly, for X = {0, 1 }" and W = (Rn)*I ix R, a function f : X —> R is finitely W-convex if and only if it admits a polyhedral convex extension R n —> R. Thus Theorem 3.13 shows that every function f : X —> R is finitely ((Rn)*Ix R)convex, and hence, as has been observed in [269], remark 4.2b, this is not a good concept of convexity of functions f : {0, l —> R. Along the same lines Qi [232] has introduced and studied finitely (R")*I x -convex functions f : X ---> R (where X = {0, 1}"), satisfying f (0) = 0, which he has called discrete convex functions. For 0 < a 1, 0 < N < +cc, the set Va,N C R X of (3.123) has been considered by Dolecki and Kurcyusz ([63], p. 288), even with the distance function p:XxX—> R + replaced by a more general function. For X = R n endowed with the Euclidean distance p, Theorem 3.14 and Corollary 3.3 have been given by Martfnez-Legaz ([173], prop. 2.2 and, respectively, p. 200), using conjugations. Penot and Volle have noted ([221], p. 204) that the equivalence 1° 3° of Theorem 3.14 remains valid for any metric space (X, p). The set Vo, c R x of (3.133) has been considered by Dolecki and Kurcyusz ([63], p. 288). For a = 2, the a-growth condition has been introduced by Rockafellar ([237], p. 272, §3) (called there "quadratic growth condition"), and for arbitrary a E R by Martfnez-Legaz ([173], pp. 203-204), who has also noted Remark 3.22 ([173], p. 204). The implication 3' = 1° of Theorem 3.15, and its extension 3' = 1', are due to Dolecki and Kurcyusz ([63 1, p. 286, remark 4.10iii), and the other implications of Theorem 3.15, as well as Corollary 3.4, have been given by Martfnez-Legaz ([173], prop. 3.1 and cors. 3.3, 3.2), using conjugations. Theorem 3.16 is due to Rockafellar ([2371, thm. 2; see also Dolecki and Kurcyusz [63], p. 286). Theorem 3.17 has been given, essentially, by Dolecki and Kurcyusz ([63], thm. 4.2 and pp. 286-287; see also Martfnez-Legaz [175], p. 185).

7:

} n

3.3 The concept of (Vw,,,, + — 1? (defined by R)-convexity of a function f : X —> — formula (3.161)), where X and W are two sets and cp : X x W —> R is a coupling

Notes and Remarks

407

function, has appeared earlier than that of W-convexity, where W C R X . Indeed, Moreau (see [205] and the references therein) has studied the functions f : X —> R which are the supremum of a family of ( W, go)-elementary functions p(., tv) + r (see Remark 3.25(d)), that is, the functions f E C(Vw, ço ± R), and has denoted the set C( Vw, g, + R) by F' (X , W); among other results he has shown that f c [' (X, W) if and only if f = fc (Oc (01 , i.e., Corollary 8.5, equivalence 1" 3 0 . Weiss (see [305], p. 544, for X, W C Rn and go : Rn x R" —> R continuous and [306] for X, W locally convex spaces and ço : X x W R separately continuous in each variable) has called "go-convex" the functions f : X —> R satisfying f = f c 4"(P ) ' . Lindberg [159] has called "regular" the functions f that coincide with their second conjugate (associated to go :XxW—>R, where X and W are arbitrary sets), but in the sense of a different "conjugation;" for Lindberg's conjugation, see the Notes and Remarks to Sections 8.2 and 8.6. A different approach has been proposed by Schrader [248], [249]. Namely, if X and W are two sets and go :XxW —> R, then, following Schrader ([248], def. 2.12; [249], def. 2.4) a function f : X —> R is said to be "(p-convex" provided that the set Epi f c X xR is e (v) -convex in the sense of Schrader (see the Notes and Remarks to Section 2.6), where e ((p) : (X x R) x W —> R is the function e m ((x, d), w) = yo(x, w) — d

(x E X, d E R, w E W);

(3.174)

note that this happens if and only if f is (V w.(p + R)-convex (by Theorem 3.4, formula (3.48), applied to W replaced by

Chapter 4 4.1 Definition 4.1 has been used in [266], Definition 2.1, but the idea of defining "quasi-convexity" of f by requiring that all level sets Sd ( f) (d c R) should be convex, for some more general concepts of "convex sets," has appeared earlier. For example, for a locally convex space X, .A4 of (2.98) satisfying (2.100) and the parametrization (2.101) of Ar of (2.89), a function f : X —> R has been called [261] 0-quasi-convex (or surrogate convex), if all level sets Sd( f) are 0-convex in the sense of Remark 2.16(c) (see (2.103)); for some other examples, see e.g., Martfnez-Legaz ([168], p. 77) and the references in Soltan ([279 1, ch. I, §4). Formulas (4.5), (4.7), and Proposition 4.1 can be found in [266], remark 2.1, prop. 2.1 and prop. 2.2, formula (2.6). Definition 4.2 has been used in [266], def. 2.2. Proposition 4.2, Corollary 4.1, and Theorem 4.1 were given in [266], remark 2.2 and prop. 2.3 respectively. The function /o{u, } of (4.21) was introduced by Crouzeix ([41], p. 10), who gave Lemma 4.1 ([41], p. 10) and, in the particular case of usual convexity of subsets of Rn , also Corollary 4.2 ([411, p. 11, prop. 4a). Theorems 4.2-4.5 were given in [266], thms. 2.1-2.3 and prop. 2.5, respectively. Corollary 4.3, Remark 4.4, Proposition 4.3, and Remark 4.5(a) can be found in [266], cor. 2.1, remark 2.3, prop. 2.2 (formula (2.5)) and remark 2.8, respectively. The representation function rIG of (4.40) was introduced by Flachs and Pollatschek [88]. Theorem 4.6 and Remark 4.6(a) were given in [266], thm. 2.4 and remark 2.6. Definition 4.3 and Proposition 4.4 are due to [266], def. 2.3 and prop. 2.4; some particular cases of Proposition 4.4 have been given previously in [259], [258], and [261].

Notes and Remarks

408

In connection with Theorem 4.3, Voile ([303], P. 110) has observed that, for any (fixed) X E X, the family of sets Arx ={X\Me 2 x IME.A4,xeX\M},

(4.205)

occurring in (4.30), is a base of a semifilter, in the sense of Greco [101] and that (4.30) is nothing else than the lim inf of f at x along Ç. Hence, using some results of Greco [101] on limits along semifilters, Voile has deduced ([303 1 , p. 111, prop. 8) that if com 0 = 0, then for any f : X —> R we have also fq(m)(x)

=

sup

inf f (X \ G) =

GCX xEX \co .m G

inf

GCX xEcom G

sup f (G)

(x

E

X). (4.206)

The "unilateral limits" of a function f : X —> R along a nonempty family A of subsets of the set X, with 0 g A, defined by f —> lim infA f = sup inf f (A), lirn supA f = inf sup f (A), AEA nEA

(4.207)

have been characterized by Greco [101] as being the "limitoids," that is, the functions R satisfying T : Rx f

h = T(f)

T(k o f) = k(T(f))

T(h) (f

(4.208)

R X , k e C),

E

(4.209)

with C c e of (4.203). Moreover Greco [101] has noted that the functions T : R x ---> R constructed with the aid of successive extremizations, such as the "F-limits" of De Giorgi (e.g., see [47]) are also examples of limitoids. The following more general class of functions T : R x —> R has been introduced and studied by Dolecki and Greco [61]: A function T : R x ---> R is called a niveloid, if it satisfies (4.208) and T(f + d) = T(f)+d

(f

E

Rx, d

E

R).

(4.210)

Clearly, every limitoid (or, equivalently, every unilateral limit (4.207)) is a niveloid. Dolecki and Greco [61] have noted that for any sets X and W, any xo c X and any (p:XxW—>R, the function T(f) =fe (`P)c(Or (xo)

(f

E

R X ),

(4.211)

where f0)' is the biconjugate of f associated to ça (see Sections 8.2 and 8.3), is another example of a niveloid (which need not be a limitoid). Also they have characterized the niveloids that are limitoids and have obtained representation theorems for niveloids. Let us also mention here the following particular class of niveloids (although, in general, the niveloids (4.207) and (4.211) do not belong to this class) introduced and

Notes and Remarks

409

studied in [185], [186], with applications to duality in optimization (e.g., see the proof of Theorem 0.1): A function T : Te —> T? is called a strong niveloid if we have (4.210) and, for any index set I, T(inf fi ) iE/

inf T(fi) iEz

({fi

LE/

c R x ).

(4.212)

4.2 For the concept of quasi-convex functions on linear spaces see e.g. the references in Mangasarian [166], and Martos [194] (see also Hammer [109], p. 106); a review of their properties was given in Greenberg and Pierskalla [104]. The observation that for the family .A4 of all (usual) semispaces in a linear space, Q(M) is the family of all (usual) quasi-convex functions can be found in [266], p. 60. For X = R" the first part of formula (4.55) is due to Crouzeix ([41], p. 11, prop. 5). The second part of (4.55) has been shown in [258], prop. 4.5. Formulas (4.56) and (4.58) were given in [266], formulas (2.29) and (2.31). Formula (4.59) is due to MartinezLegaz ([171], prop. 2.4 and cor. 2.5), who has called "generalized half-spaces" the sets ( y G R n I U(Y) L Z) with u E ,C(R") and z E R" , occurring in (4.59); for a study of these sets and their complements, see [171] and [180]. For X = I?' the first part of formula (4.60) is due to Crouzeix ([42], prop. 2). Lemma 4.2, Corollary 4.5, and Remark 4.8 were given in [258], 1mm. 4.2, cor. 4.4, and remark 4.9, respectively. The observation that for the family M of all closed half-spaces in a locally convex space, Q(M) is the family of all lower semicontinuous quasi-convex functions was made in [266], p. 60. The first part of formula (4.67) is due to Crouzeix ([41], p. 11, prop. 5) and formula (4.68) was given in [266], formula (2.27). Formulas (4.69), (4.71), and (4.72) have been obtained, with other proofs, in [260], formulas (13), (9), and (11), respectively. For X = R", the first part of (4.73) is due to Crouzeix ([42], prop. 2). Evenly quasi-convex functions have been introduced and studied independently by Martfnez-Legaz [168] (who has called them "normal quasi-convex functions" ) and Passy and Prisman [212]. The observation that for the family M of all open half-spaces in a locally convex space, Q (M) is the family of all evenly quasi-convex functions was made in [266], p. 60. The first part of formula (4.75) has been obtained independently by Martfnez-Legaz ([168], prop. 4.3) and Passy and Prisman ([212], thm. 2.1). Formula (4.76) was given in [266], formula (2.26). Formula (4.77) is due to Martfnez-Legaz ([168], p. 59) and Passy and Prisman ([212], thm. 3.1b). For X = R", the equality between the right-hand sides of (4.77) and (4.78) has been given by Greenberg and Pierskalla ([105], thm. 2i); formula (4.78) can be found in [266], formula (2.29). Finally, (4.79) was obtained in [263], formula (43). The observation that for an intersectional base M for the closed subsets of a topological space, Q(M) is the family of all lower semicontinuous functions was made in [266], p. 60. For X = R" , the first part of (4.83) is due to Crouzeix ([42], Proposition 2), and (4.84) has been given in [266], formula (2.33). For X = R", the first part of (4.85) can be found in Crouzeix [42], proof of Proposition 2. For X = R", the observation that (4.86) holds if and only if f is nondecreasing, is due to Chabrillac ([36], pt. 2, prop. 6; see also Chabrillac and Crouzeix [37], prop. 2). Formula (4.88) is due to Volle ([3021, p. 293).

410

Notes and Remarks

The method of proving the results of Section 4.2 as particular cases of the results on Q(M), for suitable M c 2, is due to [266].

4.3 W-quasi-convex functions f : X —> R, where X is a set and W c R x , have been studied in [266] (see also Martfnez-Legaz [168], p. 77), and for W c R x , in Soltan [279], ch. I, §4. Formulas (4.91)—(4.94) can be found in [266], props. 2.6-2.8 and cor. 2.4. Remark 4.12(b) and Proposition 4.7 were given in [266], remarks 2.11 and 2.9. Definition 4.6 has been used in [266], def. 2.5. Theorems 4.7 and 4.8 have been obtained in [266], formula (2.60) (and the remark made before that formula) and thm. 2.6. Remark 4.13(b) and Corollary 4.6 have been given in [266], the remark made before thm. 2.6 and, respectively, cor. 2.5. Definition 4.7 can be found in [266], the remark made after formula (2.60). It is natural to ask, under what conditions the inclusion (4.92) becomes an equality. Let us first give Lemma 4.3 Let X be a set, and let W c Te satisfy aWd-Rcw

41-7 c W

(0

< ±oo),

sup 1717 c W,

(4.213) (4.214)

with the convention 0 W = 0 (so (4.213) for r = 0 means that R c W). Then we have the implication

G c 1C(W)

(4.215)

xc c W.

Proof Assume first that G = Sd (W) for some w E W and d E R (so G G /C(W), by (2.116)). Define a function wo:X—>R by

wo (x) =

sup n(w(x) — d) LQ/ d, we have

wo ( x )

_ +00

(x e X \ G).

(4.217)

Also, by (4.216), So(Wo) = fx E X I sup

n(w(x) — d)

0) = Sd (w) = G.

(4.218)

Thus from (4.218), (4.217), w o G W, OE W (by (4.213) for a = 0), and (4.214) (for IN = (wo, 0)), it follows that XG = X{x EX I wo(x)D} = MaX(Wo, 0) c W,

Notes and Remarks

411

which proves (4.215) for G E iC(W) of the form G = Sd(W) with w E W,d e R. Hence, also for G = S_00 (w) = ndER Sd(W) we have xG = sup dER Xs d (w) E W (by 4.214)), and for G = S+00 (w) = X we have xG = +Do = supdER d c W (by R C W and (4.214)). Now let G e k(W) be arbitrary. Then, by Corollary 2.3, we have G = nwew Ssupw(G)(w) , where, by the above, each xs, up w(G),w) _ W. Hence, by (4.214), we obtain XG = X weW n

S‘up w(Ci) (D) = SU P XS,up w(G)(W)

E

W.

WEW

Theorem 4.16 Let X be a set. For W C R x the following statements are equivalent:

1°. We have (2.118) and W = Q(W) (i.e., equality in (4.92)). 2°. W satisfies (4.213), (4.214) and the implication Sd(wi) c k(W) (d E R) = inf wi E W.

fwiher g W,

(4.219)

iEI

lEI

Proof 1' = 2'. If 1° holds, then, by 1" and Proposition 4.5(b), we have (4.213) for a = 0. If 0 < a < +co and f E W, fi c R, then, by an obvious extension of (2.154), we have

(d E R),

Sd(af fi) = Sw-pvce(f) E iC(W)

(4.220)

so af fi E Q(W) = W (by 1°), which_proves (4.213). Furthermore, if fi-7 c W = Q(W), then, by Proposition 4.5(a), sup W E Q(W) = W. Finally, if the first part of (4.219) holds, then, by Proposition 2.9(a), we obtain

Sd(inf w i ) = iE,

n (u sd+E(wo)

(d c R),

E 1C(W)

(4.221)

e>0 iel

whence infi E iWi E Q(W) = W, which proves (4.219). 2' 1°. If 2° holds, then, by (4.213) for a = 0 and Proposition 4.5(b), we have (2.118). Furthermore, let f E Q(W), so Sr(f) E k(W) (r E R). Then, by (4.213), (4.214), and Lemma 4.3, we have xsr (f) E W (r E R), and hence, by (4.213), for wf , defined by f,r(x) = xs,(f)(x)

r

(x e X, r E R),

(4.222)

we have W E W (r E R). Observe now that for any f E R X there holds the following formula, similar to (3.17) and (3.23): inf (Xs r (f) f = reR

r} = inf

rER

W f r•

(4.223)

412

Notes and Remarks

Indeed, by

XS, (MX)

+r =

i

if f (x)

r

r

(4.224) -koo

if f (x) > r,

we have inf

f (x) =

= inf Ix s, ( f ) (x)

reR

(x

r)

E X).

rER

f

Also we claim that for any f

R x we have

Sd(wf,r) = St(f

)

(d E R)

(4.225)

.

rER

Indeed, if there exists r e R such that wr r (x) d, then, by (4.222), we have d — r, whence xs, (f)(x) = 0 and f (x) r 2 w . Thus formulas (6.13)—(6.19), the equivalence (6.20).(6.21), and formulas (6.22)—(6.24) have been given in [272], formulas (4.5)—(4.11), the equivalence (4.12) (4.13), and formulas (4.14)—(4.16), respectively. For some comments on the particular case E = (2x , D), F = (2w , D) (where X and W are two sets) of Theorems 5.6 and 5.7, see the Notes and Remarks to Chapter 5, Section 5.3. Formula (6.30) can be found in [272], formula (5.110). Formula (6.34) for f (i.(A , A) has been given by Voile ([303], thm. 1.1.5 and formula (18)), but the simple proof given here can be found in [272], remark 5.9b, where it has been observed that this formula, as well as a number of other results of Volle [302], are simply the particular case .A4 = (6.29) of some results of [266]; this holds, for example, for Corollary 6.2(b) (see [302], lmm. I.2.3ii) and Remark 6.3(a) (see [302], p. 288). Moreover, this method of applying some results on Q(M) or Q(M; xo) (e.g., of [266]) to M of (6.29) leads to further results on Q (A / A) or Q (A' A ; x0), respectively, such as Corollaries 6.3 and 6.4. For the case where 0 : W —> M is bijective or where 0 is as in (2.241), the parametrization (6.44) of the Minkowski-type duality Am has been given by Evers and van Maaren ([80], §2.12) and Volle ([303], p. 208), respectively; in the latter case, Volle has also observed that A 1 ({w}) = 0(w) (w E W) and A' A = u ([3031, p. 209). The fact that for any sets X and W every duality A : 2 x —> 2 w is equivalent to a Minkowski-type duality A m , namely, with M of (6.29) (see Remark 6.5) has been shown by Evers and van Maaren ([80], prop. 2.9), but Theorem 6.2 says more about M = MA of (6.29), and it can be applied to all particular dualities A : 2 x —> 2 w encoutrd in the literature (see the Notes and Remarks to Section 6.2).

416

Notes and Remarks

The results of Section 6.1 on partial order and lattice operations for dualities A : 2x —> 2 w , where X and W are two sets, and some related results have been given in [272], §4.

6.2 A Minkowski-type duality similar to (6.74), namely, considering only semispaces at 0, can be found in Evers and van Maaren ([80 ] , §1.22), with the mention that it seems to have been introduced by Motzkin [209] and studied in extent by Jamison [128]; in that example A im Am (G) is either X or the smallest convex cone C with vertex 0 g C, containing G. Formula (6.75) is equivalent to the theorem on the separation of convex sets from outside points by semispaces. Similarly, if X is a locally convex space and M is the family of all closed half-spaces in X, then for the Minkowski-type duality Am of (6.25) we have A im A .A4 (G) = 5G for all G C X, which is equivalent to the separation theorem of closed convex sets from outside points by closed half-spaces. For the next references concerning some of the examples of dualities Ao of Section 6.2, the following remarks are valid: The expressions of Aot A o (G) have been computed (in fact these hulls have been one of the reasons* for introducing Ao), but it has not been observed that these A o are parametrized Minkowski-type dualities. Thus the duality Ao of (6.79) can be found in Voile ([304 p. 303), together with the remark that Aio A o (G) = co G, but there it has not been observed that this duality Ao is a parametrization of the Minkowski-type duality (6.74), via (6.78). For X = Fr-H I , the duality A 9 (G) U ( 0), with A o of (6.82), can be found in Evers and van Maaren ([80], §1.16) and, in the general case (of a locally convex space X), in Voile ([302], p. 298). The set A 9 (G) U (0), with A o of (6.83), which may be called the "usual" polar of G, has been used, for example, in Bourbaki [32], Kiithe [148], Moreau [205], Rockafellar [235], Barbu and Precupanu [21]; note that, in particular, if G is a convex cone with vertex 0, then this polar of G coincides with A9 (G) U {0 } , with A o of (6.85), and, by Remark 6.12 (c), it is a weak* closed convex cone in X* with vertex 0. For X = R" , the duality G —> A 9 (G) U ( 0), with A 0 (G) of (6.85), can be found in Evers and van Maaren ([801, §1.23) and, in the general case, in Voile ([302], p. 297), with the observation that A9 A 9 (G) = cone G. Some authors (Grothendieck [107], Day [46], etc.) call "polar" of G the set (— A o )(G) U ( 0), with A o of (6.83), i.e., the set (—A 0 )(G) U ( 0) = {(1) e X* I inf cl(G)

— 1);

(6.193)

note that, in particular, if G is a convex cone with vertex 0, then this polar coincides with {0 G X* I inf (1)(G) ?, 0). For X = , the set "À0(G), with 2K 0 of (6.89) (which is different from "À0(G) U {0}), has been called, by Tind ([288], [2891), the "reverse polar" of G, and, by Ruys and Weddepohl [242], the "lower polar" of G (the term "upper polar" of G has been used in [242] for the set A9 (G) u ( 0), with Ao of (6.83)). Combining (6.83) and (6.89), El Qortobi [72] and Attéia and El Qortobi [5] have defined the "projective polar" of G C X by G# = G° U G v , where G ° = A 9 (G) u {O), with NAG) of (6.83) and G v = À o (G) of (6.89), and the *Another motivation lies in the conjugations of type Lau associated to these dualities (see Section 8.8).

417

Notes and Remarks

"projective bipolar" of G by G" = G oo n Gvv, and have observed that G " =(G#)# . The duality (6.91) has been considered by Voile ([302], p. 296). For C7) G evenly convex subsets G of X, the duality (6.92) has been introduced and studied by Fenchel [84]; a version of the duality (6.92), namely with X = 1V+ , W = (R")*+. and < replaced by >, can be found in [302], p. 304, with an application to utility functions of mathematical economics. A duality similar to (6.94), namely with X being a linear space and W = X 4 , has been mentioned in Evers and van Maaren ([80], §1.17); this duality, as well as (6.94), are often used in linear algebra and functional analysis, respectively. The duality (6.95) has been introduced by Volle ([3021, p. 302), in connection with the pseudoconjugations of [257]. The dualities (6.97) and (6.102) can be found in Volle ([302], pp. 293 and 294, respectively). The set A ov (G) of (6.107) is nothing else than "the (p-conjugate set" G'Ço` to G, of formula (2.242). For some other examples of dualities A : 2x --> 2 w , see e.g. Birkhoff [27], ch. V, §7, Araoz [3], Griffin [106], Evers and van Maaren ([80], §1), Araoz, Edmonds and Griffin [4] (and the references therein), and Voile [302]; see also Remark 6.13(d). 6.3 In Chapters 6-9 we have used the term "representation" in the sense of functional analysis (e.g., see 146], p. 35, or [148], p. 201, on "the Riesz representation theorem" ), i.e., in the sense of finding an expression for the "general form" of the mappings A belonging to a certain set of mappings (see also Remark 9.22 below). Theorem 6.4 (in an equivalent version, namely, with e Q and e A ({x }) instead of g Q and gi A ({x }), respectively), and Remarks 6.13(c), (d) were given in [270], thm. 1.1 and remark 1.1, respectively. In view of Remark 5.1(b), let us also mention the following representation theorem of similar type, due to Choquet [38]: Theorem 6.6 ([38], pp. 196-197) Let X be a topological space, W a compact space, Q a closed subset of X x W, and define A : 2 x --> 2 w by A(G) = {w E W I3g E G, (g , W) E S-2} = W \ (W E W

I (g, w)

S2 (g E G))

(G C X).

(6.194)

Then A is a U-homomorphism, continuous on the right (i.e., A (G1 U G 1 ) = A (G 1 )U A(G2) for all G 1, G2 C X, A(G1) C A(G2) forailG 1 C G2 C X, and for each Go C X and each neighborhoodllo of A (G 0 ) there exists a neighborhood Vo of G0 such that A (G) C 1,10 for all G C V0). In the converse direction, if g is an additive hereditary family of compact subsets of X (i.e., G1 U G2 E g for all GI, G2 E g and {G 1 E 2 X IG1 C G) C g for all G E g), and if A is a U-homomorphism, continuous on the right, of g onto a family ,F of compact subsets of a Hausdorff topological space W, then there exists a closed subset Q of X x W such that we have (6.194). As has been noted by Choquet ([38], p. 198), it is an open problem to find such a representation for n-homomorphisms continuous on the right, from the family g of all compact subsets of a compact space X into the family T of all compact subsets of a compact space W.

418

Notes and Remarks

Remark 6.13(b) and the first part of Remark 6.13(d) can be found in Dolecki ( 11 58 1 , prop. 1.2 and formula (1.7)). The equivalence 1' -#>- 2° of Theorem 6.5, for yo(X x W) c R and with 0 instead of —1 in (6.130), can be found in Evers and van Maaren ([80], §2.1); in the proof of the implication 1' = 2°, they gave a coupling function ço "of type [0, 1)" (namely, (p(x, w) = 0 if w E A ({ O) and 1 if w g A ({x ))), whose uniqueness was noted in [270], remark 2.3b. The complete Theorem 6.5 has been given, with —1 instead of —1 in (6.130), in [270], thm. 2.1, and, in the present form (with —1 in (6.130)), in [272], remark 5.10. The first part of Remark 6.14(c) was made in [270], remark 2.3a. Remarks 6.14(e) and (f) can be found in [270], def. 2.5 and remark 2.3c, respectively. Proposition 6.4 was given in [270], thm. 2.2. For Aow of (6.107), occurring in Proposition 6.5, the hull operator A /0 „, A 64, : 2 x --> 2 x has been called in [270], def. 2.4, "the hull operator associated to the coupling function ço," and the inclusion A A(G) C A iço A y,(G) (G c X) has been observed in [270], formula (2.19). Let us also mention that generalizing A y, of (6.134), in [272], def. 6.1, there has been defined the duality A ço : E --> F associated to ço Y x T --> R, where E and F are two complete lattices, with infimal generators Y and T, respectively; for some results and remarks on this concept, see [272], §6. Definition 6.3, Proposition 6.6, and Remarks 6.16 and 6.17(a) were given in [270], def. 2.3, cor. 2.1, and remarks 2.4 and 2.5. Remark 6.17(b) is a part of [272], thm. 4.4. 6.4 Remark 6.19 can be found in [270], example 2.4. Finally, let us mention that there have been given several sufficient conditions and characterizations for the continuity of arbitrary dualities A : 2 x --> 2 w , with respect to various topologies and convergences on 2X, 2 w (see Dolecki [59] and the references therein; for a recent survey and classification of set convergences, see Sonntag and nlinescu [280]).

Chapter 7 7.1 Remark 7.1(a)—(c) and Proposition 7.1 were given in [272], remark 1.5c and prop. 1.4, respectively. Formula (7.13) can be found in [267], formula (4.1). Formula (7.15) has been observed, for conjugations A= c : R x —> R n', in [267], formula (4.5) and, for arbitrary dualities A : Rx —> T?w , in [272], formula (5.4). Formulas (7.17)—(7.26), with R and ± replaced by R and ±, respectively, were given in [272], formulas (5.6)—(5.10), (5.12), (5.14)—(5.16), and (5.18). For W c RA', the duality A w : R x --> 2 w of (7.27) and Remark 7.2(a) are due, essentially, to Evers and van Maaren ([80], §8.2). Still for W c R x , the problem of the existence of a duality A : R x R w such that f AA' _ fc0(w) (f e was raised in [271], pp. 108-109; the duality A w of (7.32) has been considered in [271], formula (3.76), as a candidate, but the fact that it is indeed a solution has not been observed there. For some results of [181] on this problem, see the Notes and Remarks to Section 8.3. The fact that the set (7.33), with R and + replaced by R and ±, respectively is a generating class for A : R x —> Te was observed in [272], formula (5.24), using a different terminology. Formulas (7.34)—(7.37) were given in [272], formulas (5.27)—(5.30). )

Notes and Remarks

419

7.2 Formulas (7.38)-(7.40) can be found in [181], formulas (1.8)-(1.10). For A = R, (7.45)-(7.48) have been observed in [272], remark 1.5c. Theorems 7.1, 7.2, Lemma 7.1, and Remark 7.4 were given in [181], thms. 2.1, 2.2,1mm. 2.2, and remark 2.1, respectively. For A = r?, Remark 7.3(a) can be found in [272], p. 297. 7.3 Theorems 7.3-7.5 and Remark 7.5(a) were given in [181], thms. 3.1, 3.2, 3.5, and remark 3.1, respectively. For A = B = R, Corollary 7.2 and Remark 7.5(b) were given in [278], thm. 1.1 and its proof. For A = B = T2, Proposition 7.2, Corollary 7.3, Proposition 7.4, Definition 7.2, and Remark 7.8 can be found in [188], Imm. 1.1, cors. 1.2 and 1.1 (together with prop. 1.1) and remark 1.3b, respectively. Theorem 7.6 was given in [181], thm. 3.6. Remarks 7.10(a) and (b) were made in [188], remarks 1.1 and 1.5a, respectively. Let us note that those results of Chapter 7 that do not involve the topology of R or the special properties of the order of T? (Theorems 7.3, 7.5, 7.6, etc.) remain valid, with the same proofs, for arbitrary complete lattices (A, and (B, (not necessarily contained in 2 w R , using the identification of functions with their epigraphs. This observation has been further developed, in a more explicit form, for arbitrary X, W, ço : X x W R and c(yo) : R x —> R w , by Voile [302] (see the Notes and Remarks to Section 8.11). Next Elster and Wolf ([76]-[79]), and Elster and Giipfert [73] have studied more general mappings than conjugations, from -R-x into R w , where X and W are two sets, with similar methods but replacing the epigraphs by graphs; we recall that for any function f e R x the graph off is the set

420

Notes and Remarks

Graph f cXxR defined by Graph = {(x, f (x)) E X x Rix E X}.

(7.110)

Initially these authors considered dualities r : 2xx R —> 2 W x R and mappings A (F) : Rx > W, and later, in [78], they passed to dualities r : 2xx —R —> 2 WxR and KW mappings A( r) .. R X —> Tr Definition 7.6 Let X and W be two sets and r : 2 x R —> 2 W x R a duality. The mapping A (T) : R x —> W W defined by f A(F) ( p

)

inf

r

rER (w,r)E [(Graph

(f E

RX ,

tV E

W)

(7.111)

f)

is called the mapping associated to the duality r. > 2xxR Elster and Wolf [78] have called f '1 of (7.111) (with r E R replaced by r c R) "the upper P-conjugate (function) of f with respect to r" (where P stands for "polarity" ); we prefer to use here a different terminology, since

For r 2 xxR

A (F) need not be a "conjugation" in the sense of (8.1), (8.2) (it does not even need to be a duality in the sense (7.1)). Elster and Wolf ([78], §3) have noted that the (FenchelMoreau) conjugations c(p) of (8.35) and the (D-conjugations mentioned above are particular cases of mappings A (r), for suitable r, and have shown ([78], §4) that the families of all conjugate functions fe (0 and of all (D-conjugate functions f `I' are not identical, and neither one is a subfamily of the other one. Also they have shown that the intersection of these two families is not empty, and they have characterized the elements of this intersection ([78], thm. 4.1). 2wxR that will be used Now we will introduce a class of dualities r : 2xxR in the sequel. Definition 7.7 Let X and W be two sets and e:XxWxR The mapping r (e) : 2 x R —> 2 w x R defined by

r(e)(m),_

n

{(w,r)E W x Rle(x,w,d)

—> R a function.

(M c X x T?)

(x,d)EA4

(7.112) is called the duality induced by the function e. For simplicity, we will denote r(e) ({(x , d))) by 1- (e) (x, d). Remark 7.15 (a) r (e) is indeed a duality, since r(e)(M) = Rx,d)Em r(e)(x, d) (m c X x --k). If we define a coupling function 7 = 7re : (X x R) x (W x R) --> R by 7((x, d), (w, r)) = e(x, w, d) — r — 1 ((x, d) c X x R, (w, r) c W x R),

(7.113)

421

Notes and Remarks

then, by (7.112) and (7.113), we have F(e )(M) = ((D, r) e W x RI sup

((x, d), (w, r))

—1)

(x,d)EM

(M c X x T?),

(7.114)

SO r(e) is the duality associated to the coupling function 7 (in the sense of Remark 6.14(a). Also, considering the set Cie

=

MX/

d), (w, r)) c (X x R) x (W x R)

I e(x, w, d)

r}

(7.115)

(which can be identified with Epi e C (X x W x T?) x R), ro is nothing else than the duality 2x x R -± 2' associated to the set Q, = ((X x R) x (W x R)) \ 0, (in the sense of Remark 6.13(a)). (b) If we define, for any Mc X x R, a function Ifrm ,, : W —> R by Vfm,e(w) =

sup e(x, w, d)

(w E W),

(7.116)

(x,d)EM

then, by (7.112), we obtain (M c X x R),

F(e )(M) = EPi *m,e

(7.117)

and thus r(e) is an epigraphic subset of W x R (see Definition 3.4). Hence

r (e) (2X x R)

c

(7.118)

where E(R w ) is the family of all epigraphic subsets of W x R (see (3.27)). (c) Identifying XxWxR with (X x R) x W by (x, w, d) —> ((x, d), w), one can identify e with the coupling function ((x, d), w) —> e((x, d), w) = e(x, w, d). Then the duality associated to e, in the sense of Remark 6.14(a), is the mapping 2X x R 2 W defined by Te: T e (M) = 1w C W I e(x, w, d)

—1 ((x, d) c M)} (M c X x R),

(7.119)

which is different from the duality r(e) : 2 X x R —> 2' of (7.112) induced by e.

2wxR is induced by some It is immediate that not every duality r : 2xxT? function e:Xx Wx R —> T? (i.e., of the form 1- (e) of (7.112) for some e). Indeed, by Theorem 6.5, every mapping r : 2X x R —> 2 W x R defined by the right-hand side of (7.114), for some coupling function 7 : (X x R) x (W x R) —> T?, is a duality (called the duality associated to 7), but Remark 7.15(a) shows that we have r = r (e) for some e:Xx WxR R (if and) only if this 7 can be chosen of the form (7.113) for some e:XxWxR R. Therefore it is natural to introduce:

422

Notes and Remarks

Definition 7.8 ([781, defs. 1.5 and 1.6) Let X and W be two sets. (a) A coupling function 7 : (X x R) x (W x R) —> R is said to be epigraphic if there exists a function e:Xx WxR —> R such that 7 =The of (7.113). 2 wxR (b) A duality r : 2xx -R- -> 2 W x R is said to be epigraphic if F(2R) and there exists a function e:XxWxR R such that F = F( e) of (7.112) (i.e., the duality induced by e) or, equivalently, if there exists an epigraphic coupling function 7 : (X x R) x (W x R) —> R such that r is the duality associated to 7 (in the sense of Remark 6.14(a)).

We have the following "intrinsic" characterizations of epigraphic coupling functions and epigraphic dualities: Proposition 7.6 ([78], thm. 1.3) Let X and W be two sets. (a) A coupling function 7 : (X x R) x (W x R) —> R is epigraphic if and only if we have the implications 7r((x , d), (w, r))

0 = jr ((x, d), (w, r

—1, s

s)) i —1,

(7.120)

r„=i-ER r((x d), (w, rn)) R is epigraphic if and only if there exists a function e : X x W x R —> R such that for the set e e of (7.115) (which can be identified with Epi e C (X X W X R) x R) we have

Oe = {((x, d),

, r)) c (X x R) x (W x R) I 7 rqx , d), (w, r))

—11. (7.124)

Hence, by Proposition 3.4, 7 is epigraphic if and only if we have the implications (7.120), (7.121). 2wxR is epigraphic if 2xx 1? (b) By Definitions 7.8(b) and 7.7, a duality and only if there exists a function e:Xx WxR —> R such that for the set e e of (7.115) we have ee = {((x d), (w, r)) e (X x R) x (W x R) I (w, r)

E

F(x, d)}.

(7.125)

Hence, by Proposition 3.4, F is epigraphic if and only if we have the implications (7.122), (7.123).

423

Notes and Remarks

The following proposition and theorem are due to Elster and Wolf in the context of epigraphic dualities ([78], cor. 2.2). Proposition 7.7 Let X and W be two sets and el, e2 : X x WxR —> R = F(e2 ) (if and) only if ei = e2.

two functions. Then

Proof By Remark 7.15(a), we have r (ei) F(e2 ) (if and) only if ge , = 0,2 , or, equivalently, Epi e l = Epi e2. But, by Proposition 3.3, this happens (if and) only if el e2.

Theorem 7.10 Let X and W be two sets and e:Xx WxR ---> (7.68) for A = B = T?)

a function. Then (with f °(e) of

(f E R X ,w E W). (7.126)

= sup e(x, w, f (x)) = f(w) xeX

Proof By (7.111) and (7.112) (for M =- Graph f of (7.110)), for any f G R x and w E W we have

f A (In (0 )( w )

inf

reR (w,r) E P(e) (Graph f)

r

=

inf

reR e(x,w,f(x)). —R-w of Definition 7.6 is a dualitylike mapping (in the sense of Definition 7.5). (b) By Proposition 7.7, for a duality F : 2x R —> 2 W xR there exists at most one function e:XxWxR —> R such that F = r (e) . Hence, if F is an epigraphic duality, then the function e in Definition 7.8(b) is unique. Thus we have a one-toone correspondence between epigraphic dualities r : 2 Xx R —> 2 w x R and functions e:XxWxR---> R. (c) If F1, r2 : 2xxR > 2 w x R are two epigraphic dualities, the relations A (F 1 ) = A(r2) need not imply that F1 = r2 . Indeed, we have A (F(e ,)) = A(r(e 2 )), where e 1 ,e2 :XxWxR—> R, if and only if A (e i ) =- A(e2), while F(e ) = F(e2 ) if and only if el = e2, so a counterexample to the above implication can be obtained by taking any functions el, e2 :X x WxR ---> R, el e2, with A(e 1 ) = (e2 ) (or, equivalently, with supx , x e i (x, w, d) = supxEx e2(x, w, d) for all wEW,dc R). However, if e:XxWxR —> R is such that A (e) is a duality, then e, and hence also F(,), is uniquely determined by A (F(e) ) = (e). Therefore in this case one can call F(e) the duality induced by the duality A (e), and one can denote it by F(e) . Remark 7.16(b) has been given by Elster and Wolf ([78], cor. 2.2, and [79], thm. 3.1 i), who have also claimed ([78], cor. 2.1, and [79], thm. 2.1 iii) that if F1, r2 are two epigraphic dualities, then the relations A (F1) = A (F2 ) imply that r i = F2 (i.e.,

424

Notes and Remarks

that the duality A() associated to an epigraphic duality F determines r uniquely); however, counterexamples to the latter claim are given in Remark 7.16(c). The next proposition has been noted implicitly in Elster and Wolf ([78], p. 73). Proposition 7.8 2W xR

Let X and W be two sets and F : 2 x xR

an epigraphic duality. Then

Epi f A(r) = F (Graph f)

(f e R x ).

(7.127)

Proof Led' = F(,), where e:XxWxR —> R. Then, by (7.126) and (7.112) (for M = Graph f of (7.110)), we have for any f e RA', Epi P (r (o ) = [(w, r) E W x RI sup e(x, w, f(x))

r}

xEX

=- {(w r) E W x RI e(x , w, f (x))

r

E X)} -=-

(e) (Graph f).

The following theorem and corollary are due to Elster and Wolf ([79], thm. 3.1 iii), who have also given some results on the functions f G –1?x - for which the equality sign holds in (7.132), which they called in [79] "P-convex" functions (see [79], def. 3.2 and thms. 3.2 and 3.3).

Theorem 7.11 Let X and W be two sets and A of (7.111) satisfies

r:

2x

—> 2WxR an epigraphic duality such that

()

(f e RA', w e W).

f °(r) (w) E R

(7.128)

Then we have f A (F) A (F') )

inf

d

(f E R A', X E X).

dER (x,d)E F' F (Graph f)

(7.129)

2 w xR Proof Let r = r (e) . By Definition 7.6 (applied to the duality 2xxR, which need not be epigraphic, and to f'())e R w , where f c –fix ), we have f A (r(e) ) A (ro ) (x

) —

inf

d

dER (x,d)Er e) (Graph f'")

(f cTe, x c X).

(7.130)

Graph f),

(7.131)

We will show that, for any f E R X , r(e) (Graph f A(r (')) )

= 1- e) r( e )

which, together with (7.130), will prove (7.129).

(

Notes and Remarks

425

By (7.128) and (7.127), we have Graph f A(e) c Epi f °( e) = r (e) (Graph f), whence, since r(e) is antitone (e.g., by Proposition 5.2), we obtain the inclusion D in (7.131). Conversely, let (xo,do) c rq(e) (Graph f A W(0 ) ). Then, since (by Remark 7.15(a)) F (e) is the duality associated to the coupling function n- = n(e) of (7.113), from (6.135), (7.128), and (7.113) we obtain ((xo, d0), (w f A(r (e)) (w))) = e(xo, do, w)

1

f A(I. (e)) (W) — 1

(w E W), whence also e(x o , do, w) — r — 1

(w E W, r G R, f A(r (e) ) (w)

—1

r);

that is, (xo, do) E r i(e) (EPi f (e) ) ) = r e) r(e) (Graph f) (where the last equality holds by (7.127)), which proves the inclusion c in (7.131), and hence the equality LI (7.131). Remark 7.17

In general, A(F)' 0 A ( 1- e))•

Corollary 7.4 For X, W, and r as in Theorem 7.11 we have

f

f") Proof Since Fir : 2xx —R-

>

(f E —fiX) .

(7.132)

x x-R- is a hull operator, we have

Graph f c F(Graph f)

(f E R X ),

whence, by Theorem 7.11, we obtain inf

d = f (x)

d€ R

(f E R X , x E X).

LI

(x.d)EGraph

Some parallel methods using dualities F 1 : 2x R —> 2 W x R and epigraphs, instead of dualities F : 2 x x -R- —> 2 w x T? and graphs, which permit us to obtain results also about f A(1"1)A(Fir have been developed in [278]. Note that for a duality r, : 2xx R R w by (7.111) with r replaced by F 1 , since 2 W x R one cannot define A (F 1 ) : Fi (Graph f) is not defined (indeed, Graph f e2 X x R- , while r, is defined only on 2X X R

).

Definition 7.9 (1278], def. 2.2) Let X and W be two sets and r 1 : 2 x x R a duality. The mapping A (F ) : Rx > R w defined by f

(F1)( w )

inf rER (w,r)Eri (Epi f)

r

(f E R X ,w E W)

(7.133)

Notes and Remarks

426

is called the mapping associated to the duality F 1 . Definition 7.10 ([278], def. 2.1) Let X and W be two sets and e l : X xWxR ---> R a function. The mapping r (ei) : 2XX R —> 2" defined by

r(ei)(m)

=

n

{(w, r)

E

W x R)

(Ai c X x R)

w, d) r)

(x,d)EM

(7.134) is called the duality induced by the function el. Remark 7.18 (a) Definitions 7.9 and 7.10 correspond to Definitions 7.6 and 7.7. Similarly, replacing, in Remark 7.15, X x R and e by X x R and e 1 , respectively, we obtain corresponding remarks for e 1 :XxWxR---> R, F(e) 2xxR 2 wxR , 71= Tre : (X X R) x (W x R) ge , C (X X R) x (W x R) and *m , e , (with M c X x R). (b) One can define epigraphic coupling functions n- 1 : (X x R) x (W x R) and epigraphic dualities r, : 2xxR —> 2' replacing, in Definition 7.8(a),(b), X x R, W x R and e by X x R, W x R and el, respectively, and using r (e , ) of (7.134). Then, making the same replacements in Proposition 7.6, one obtains "intrinsic" characterizations of such jr 1 and r,. Also, Proposition 7.7 remains valid for el, e2 :XxWxR---> R. The following fact corresponds to Theorem 7.10: Theorem 7.12 Let X and W be two sets and el :XxWxR —> R a function. Then sup

ei(x , w, d) = fA(e1) (W)

(f

E

RX ,

WE

W).

(x,d)EEpi f

(7.135) Proof By (7.133) and (7.134) (with M = Epi f), for any f G -"ix and we have f A(r 0, 0 )( w )

inf

rER 040E1' ( ,, i) (Epi f)

r=

inf rER e i (x,w,d)r ((x,d)EEpi f)

r=

sup

WEW

e 1 (x, w, d).

(x,d)EEpi f

D

Remark 7.19 From (7.135) and Remark 7.6 it follows that ife i : XxWxR ---> R is such that each e 1 (x, w, : R —> R (x E X, w E W) is nonincreasing on R (in particular, if el is such that A(r( e) ) = A (ei) is a duality), then el, and hence also F (e , ) , is uniquely determined by A (F (e 0). Therefore, if A (r (e , ) ) = A (e i ) is a duality, then one can call F( e ,) the duality induced by the duality A (ei), and one can denote it by r6(e,)•

Notes and Remarks

427

Proposition 7.8 cannot be applied to dualities r, : 2xxR 2wxR, since F 1 (Graph f) is not defined (because, in general, Graph f g 2x x R ). However, we have now the following result, corresponding to Proposition 7.8 and Theorem 7.11: Theorem 7.13 ([278], thm. 2.1)

Let X and W be two sets, e l :XxWxR-> R a function satisfying the conditions of Corollary 7.2 with A = B = R (or, equivalently, such that A = A (e l ) : R x -> of (7.75) is a duality), F1= r(ei) 2 xx R —> 2' the duality (7.134) induced by e l , and A(F 1 ) : Rx R w the mapping (7.133) associated to F. Then A (F 1 ) is a duality, satisfying Epl f A(FI) = ri(EPi f) Au-0Am)/ = ri r,(Epi

Epi f

(f

E

R X ),

(f E Rx),

f)

(7.136) (7.137)

and hence f A (F )A(F ( x )

inf

dE R (x,d)EF 1 (Epi f)

(f e kx , x e X).

d

(7.138)

For the proof, see [278], proof of Theorem 2.1. Remark 7.20

(a) By the definition of IkEpi f

i (see Remark 7.18(a)) and by (7.75)

and (7.135), for any f c R x and w E W we have lEpi f,ej

=

sup

el (x, w, d) = f°('' ) (w) = f A(r1) (w),

(7.139)

(x ,d)EEpi f

which yields another proof of (7.136); indeed, by (7.117) (with e replaced by e1 and M = Epi f) and (7.139), we obtain T I (Epi f) = r (e,)(EPi f) = Epi *Epi

= Epi f A(F1) .

(b) Formula (7.136) also shows that if for any set X we denote by Ex the oneto-one mapping f -> Epi f of -fix into 2 X x R (see Proposition 3.3) and by Ex-1 its inverse mapping Epi f f, from (R x ) (the family (3.27) of all epigraphic subsets of X x R) onto R x , given by (3.51), then (7.140)

mr,) = (Ew) iriEx•

For the conjugation of type Lau L (F i) : R xx --> R 2wxR, given (see Definition 8.8) by 2 xxR h L(r ' ) (w, r) = -

inf

(x,d)EXx R (w,r)E(W x RAF i (x,d)

xR

associated to a duality

h(x, d)

- inf h((X x R) \ (w , r))

Notes and Remarks

428

(he R

R , (w, r)

R),

(7.141)

R x ).

(7.142)

((x, d) c X x R),

(7.143)

R x ).

(7.144)

EWX

we obtain:

Corollary 7.5 Let X, W, el, and (a) We have

r,

be as in Theorem 7.13. Then

X(WxR)\Epi

(f

f A(r i ) = — (XEpif) L(ri)

E

(b) If sup ei (x, w, d) > —co 1.1.1 E

W

then XEpi f A(1 - 1)Acr l Y = (XEpi f L(F

1 )L(F

(f

E

Proof (a) By (8.258) and (7.136) we have for any f e R x , (XEpi f)

L(Fi) —

X(W

xR) \ F 1 (Epi f)

—

X(WxR)\Ep•fwri ) •

(b) Assume now (7.143), and let (x, d) e X x R. Then, by (7.143), there exists wo e W such that e 1(x , wo, d) > —oc, and hence for any ro el (x, wo, d) > ro, we have (wo, ro)

{(w, r) e Wx RI ei(x ,

d)

E

R such that

r} = F i (x , d).

Thus F i (x, d) W x R ((x, d) e X x R), whence, by Remark 6.3(a), n r i m Consequently, by Corollary 8.19(b) and (7.137), we obtain (XEpi f

L(FI)L(Fi)' Xrj)(Epi f)

XEpi f Au'i)A(ri)/

(f E

R x ).

=

0.

111

Finally, let us give another application of the set ee , of Remark 7.18(a), that is, of the set

ge , = {((x, d), (w, r)) e (X x R) x (W x R) I ei(x, w, d)

r}.

(7.145)

To this end, let us note that if el:XxWxR----> R is any function such that A (e i ) : R x —> —k w is a duality, and if e:Xx WxR —> R the function defined by (7.76), (7.67), and (7.77) (with A = B = R), then, by the proof of Corollary 7.2, we have A(e) = A(e1),

(7.146)

Notes and Remarks

429

and thus, by Proposition 7.3, f E R x and g E R w are "dual with respect to e1" (i.e., f l(el ) g) if and only if they are dual with respect to e (in the sense of Definition 7.3). Proposition 7.9 Let X and W be two sets and e l :XxWxR—>Ra function such that Lx(e 1 ) : —> T?w of (7.75) is a duality. For two functions f e T? x and g E T?w the following statements are equivalent: 1 0 . f and g are dual to each other with respect to e l (i.e., f A( el ) 2°. We have Epi f x Epi g

g).

(7.147)

C O ei .

Proof 1° = 2°. If 1 0 holds, then for any (x, d) E Epi f, (w, r) E Epi g we obtain, using that e l (x,w,.) : R —> R is nonincreasing and (7.75) (with A = B = R), e i (x, w, d)

e i (x, w, f (x))

f' ) (w)

g(w)

r,

so ((x, d), (w, r)) E 0 e, 20 = 1°. Assume 2', and let (x, d) E Epi f, w E W. If g(w) = —so, then (w, r) E Epi g for all r E R, whence, by (7.147) and (7.145), we obtain e i (x, w, d) r (r E R), so e i (x, w, d) = —oo = g(w). If g(w) = then e 1 (x, w, d) — R w of (7.75) is a duality, then for any f have Epi f x Epi f A(e,) c Epi g A(el)'

X

E

7— R x and g

oei

Epi g c 0 e ,.

E

Te we

(7.148) (7.149)

Proof By Proposition 7.9 applied to g = f ), we have (7.148). Hence, by A(e i YA(e i ) = g A(ei )'A(ei)" (7.148) applied to f = g A(e1)' and by g g, we obtain Epi g' )/ x Epi g c Epi g °(e 'Y x Epi ee 1)/°( e1) c

0

Chapter 8 8.1 Definition 8.1, Lemma 8.1, Proposition 8.1, and Remark 8.1(a), (b) were given in [267], def. 2.1, lmm. 2.1, prop. 2.1, remark 2.3, and formula (3.4), respectively.

430

Notes and Remarks

Remark 8.1 (c) was made in [271], p. 109. Theorem 8.1 can be found in [267], thm. 4.1. Remarks 8.2(a), (b) have been observed in [267], remark 4.2 (see also [272], p. 331) and [267], remark 4.1a, respectively. Formulas (8.11)-(8.15) were given in [267], formulas (4.22), (4.21), (4.24), (4.23), and [272], formula (5.13). Formulas (8.16) and (8.19), with R and - replaced by R and ± -, and formula (8.21) can be found in [272], formulas (5.19), (5.17), and p. 332, line 4, respectively. Formula (8.22), with R and - replaced by T? and ± -, has been given in [272], formula (5.25). The partial order (8.23) for conjugations has been studied in [272], §5. Some authors use for the operations + - and ± - on R (which may be called "upper subtraction" and "lower subtraction," respectively) occurring, for example, in (8.3), (8.2), the notations ± and ±, respectively. However, we use here only + - and ± of Moreau ( 1205], [206]), since we need also the binary operations V-, A-, I-, and T- on R (see the Notes and Remarks to Chapter 9). 8.2 Most of the rules of computation with jk, ±, etc. on R (such as Lemmas can be found in Moreau [206]. Theorem 8.2, which has been given 8.2 and 8.3) in [267], thm. 3.1, has suggested an "axiomatic" approach to the theory of FenchelMoreau conjugations via the "axioms" (8.1), (8.2). This approach has led to a study of general dualities A : R x R w , where the "second condition" (8.2) is omitted (see the Notes and Remarks to Chapter 7) and to the discovery of new classes of dualities A : R x > R w , where (8.2) is replaced by various other "second conditions" (see the Notes and Remarks to Chapter 9), with applications to optimization theory (e.g., to fractional programming duality). Formulas (8.29)-(8.31) of Remark 8.5(a) were given in [181], formulas (5.3)-(5.5). Remark 8.5 (d) was made in [267], remark 3.2b. Note that if W c R x and ço = (pnat, then, using the canonical embedding of R x x R into R x x R (given by (3.45)), formula (8.34) can be also written in the form f* (w) =

sup (w, -1)(x, d) (x,d)EEpif

(2.1)

E

W).

(8.407)

We will not consider here the history of the (usual) Fenchel conjugation (its beginnings are sketched, e.g., in Moreau [205], p. 1, and Rockafellar [235], pp. 426-427), nor the results on the usual conjugate for the cases where X is endowed with some structure. For example, when X is a locally convex space, there are a large number of results on conjugates involving the structure of X (conjugates of infimal convolutions, etc.), that can be found in the books on (usual) convex analysis. Also many efforts have been devoted to the discovery of topologies, convergences, or metrics, on R x , R w , where X and W are, for example, locally convex spaces or Banach spaces, for which the Fenchel conjugation from R x into R w is continuous, or a homeomorphism, or, respectively, an isometry (e.g., see Wijsman [313], Mosco [207], [208], Joly [135], [136], Dolecki [59], Attouch and Wets [9]-[11], Beer [23], Penot [216], [217], and the references therein; for applications of these concepts to optimization, see e.g. Attouch [6], Dontchev and Zolezzi [64], and the references therein). The idea of generalization of the Fenchel conjugate, defining the conjugate of a function f : X -> R associated to an arbitrary coupling function yo as in Definition 8.2(a), is due to Moreau ([205 1, p. 93, formula (14.7), and [206], R), p. 122, §4c), and it has been discovered independently (for yo : X x W

Notes and Remarks

431

by Vogel [297], and studied by several authors (e.g., see Weiss [305], [306], Seidler [251], Kobler [144], [145], Elster and Nehse [74], Schrader [248], [249], BrOndsted [34], Balder [19]). For example, Brondsted [34] has shown that the duality theory for triples (X, W, yo), where X and W are two sets and ço :XxW—> R, can be "embedded," in a certain sense, in the "standard" theory for triples (U, V, *), where U and V are linear spaces and 1P. : U x V —> R is a bilinear function on U x V (or, equivalently, V C U# and IA' (u, v) = v(u), the natural coupling function). The particular case of (8.35) where W c Rx and go = (Pnat, i.e., the conjugate (8.38), has been studied by Kutateladze and Rubinov ([152], p. 44) and Dolecki and Kurcyusz [62], [63], and the case where W c R x and ço = (Pnat, i.e., the conjugate (8.37), in [266], §4 (especially for W = Jm of (2.157), where »1 c 2 x ). One may call fc (`P) of (8.35) the upper conjugate off associated to ço and define the lower conjugate of f associated to yo by (8.35) with sup x , x replaced by inf x , x Elster and Nehse [74], where, however, the reverse terminology is used).(e.g,s In the particular case where X is a locally convex space, W = X* and ço = (pnat, the upper (resp. lower) conjugate of f associated to (pnat is also called the convex (resp. the concave) conjugate off (e.g., see'Rockafellar [235], Barbu and Precupanu [21]). Let us mention that the convex conjugate of f has been often called the "polar" of f (e.g., see Moreau [205], Laurent [156]), but the latter term has been used by Rockafellar [235] in a different sense (see (8.475) below). For two sets X and W and a coupling function ço : X x W —> R, a modification of the upper and lower Fenchel-Moreau conjugates has been introduced by Lindberg [159] in order to obtain a "symmetric Lagrangian duality theory" for optimization problems. Namely the upper Lindberg conjugate f u(0 and the lower Lindberg conjugate f n(9') off E R x , associated to yo, are defined [159] by f u('') (w) = sup {—ço(x, w) ± f (x)) = xex

f(w) = inf fço(x, w) xoc

inf

deR f <w(-,w)+d

f (x)) =

sup

d

d

(w e W), (8.408)

(w

E

W), (8.409)

del?

where the second equalities hold by (8.8). Also Lindberg has mentioned in [159], remark 2.2, that one can obtain more general concepts and results, for example, replacing in (8.409) the minorants d — (pe, w) of f by minorants 4:1)(-, w, d), where (I) : X x W x R —> R is such that each (1)(x, w, .) : R —> R is nondecreasing and continuous from the left. As has been noted by Lindberg ([1591, remark 2.2), the extra generality would allow one to study within this framework the "r-convexity" of Avriel [16] and the "F-convexity" of Ben-Tal and Ben-Israel [24]. In [270], p. 21, addendum, it has been shown that the theories of Fenchel-Moreau conjugations and Lindberg conjugations are equivalent. Indeed, by (8.409), (8.25), and (8.35), we have f(w) = inf Iço(x w) xeX

fC ( - (p)( w )

f (x)) = — sup {—ço(x, w) ± — f (x)} x EX (ID E

W)

(8.410)

432

Notes and Remarks

([270], formula (5.14)), and a corresponding relation holds between f u( w) and the lower Fenchel-Moreau conjugate of f. Also, in [270], remark 3.2a, it has been observed that n = n((p) : TV' T?w of (8.409) is not a "conjugation" in the sense (8.1), (8.2), but it satisfies, instead, (inf fi ) n = inf fin iEr iE/ (f

d) n = f n d

({fi}iei g R x ),

(8.411)

(f e -1-2 x , d c7R),

(8.412)

and, conversely, for every mapping n Rx> Rw satisfying (8.411) and (8.412), there exists a unique coupling function yo : X x W —> R such that n = n(yo) of (8.409), namely yo of (8.28), with c replaced by n. As has been noted in [270], p. 13, for Lindberg conjugations one can prove results corresponding to those for FenchelMoreau conjugations (e.g., the above characterization (8.411), (8.412) of the lower Lindberg conjugates corresponds to Theorem 8.2). We have not considered Lindberg conjugates in Chapters 8-10, since it is the theory of Fenchel-Moreau conjugates that is widely used in the literature (and, as observed above, the two theories are equivalent). Along the lines of the axiomatic approach to conjugation operators, Rubinov (see [241] and the references therein) has introduced and studied the following concept: If E and F are two "c2 -lattices," a mapping A : E F is called [241] an antihomogeneous conjugation operator if A is antitone and :

1 Ax) = — A(x) À.

(x E E, X > 0);

(8.413)

by [241], def. 2.1, a c2 -lattice E is a complete lattice in which there is defined a multiplication by positive scalars such that 1.x = x, À.(p,x) = (À.,a)x

(x c E, X, 11.t > 0).

(8.414)

Definition 8.2(b) was given in [267], remark 3.2a. Remark 8.6(c) has been observed in [267], cor. 3.1a. For W = Jm of (2.157), where .A.4 c 2 x , Remark 8.6(d) was made in [261], thm. 5.5 (see also [267], formula (2.16)). Remark 8.6(e) was given in [267], remark 2.2. Remark 8.7(a) has been noted in [267], remark 3.2a. Remark 8.7(c) can be found in [272], formula (5.38). Proposition 8.2 was given in [267], prop. 3.1. Theorem 8.4 is contained in [267], cors. 4.3 and 4.2. For W c R x , (8.55) can be found in Dolecki and Kurcyusz ([63], formula (1.13)), where g* (of (8.55)) is called the "X-conjugate of g." Proposition 8.3, Definition 8.3, and Remark 8.10(a) are due to Moreau ([205], pp. 92-93). The part of Remark 8.9(b) concerning the Fenchel-Young inequality was observed in [188], remark 1.2. Theorem 8.5(a) and the second equality of (8.61) have been given in [267], cor. 4.5 and thm. 5.1, respectively. The first equality of (8.61) is due to Moreau ([206], p. 123). Remark 8.11(a) is contained in [266], formulas (4.2) and (5.4). In the general case, Proposition 8.5(a) was given in [267], formula (3.9). In the particular case considered in Remark 8.12, formula (8.66) and the equivalence G E k(W) (8.67) can be found in Kutateladze and Rubinov ([152], ch. I, formula

Notes and Remarks

433

(5.2) and prop. 5.7). Formula (8.66) is useful, since it permits one to obtain results on the support functions aG as particular cases of results on conjugate functions f*. Let us mention that in the particular case when X is a locally convex space, W = X* and OeG CX, ac = (XG) * can be also expressed as the "Minkowski function," i.e., the "gauge" ([205 1 , [2351), of the set G° = {(1) E X* I sup 43.(G) 1) (that is, of A0 (G) U (0), with A0 (G) of (6.83)) and, dually, the conjugate of the Minkowski function of G is the indicator function x Go of G ° (e.g., see [152], ch. I, prop. 6.1 or [21], p. 95). 8.3 Theorem 8.8 and Corollary 8.5 are due to Moreau [205], p. 123. Their particular cases given in Theorem 8.6 and Corollary 8.2 can be found, for example, in Dolecki and Kurcyusz ([63], thm. 1.19; however, there it is assumed that W+R= W C R x ). By the particular case of Theorem 8.6, where X is a locally convex space and W = X*, we have f **

—

fco(X* R)

(f c le);

(8.415)

this formula, combined with Theorem 3.7, yields the "Fenchel-Moreau theorem" of usual convex analysis. For W-I-R=W c R x , a slightly less complete form of Corollary 8.3 can be found in Dolecki and Kurcyusz ([63], p. 280). Corollary 8.4 is a well-known result of (usual) convex analysis; for example, see Barbu and Precupanu ([21], ch. 2, prop. 1.8). Theorem 8.7 and Remark 8.14(a), (b) were given in [181], thm. 3.7 and remark 3.3. Remark 8.14(c) was observed in [271], formula (3.78). Parametrizations of W c R x , more general than (8.73), have been used by Evers and van Maaren ([80], §8.6). 8.4 Theorem 8.9 and formula (8.98) of Remark 8.16(b), for X as in Remark 8.16(a) and for any submodular function f : X —> R, have been given by Fujishige ([92], thm. 3.1) and, in the general case (i.e., for any f:X—>RU {-Foo}), in [181], thm. 5.3 and remark 5.6. Remark 8.16(c) can be found in [181], remark 5.8 and proof of cor. 5.3b. For X = W = R", the coupling function (pa,N of (8.102) and the conjugate (8.103) have been studied by Martfnez-Legaz ([173], p. 199), but for any metric space X = W, the subset Vw,,,, of R x (of (2.188), with ço = (pa, N ) has been introduced earlier by Dolecki and Kurcyusz ([63], p. 288). Proposition 8.6, Theorem 8.10, and Corollaries 8.7, 8.8 can be found in Martinez -Legaz ([173], prop. 2.1, 2.2, cor. 2.4 and the last formula on p.200), together with some additional remarks ([173], p. 201). For X = R", W = R n X (R + \ {0 } ), the coupling function (pa of (8.111) and formulas (8.113), (8.114) have been studied by Martfnez-Legaz ([173], p. 203), but for any metric space X, the subset Vw, ç,a, of R x (of (2.188), with W = X x (R + \ {0 } ) and ço = ç) had been introduced earlier by Dolecki and Kurcyusz ( 1 63], pp. 284 and 288). Remarks 8.18(a) and (c) were made by Martfnez-Legaz ([173], p. 203). Theorem 8.11 was given, essentially, by Martfnez-Legaz (see [173], prop. 3.5, cor. 3.6 and the last remark on p. 205). For 1 < a < -I-oo, the expression of the conjugate .f( ( P- N ) (with (pa, A, of (8.102)) is still the same, namely (8.103). Recalling that for a metric space X = (X, p), the Moreau-Yosida approximates and the Lipschitz approximates of a function f e Rx

434

Notes and Remarks

are (see Brézis [33], McShane [195], Whitney [3101) the functions fr fid e R x defined, respectively, by 1 fr(w) = xinjc

f (x)

P(x ' "21 1

firi(w) = inf f(x) xex

p(x,w)1

E

R X and

(w e X, r > 0),

(8.416)

(w

(8.417)

E

X, r > 0),

Martfnez-Legaz has observed ([175], p. 181) that, by (8.103), we have f

fC(c2,N)

2N

(N > 0),

(8.418)

(N > 0);

(8.419)

and similarly fe(S01.0

=

fr

j L N]

thus the approach via Fenchel-Moreau conjugates sheds a new light on these approximates. 8.5 Lemma 8.4 and Remark 8.19(a) have been observed by Martinez-Legaz ([175], lmm. 4.1a, b and the remark made after lmm. 4.1). The main part of Proposition 8.8, namely the implication 1 0 3 0 , has been given independently in [273], thm. 5.2, and Voile ([303], p. 117). In the particular case where X is a locally convex space and W = X*, this result has been obtained independently by Toland [290], via some problems of the calculus of variations and mechanics (see also Toland [291]), and in [255], coming from some problems of best approximation and optimization (see [90], [254], [256]); therefore some authors (Martfnez-Legaz [175], MartfnezLegaz and Seeger [178], Michel and Voile [198], Attouch and Théra, who gave a far reaching generalization in [8], and others) call it the "Toland-Singer theorem," while some other authors (apparently unaware of [255], [90], [254], [256]) have called it the "theorem of Toland." Note also that a particular case has been discovered earlier by Pshenichnyi ([230], p. 180, lemma). The other parts of Proposition 8.8 are due to Martfnez-Legaz ([175], thm. 3.1), but here condition 6° of [175] is slightly reformulated and condition 5° is added. Remark 8.20 was made by Martfnez-Legaz ([175], the remark made after cor. 3.2). Proposition 8.7, Corollary 8.9, and formula (8.139) of Remark 8.21(b) (with q(M) replaced by q(u), where u : 2x —> 2x is a hull operator) have been given by Voile ([303], pp. 117 and 120). Propositions 8.9, 8.10, and Corollary 8.10 have been given by Martfnez-Legaz ([1751, lmms. 4.2, 4.3, and cor. 4.2). In the particular case where X is a locally convex space and W = X*, the main part of Theorem 8.12, namely (see Remark 8.21(a)) the implication 1° 3°, has been proved by Hiriart-Urruty ([1171, thm. 2.2), who has also noted Remark 8.19(b) for this case ([117], remark 1), and for X = R", an alternate proof, using the subdifferentiability of h, has been given by Ellaia and Hiriart-Urruty [70]. In the more particular case where W =- X* and both f and h are finite-valued convex functions and h is lower semicontinuous, the main formula (8.135) has been

Notes and Remarks

435

proved earlier by Pshenichnyi ([230], pp. 180-182). The other parts of Theorem 8.12 are due, essentially, to Martfnez-Legaz ([175], thm. 4.1 and remark 4.1). 8.6 Definition 8.4, with —1 instead of —1 in (8.146), has been given in [270], def. 3.1 and, in the present form (with —1 in (8.146)), in [272], remark 5.10. This difference holds for all results of [270] involving coupling functions (and often we will not mention it); for example, see the Notes and Remarks to Theorem 6.5. Remark 8.22(b) has been made in [270], the remark after def. 3.1. Proposition 8.11, Remark 8.23, Theorem 8.13, Corollary 8.11, and Remark 8.24 can be found in [2701, prop. 3.1, remark 3.1, thm. 3.1, cor. 3.1, and the remark made after it, respectively. Definition 8.5, with ?, —1 instead of > —lin (8.153), has been introduced in [270], def. 3.2, and, in the present form (with > —1 in (8.153)), in [272], remark 5.10e. Remark 8.25(c), with —1 instead of > —1 in (8.153), can be found in [270], remark 3.2a, with the following additional comments: For ça "of type {0, -I-oo)," the lower Lindberg conjugate (8.409) becomes f(w)

=

inf xEx

(f E R X , W E W).

f (x)

(8.420)

cp(x

The mapping n((p) : Rx —> R w of (8.420) may be also considered for an arbitrary coupling function yo : X x W —> R and may be called "of type Lou." In the particular case where X = R", W = X*, and (pi = (pnat , the mapping n(yonat ) of (8.420) has been considered by Lau [155] (see also Crouzeix [41], ch. I, §4); this motivates the term "of type Lau." Let us also mention two related concepts, although they are not conjugations in the sense of Definition 8.1. For a locally convex space X and a function f : X —> R, El Qortobi [72] and Attéia and El Qortobi [5] have introduced the "projective polar" : X* —> R of f and the "projective bipolar" f" : X —> R of f defined, respectively, by f ff (0) = min I— inf f (x), — xEx f"(x) = max { sup cl)ex* (10(x)>1

inf f (y),

VEX

inf xEx

sup (Per

(Y)>l

ct(x) T? of f , defined by

— f x (0)

inf xEx

f (x)

if (I) E X * \ 10), (8.423)

— sup f (X)

if (I) = 0,

436

Notes and Remarks

and has observed that (inf,, / f ) 11 ((1)) =- sup ia ((1)) for all (1) E X* \ (0 ) , but not for (1) = 0, and that (f d) H = f H — d (d E R). These "quasi-conjugates" have applications to quasi-convex and reverse convex optimization ([283]—[285]). In [270], remark 3.2a, it has been observed that for ço of type (0, +oo), n(yo) of (8.420) coincides with f n ((p) w )

inf

f (x)

xe X

(f E

Te,

2.1) E

W)

(8.424)

yo(x ,

(since (x E X I (p(x , w) 0)). For a locally convex space 1 ) =[.x E XI 40 (x w) X, W = X * , and = (Pnat, the conjugation n(go) of (8.424) has been studied by Oettli [210] and, with 0 replaced by 0 in (8.424), by Martfnez-Legaz [172]. For X = R', W = (R n )* (i° = (Pnat, and homogeneous functions f, the conjugation a((p) of (8.160) has been used by Passy and Prisman ([213], [215]). Remark 8.25(d) was made in [270], remark 3.2b. For a locally convex space X and a function f : X ---> R, Dragomirescu [65] has defined the "H-conjugate" f H : X* \ {0} —> R of f, by

f H ((I)) = — inf f (x) xEx\fol

(4) E X * \ (0)).

(8.425)

Note that this is different from f" of (8.423); actually, in (8.425), H stands for "hyperplane" ([651, p. 139). Another concept of "conjugate" to a function f, introduced by Crouzeix [41], with the aid of the level sets of f, will be mentioned below (see the Notes and Remarks to Section 8.7). Corollaries 8.12, 8.13, and Remarks 8.26(a)—(d) and (e), (f) were given in [270], cors. 3.2, 3.3, and remarks 3.3 and 3.4, respectively. The equivalence 1' 2x is an arbitrary hull operator, have been given by Voile ([302], pp. 120-121). In the particular case where X is a locally convex space, W = X*, and = (Pnat, Theorem 8.24 is due to Crouzeix ([411, p. 29); in fact in this case, formula (8.341) can be written in the form 8.11

sup dER

g

(f c R x ),

(8.436)

440

Notes and Remarks

where fc(,) : R is the function (8.428). In the same case, part of Theorem 8.25 (namely, the equality of f *( w ) with the last term of (8.344), is due to Rockafellar ([234], formula (1.8a)). Lemma 8.6(a), (b) has been given in [258], prop. 1.1 and cor. 1.1ii, and Lemma 8.6(c) in [263], thm. 1. In the particular case considered in Remark 8.61(b), namely where X is a locally convex space, W = X = (pnat , and d is replaced by d - 1 in (8.226) (i.e., where f'" is the semiconjugate (8.212) of f), Theorem 8.28 was given in [263], thm. 2 and the remark made after formula (31); also, for this particular case, the first part of (8.357) can be found in [263], formula (32). For the case of Remark 8.61(a), namely for a locally convex space X, W = X*, and ço = (Pnat (i.e., where f L(17)(1 '5-‘P) of (8.233) is the quasi-conjugate (8.200)), Theorem 8.29 was given in [263], thm. 2 and formula (31). For the same particular case, Theorem 8.30 can be found in [263], formula (37). Similarly for Remarks 8.65(c)-(e), see [263]. A unified approach to these results on conjugate functions and level sets for a locally convex space X, W = X* and (p = (pnat , has been given in [261], where corresponding results have been proved for the surrogate conjugate functions f (IV '()) (. , d) : X* \ {0) -> R (see the Notes and Remarks to the conjugations L(W , 0) of (8.190)) and their level sets. For X = R" and the "quasi-conjugate" f H of (8.423), Thach ([283], thm. 3.1i) has proved that if f X -> R is upper semicontinuous and f (0) = inf f (X), then ,

f H (.4)) = inf ty E R I (D(x)

1 (x E A,(f)))

( E

X * {0}); (8.437)

note that this corresponds to the particular case X = R" , W = X* \ {0), and ((I)) and ( 1 (x) -1 of = (Pnat , of Theorem 8.26, with (I)(x) > -1 of f (8.350) replaced by 11)(x) 1 of f H ((I)) and (D(x) 1 of (8.437), respectively. Also Thach has noted ([2831, formula (21)) that sd(fH).

n {,DEx*,.(x),,1

}

(f E TjX, d E R),

(8.438)

with the mention that this a polarity relationship between the level sets of f and those of f H , since the right-hand side of (8.438) is the "polar" (see the Notes and Remarks to formula (6.83)) of A—d(f); note that this corresponds to the particular case X = , W = X* \ {0 } and cp = (N at of Theorem 8.31, with 43.(x) > -1 of f L( (P-i ) ((I)) and {(I) E X* I (I) (x) -1) of (8.370) replaced by cD(x) 1 of f H (4)) and {0 E X* (I) (x) 1) of (8.438), respectively. Formula (8.401) of Theorem 8.37 has been given with a different proof (namely, as a consequence of Theorem 8.38, formula (8.403)), by Volle ([302], cor. I.1.7i). Theorem 8.38 can be found in Volle ([302], thm. I.1.6i and [303], p. 210), who has also given formula (8.406) of Remark 8.66(b) ([3021, thm. 1.1 .6i, ii). Since the quasiconjugates, semiconjugates, and pseudoconjugates are particular conjugates of type Lau L(A) (by Remark 8.51(a)), Theorems 8.37 and 8.38 are generalizations of the results of [263]. If X and W are two sets and ço : X x W R is a coupling function, one can apply the results given at the end of the Notes and Remarks to Section 7.6 to the function e : X x WxR -> R defined by (8.30). In this case the function

Notes and Remarks

441

el:XxWxR-->- -R- of (7.76) (with A = B = R) becomes e i (x , w, d) = ça(x, w) - d

(x c X, wc W, dE R)

(8.439)

(this is, essentially, e () of (3.174)), and, by Remark 8.5(d) and (7.75) (with A = B = R), we have fc(cp)

= f 6,(e) = p(ei)

(f

E

(8.440)

R x ).

For e 1 of (8.439) we will denote the duality F (e , ) : 2xx R ---> 2W x R of (7.134) by rw , + ; that is,

F(M)=

n

{(w, r) e Wx RI ço(x, w) ( d r)

(M c X x R).

(x,d)Em

(8.441) Remark 8.67 (a) For e l of (8.439) the coupling function 71 = 7re , : (X x R) x (W x R) R of Remark 7.18(a) becomes

7 1 ((x, d), (w, r)) = ço(x , w) - d - r - 1 ((x, d)

E

X x R, (w, r)

EWX

(8.442)

R),

and the corresponding set 0e , c (X x R) x (W x R) of Remark 7.18(a) (see (7.145)) becomes the so-called (see Dolecki [58]) diepigraph of ço, denoted by Diepi go:

Diepi ça = {((x, d), (w, r))

E

(X x R) x (W x R) I ço(x , w)

d ± r); (8.443)

thus r=, of (8.441) is nothing else than the duality A Q : 2x x R —> 2 w x R associated to the set Q = (X x R) x (W x R) \ Diepi ço (in the sense of Remark 6.13(a)). > 2wxR : 2' --> 2 w x R of (8.441), M = (G, 0) and A0,0 : (b) For r of (6.107), we have

r,(G, cl) = {(w, r) EWxRkp(g, = A(G)

(G

C

w)

r (g E G))

(8.444)

X).

2 xxR 2 wxR Hence, if we identify X with the subset X x {0} of X x R, then F of (8.441) is an extension of Ao, : 2x x {()} ---> 2 w x R of (6.107). (c) For e of (8.30) and M = (G, 0) (where G C X), formulas (7.116), (8.31), and (2.198) yield f(G,0),e(W) =

sup ça(g, w) = o-G , w (w)

(w

E

W).

(8.445)

gEG

(d) For X = Rn , W = (Rn)* and ço = (Pnat, some conjugations from R X x R into TexR have been considered by Flachs and Pollatschek [88] (see also Flachs [86], [871) and Passy and Prisman ([212]-[214]) the observation that these are particular

Notes and Remarks

442

cases of conjugations L(F 1 ) of (7.141) for suitable dualities such as for F = Fço, 4, of (8.441) or for

r,(,), n {(w,r) e 1 4 fx Rlw(x)+dr

< 0)

r, :

2XxR

(M c X x R),

2W xl?

(8.446)

(x,d)EM

has been made by Voile ([302], pp. 300 and 296). (e) Applying Theorem 7.13, Corollaries 7.5, 7.6, and Proposition 7.9 above to e i of (8.439) and using (8.440), (8.441) and (8.443), one obtains results on f), and Diepi ço. For example, by (7.136), we have ,

)

Epi fc(ç'' ) =

(Epi f)

(f e R x );

(8.447)

note also that condition (7.143) for el of (8.439) is equivalent to (2.195). As has been mentioned in the Notes and Remarks to Section 7.6, for X = R n , W = (IV )* , and ço = (pnat , formula (8.447) (with F of (8.441)) is due, essentially, to Fenchel [83]; in the case where X is a normed linear space, W = X*, and ço = (Pnat it can be found, for example, in Luenberger [164], p. 198, and in the general case, in Dolecki [58], prop. 2.5. The other results mentioned in Remark 8.67(e) above, except those on Diepi ço, have been given by Volle ([302], thm. 1.2.4 and 1mm. 1.2.3), with a direct proof (i.e., not as particular cases of results on arbitrary dualities A ( e ' ) : -• Rx---> —Rw, which have not been considered in [302]). The particular case A (e i ) c(ço), e e , = Diepi ça, of Corollary 7.6 can be found in Dolecki ([58], 1mm. 2.4). For a function f : X --> T?+ = [0, d-oo] various other concepts of a conjugate function f : W R+, called in some cases a "polar" function (of f), involving multiplication instead of addition have been introduced and studied by Moreau [205], Rockafellar [235], Tind [289], Elster and Wolf [77], and others. These concepts have found useful applications in fractional programming duality (e.g., see [77], [78]), mathematical economics (e.g., see [289]), and other fields. Naturally, since these conjugations A : Te±( R dualities between the complete lattices IV( and B w where (A, = (B, = (R +,), one can apply to them the results of Chapter 7. However, we will now describe briefly a more refined approach, of [187] and [190], that exploits the multiplicative structure of R+ and encompasses the case of Chapter 8 as well. In order to introduce the general framework (of [187], [190]) for the ranges of the functions, let 11 = (11, o) be a complete ordered group, i.e. (see [26], ch. XIV), a set ri endowed with a partial order such that (11, is a conditionally complete is, every nonempty lattice (that order bounded subset of H admits a supremum and an infimum in H) and with a binary operation o for which (FI, o) is a group such that b, imply that all group translations are isotone (i.e., the relations a, b, c c 11, a cobandaoc .b oc). Then, by a theorem of lwasawa (e.g., see [261, coa ch. XIV, thm. 20), o is commutative. Assuming that 11 is not a singleton (whence, e.g. by [26], ch. XIV, §9, 1mm. 1, ri has no greatest and no least element), let us adjoin to ri = (H, o) a greatest element ±oo and a least element —oo, that is, let us '

443

Notes and Remarks

consider the set

n = fi U [-poo} U (—oc},

(8.448)

with the order extended to IT by a

—00

(a e if,

+oo

(8.449)

and let us extend the binary operation o to two different binary operations .6 and o on 1-1 (called upper and lower composition, respectively) by the rules a.6b=aob=aob

(a, b E II),

(8.450)

+oo .6 a = a .6 +oo = +oo

(a

(8.451)

—oc .6 a = a .6 —oc = —oc

(a e II U Fool),

(8.452)

+oo oa=ao +oo = +oo

(a

E

1-1 U (-Hoop,

(8.453)

—oc c a = a o —oc = —oc

(a

E

VI).

(8.454)

E

11),

Then fi = (— ri, o) is called [187] the canonical enlargement of (II , o), and it is the framework of [187] and [190] for the ranges of the functions. Various properties of (-1?- , , , ) have been extended to (FI, O, o), sometimes with different proofs, in [187] and [ .190], such as sup (a o ai ) = a o sup ai , inf (a .6 ai ) = a .6 inf ai lEI iEt iEt iEI (a

E

n,

n).

(8.455)

A duality A : H x ---> H w is called a o -conjugation, if

(f a) A =

fA

a --1

(f

E

Il x , a

E

H),

(8.456)

where O (in n x ) and o (in fl w ) are defined pointwise, each a E 11 is identified with the constant function fa (x) = a (x c X), and if a c 11, then a -1 denotes the inverse of a in the Abelian group (II, o), while the "inverses" of a E fi \ fi are defined by (+00) -1 = —oc, (—oo) -1 = +oo. Some results on o-conjugations have been proved in [187] and [190]. For example, denoting by e the unit element of the group II, and defining the generalized indicator function of a set G c X by le

if x E G, (8.457)

XG(x) = if x E X\G, we have the following representation theorem ([187], thm. 2.1):

444

Notes and Remarks

Theorem 8.39 —x —w Let X and W be two (nonempty) sets. For a mapping A : H —> 1-1 , the following statements are equivalent: 1°. 20 .

A is a o-conjugation. There exists a coupling function go : X x W > I 1 such that

f(w) = sup" {go(x, w) (f (x)) -1 )

(f E n X , w E W).

(8.458)

XEX

Moreover go of 20 is uniquely determined by A, namely So(x, w) = (X{x)) ° (w)

(x G X, w G W).

(8.459)

Let us also mention: Theorem 8.40 ([1871, thm. 2.4) x — w Let X and W be two sets and A : — —> 11 a 0-conjugation. For a function —x h E H the following statements are equivalent:

10. 2°.

h= We have

inf If (x) xEx

[h(x)]-1 ) = inf {17 ° (w) tvEw

[f ° (w)] -I }

(f E n X ). (8.460)

Theorem 8.41 ([190], thm. 2.1) Let X be a set, let W C n X satisfy Wo WC W,

(8.461)

and let * = A(Sonat) be the o-conjugation (8.458) with go = gonat : X x W ---> n. Then for any f, h G FI X with h = h**, we have (f h -1 )* (w) = sup{ f* (w

y)

[h* (v)] -1 )

(w E W).

(8.462)

VEW

Applying the results on o-conjugations to the particular case where n = R endowed with the usual total order and with the operation o = ± of usual addition, one obtains again some results of Chapter 8 on conjugations c : R x ---> R w (e.g., from Theorems 8.39 and 8.40 one obtains again Theorem 8.2 and Proposition 8.8, equivalence 1° 3 0 ; also Theorem 8.41 yields that the implication 1° 3 ° Theorem 8.12 remains valid for any W c Te satisfying W ± W c W, of with w y of (8.135) replaced by w y). On the other hand., considering n = R ± \ [0} = (0, -Poo) endowed with the usual total order and with the operation o = x of usual multiplication, we have II = R + and +Doll- = +oo, — 00 r1 = 0,

445

Notes and Remarks

whence formulas (8.451)-(8.454) become + co *a=a* +oo = +00

(a E

R +),

(8.463)

0* a =a * 0=0

(a E

R + = 1 0, -koo)),

(8.464)

+ oo xa=ax +oo = +oo 0 xa=ax0= 0

(a E (0, +001),

(8.465)

R +),

(8.466)

(a c

and _ thus * and * are nothing else than the "upper" and "lower" multiplication on +, iR n the sense of Moreau ([205 ] , p.93, formulas (14.8), (14.9)); for any set X, they extend from TLF to T?+x , poinwise on X. Thus, by (8.456) (for this case), a duality A : T?+ x --> R 4w_ is a x-conjugation if

1 (f * a) A = f A X — ' a

(f c Te+ , a c T?+ ),

(8.467)

where 1/±oo = (+ )- I = 0, 1/0 = 0 -1 = +oo. Then, applying the results on oconjugations to this case, some results on x -conjugations have been obtained in [187]. For example, Theorem 8.39 above yields a representation theorem for x -conjugations A: Te+ --> T?+w ([187], thm. 4.1) in the form 1

f A (w) = f A( w) (w) = sup I ço(x, w) x ' f (x)I xEx

(f e R x+ , w c W),

(8.468) with a (unique) coupling function yo:XxW-->TL E and with the usual sup (since X 0 0). Also formulas (8.460)-(8.462) become now, respectively, 1 1 ( xinj { f (x) * h(x) I = winE fw 1 11A(w) 5( f A l (w)I

(fG

n X)

W * W c W,

(f

* -1 )* (w) = sup h vEw

, 8.469) (8.470)

f* (w x v) h* (w)

(w c W),

(8.471)

with * = A((Pnat) of (8.458) (for (p = If X is a locally convex space, W = X*xR, and cp:XxW --> R ± is the coupling function defined by

cp(x , (c1 ) , r)) = (1)(x) ± 0+

(x c X, (I) E X* , r c R),

(8.472)

where for any a E R, a+ = max(a, 0), then (8.468) becomes fA(cp) cp, r ) (

= sup { (cD(x) ± 0 + x Ex

1

f (x) }

(f G

, cl) G X* , r e R), (8.473)

Notes and Remarks

446

whence, in particular, f

(.4) , r) =

sup xEdom f

(4)(x)

r)±

(f

f(x)

c (R + \ {0}) x , (Di c X* , r e R).

(8.474) r, has For X = , the conjugation (8.474), with (cD(x) + r)± replaced by 4)(x) been introduced and applied to fractional programming duality by Elster and Wolf [77]; see also Deumlich [50], where +r is replaced by —r in (8.474) (with X being a linear space and X* being replaced by the algebraic dual X ° of X). Rockafellar ([235], p. 136) has introduced the "polar" of any function f : Rn --> R + defined by R + with f (0) = 0 as the function f 0 : (Rn)* ( (I)) = inf {

O (I)

(4) e (e)*),

1 ± if}

(8.475)

and in [187] it has been observed that if f > 0 on R" (so, in particular, f (0) > 0), then (I) — 1

f° ( 3') = in+

(cD(x) — 1)±

= sup xER ,,

= f (99) ( CD —1)

E (R")*).

(8.476)

f (x)

Tind [289] has introduced the "antiblocking function" of any concave function f : R 1'41_ --> R+ as the function fa : (R")*± —> R + defined by fa ((I)) = inf xERT

(I)( x) k f (x)

(8.477)

E (R)),

where k > 0 is fixed (arbitrarily) and where 1/0 = +oo, and in [187] it has been observed that taking X = RT, W = (Rn)*+, and ço(x, cl)) = (1)(x)

k

(x

c R , cD c (lei)*+ ),

(8.478)

the function fa is the "lower x -conjugate" of f with respect to ço defined by (8.468) with sup replaced by inf (similarly to the "lower conjugate" of f, mentioned in the Notes and Remarks to formula (8.35)). If X is a locally convex space, W = X*, and yo : X x X* —> — R is defined by ça(x , (I)) = (0(x)) ±

(8.479)

(x E X, 14:0 E X * ),

then (8.468) becomes

[goo))

1

sup 104)(x)) + >.< f(x) I xex

(f E R X+ , (Di

e

X*).

(8.480)

The conjugation (8.480) has been introduced by Moreau ([205 1, p. 92, formula (14.4)). As noted in [205], in the particular case where f e R + x is a "gauge" (i.e., a positively

Notes and Remarks

447

homogeneous convex function with f (0) = 0), f °(ça) of (8.480) coincides with the "dual gauge" (defined by (8.480) with (c1)(x))± replaced by (I) (x)). For some related results, see [190], §6. For the coupling functions (pi : X x W -> R + of (8.472) and (8.479), some f = f"' have been given in characterizations of the functions f c R i

[187]. Some conjugation theories for functions taking values in a partially ordered Abelian group have been proposed by Dolecki [57] and Volle [300], [301]. However, these do not encompass the functions with values in R or R +, since (T?, (T?, +), (R +, X), and (14, x) are not groups but only semigroups. In fact in [300] and 1301] only coupling functions ça : X x W -> n have been considered, where 1-1 is a complete partially ordered Abelian group, and then the partial order and the group operation o have been extended to and O on n of (8.448), by (8.449)-(8.452) (but o has not been considered), which has permitted to define conjugates (and biconjugates) of functions f : X -> FI, via (8.458) with o replaced by o, and to obtain some results on them. On the other hand, the approach of Dolecki [57] has been to define the "conjugate" of any element f E E where (E, o) is a complete partially ordered Abelian group, E0 defined with a subset W and a complete sublattice E0, as the mapping f* : W by f*(w) = - sup la

E

Eo lw o a

and the conjugate of any function g : W

f}

E

W)

(8.481)

Eo by

g* = sup {w o (g(w)) -I ); WE

(w

(8.482)

W

thus this has been an attempt to generalize Proposition 8.2 and formula (8.55), with (E, o) and E0 intended to play the roles of (R x , +) and R, respectively. However, T? is not a subset of R x (it is only a subset of the semigroups (R x , ) and (R x , +)); on the other hand, assuming only that E0 is a (not necessarily complete) sublattice of E (for example, E0 = R in E = Rx ) and considering (8.481) only when the sup exists in E0 the "conjugate" f* would be defined only on a subset of W. Due to the importance of "operatorial convex analysis" in the optimization of vector-valued functions (e.g., see Akilov and Kutateladze [1], ch. II, Kusraev [150], Sawaragi, Nakayama, and Tanino [243], Jahn [126], Luc [162], and the references therein) and, more generally, of "set-valued convex analysis" in the optimization of scalar-valued, vector-valued, and set-valued functions (e.g., see Rockafellar [235], Borwein [28]-[31], Aubin and Frankowska [14], Isac and Postolicri [125], and the references therein), it may be of interest to extend the results of Chapter 8 to these cases.

Chapter 9 The results of Chapter 9 were given in [182]. If X and W are two sets and * : X x W -> R is a coupling function, one can apply the results given at the end of the Notes and Remarks to Section 7.6, to the

448

Notes and Remarks

function e = e* :X x WxR —> R defined by (9.28). In this case the function e l :X x WxR —> R of (7.76) (with A = B = R) becomes e l (x, w, d) = 0- (x, w)

A

—d

(x

E

X, w

W, d

E

E

(9.119)

R),

and, by Remark 9.5(a) and (7.75) (with A = B = R), we have f A (1/, ,v)

=

f A(e,p)

f A(ei)

(9.120)

(f E R x ). 2W x R

For e 1 of (9.119) we will denote the duality F (e , ) : 2x x R that is, using also (9.6),

of (7.134) by

r) (x,d)EM

=n

Rw, r) E W x RI 0- (x, w) ri — d}

(x,d)EM

(M

C

X x R).

(9.121)

Remark 9.20 (a) For e i of (9.119) the coupling function n- i = 7re , : (X x R) x (W x R) —> T? and the subset 0. e , of Remark 7.18(a) (see (7.145)) become, respectively, Tr, ((x, d), (w, r)) = (*(x, w)

ee

= {((-7 C

A

—d) — r — 1

((x, d)

G

X x R, (w, r) E W x R),

d), (w, r))

G

(X x R) x (W x R)

(9.122) , w)

ri —

(9.123)

(b) Applying Theorem 7.13, Corollaries 7.5, 7.6, and Proposition 7.9 to e = eo. of (9.28) and using (9.120) and (9.121), one can obtain results on f °(*') , r f L(F and 0e , (of (9.123)), respectively. ,

If X and W are two sets and i) : X x W --> R is a coupling function, one can apply the results given at the end of the Notes and Remarks to Section 7.6 to the function e = e v :XxWxR—> R defined by (9.51). In this case the function e i :X x WxR --> R of (7.76) (with A = B = R) becomes e i (x, w, d) = —dT — v(x, w)

(x

E

X, w

E

W, dE R),

(9.124)

and, by the proof of Theorem 9.2 and (7.75) (with A = B = R), we have ev,_L)

=

f(e)

=

f A(e i )

(f E R X ).

(9.125)

Notes and Remarks

449

2WxR

For e l of (9.124) we will denote the duality [' (e , ) : 2 x R r„, i; that is, using also (9.5), .1(M)

=n

{(w, r) E Wx RI — dT — v(x, w)

of (7.134) by

r)

(x,d)EM

= n {(w , r) E W x

(Ai c X x R).

RI v(x, w) A —r„.< d)

(x,d)EM

(9.126) Remark 9.21 (a) For e l of (9.124) there holds a remark corresponding to Remark

9.20(a), with fr i ((x, d), (w, r)) = (—dT — v(x, w)) — r — 1 ((x, d) Oel =

Mx, d), (D, r))

E X X

R, (w, d)

(X x R) x (W

E

R),

EWX

(9.127)

x R) I v(x , w)

A

—r

d).

(9.128)

(b) For e = ev of (9.51), there holds a remark corresponding to Remark 9.20(b). A unified approach to conjugations, v-dualities, and _L-dualities from R x into R w , where X and W are two sets, has been developed in [189] and [191]. Namely let * be a binary operation on R such that for any index set 1 (including 1 = 0) we have (inf bi ) * c = inf (bi * c)

R, c

({biLE/ C

E

R),

(9.129)

let i be the binary operation on — R defined by a >T c = —(—a * c)

(a, c

E

(9.130)

T?),

and for any set X, extend * and to R x pointwise on X. For two (nonempty) sets X and W a duality A : R x —> — R w has been called, in [189], a *-duality, if (f * d) ° = f

d

(f

Rx , d

E

E

R).

(9.131)

Some results on *-dualities have been proved in [189] and [191]. For example, calling an element e E T? the neutral element for * if e*c=c*e=c

(e

E

R)

(9.132)

(clearly, if such an element exists, then it is unique), and defining the generalized indicator function of a set G C X by

XG x (

)

e

if x

E

G, (9.133)

if x E X \ G,

450

Notes and Remarks

we have the following representation theorem ([189], thm. 4.7). Theorem 9.11 Let _X and W be two sets, and let * be a commutative and associative binary operation on R satisfying (9.129) and admitting a neutral element. For a mapping A : Te the following statements are equivalent: 1°. 20.

A is a *-duality. There exists a coupling function *: X x W --> R such that

f A (w) = sup{*(x, w) )17 f (x)} xEx

(f E R X , w

E

W).

(9.134)

Moreover * of 2° is uniquely determined by A, namely

o- x, w) = (x{x))°(w)

(x E X, w E W).

(

(9.135)

In particular, for * = + (which has the neutral element e = CI and satisfies (9.129), by (8.122)), we have T , = ±— (by (8.25)), and thus from Theorem 9.11 we obtain again Theorem 8.2. One the other hand, for * = v (which has the neutral element e = —oo and satisfies (9.129)), we have, obviously, T = A—, and thus from Theorem 9.11 we obtain again Theorem 9.1. The case of I-dualities, which is not encompassed by the above (since * = 1 does not satisfy the assumptions of Theorem 9.11), has been also treated in [189], by "dualization," using that the dual of a I-duality is a v-duality (by Theorem 9.6), and that the binary operation V on T? has more convenient properties. First, in [189] it has been shown ([189], prop. 2.6) that for any binary operation * on R satisfying (9.129), there exists a unique binary operation */ on R such that for any a, b, c E R we have the equivalence a

b* c . a *lc .. R w is a

451

-duality and only if A' is a *-duality.

If * = -I-, then */ = ± — (by (8.8)), and hence 7 = -i-- (by (8.25)). If * = v, then */ = T (by (9.5)), whence 7 = 1— (by (9.11)). If * = 1, then */ = A (by , = T— (by (9.11)), so in this (9.6)), and hence, obviously, 7 = V—; note also that T case the *-dualities are nothing else than the I-dualities. Thus for * = -j-, v, or 1, Theorem 9.12 yields again Theorem 8.1 and the first parts of Theorems 9.3 and 9.6, respectively. Using Theorem 9.12, one obtains the following representation theorem ([1891, thm. 4.25): Theorem 9.13 Let X and W be two sets, and let * be a binary operation on R satisfying (9.129) and such that >Ti is commutative and admits a neutral element. Then for each *-duality A : R x --> R w there exists a unique coupling function * : X x W --> R such that

f A (w) = sup {— f (x)

— *(x, w)}

(f E R X , w E W),

(9.139)

XEX

namely Ox, w )

= ( x{.}) °'

(

x)

(x E X, w E W).

(9.140)

Moreover the same * is the unique coupling function for which we also have

g" (x) = sup 1*(x, w) WE

g(w)}

(g E R w , x E X).

(9.141)

W

In particular, if * = 1—, then * satisfies (9.129) (by (9.19)), and we have */ = A— (by (9.6)), whence 7 = v, so * satisfies the assumptions of Theorem 9.13. Also = T (by (9.11)), so the *-dualities are nothing else than the I-dualities. Thus Theorem 9.13 yields again part of Theorems 9.2 and 9.6. The main results of Sections 8.5 and 8.10 have been extended in [191] to *-dualities as follows: For a binary operation * on T? satisfying (9.129), we will say that y E T? is a *-singular value [191] if there exist a, b E R such that

aT , 7b < y

(9.142)

Theorem 9.14 ([191], thm. 3.2) Let X and W be two sets, * a commutative and associative binary operation on R satisfying (9.129) and admitting a neutral element, and A : R x R w a *-duality (9.134) with * : X x W —> R of (9.135) taking no *-singular value. Then for a function h E R x the following statements are equivalent:

1 0.

h = h'.

452

Notes and Remarks

2°.

We have

inf { f (x) * —h(x)) = mf th A(w) * — f ° (w))

xeX

(f E R X ).

weW

(9.143)

Theorem 9.15 ([191], thm. 4.2) Let * be as in Theorem 9.14, X a set, W c R x a set of functions that take no *-singular value, satisfying W — W c W,

(9.144)

and A : R x >R w the *-duality

(f E R X , w E W).

f ° (w) = sup {w(x) f (x)}

(9.145)

XEX

If h E R x , h = h, then

(f * —h) ° (w) = sup ( f ° (w

v)

h ° (v))

(f E R X , w E W). (9.146)

VE W

Conversely, if the constant function —e belongs to W, where e is the neutral element of*, and if (9.146) holds, then h = . If 7 is commutative (e.g., if * = +, so Wi = -+), then clearly * has no *-singular value. For * = V one can show [191 ] that the set of all v-singular values is R. Thus from Theorem 9.14 one obtains, for * = and * = y, the equivalences 1° . 3° of Propositions 8.8 and 8.21, respectively. Furthermore, if * = -k then = + —, so (9.144) becomes W + W c W, and Theorem 9.15 yields again the extension of the implication 1' = 3° of Theorem 8.12, mentioned after Theorem 8.41. Finally, if * = v, then T , = A—, so (9.144) becomes WA W C W; also, if the functions w E W take no v-singular value, then the duality (9.145) becomes

f

= sup [w(x) A —f(x)) = —

inf

f (x)

(f E R X , w E W),

SEX

XEX

W(X)=-+CO

(9.147) and thus Theorem 9.15 yields a particular case of Theorem 8.23, implication 1° = 3°. In [189 ] it has been shown how some of the above can be extended to the case 6, o) of a complete where R is replaced by the canonical enlargement fl = (11, ordered group n = (FI, o) so as to encompass also the "o-conjugations" in the sense (8.456) as particular cases. Namely the following unifying framework for be a some results of [187] and [189] has been proposed in [189], §5: Let (H, complete chain (that is, a complete lattice, where is a total order on FI) and let (H, s : (n, be a bijective duality (i.e., a bijective mapping s : FI such that s(inf ai ) = sup s (ai ) iel

iel

OatilE/ g

(9.148)

Notes and Remarks

453

for every index set I). Given a binary operation * on H, define a new binary operation *s on H ([189], def. 5.1) by a *s c = s (s

-

(a, c E 11).

(a) * c)

(9.149)

Then a duality A : if X --> 11 w is called ([189], def. 5.3) a (*, s)-duality if (f * a) ° = f 1' *s a

(f

E

c II).

FI X , a

In particular, if if R, endowed with the usual total order (R, is the mapping defined by s(a) = -a

(a

E

and if s: (R,

(9.150) --->

(9.151)

R),

then s is a bijective duality, and by (9.149) and (9.130), for any binary operation * on R we have a *s c = -(-a * c) = aT, ( c

(a, c

E

T?).

(9.152)

Hence by (9.150) and (9.152), a mapping A : R x --> R w is a (*, s)-duality if and only if it is a *-duality in the sense of (9.131). On the other hand, if 11 = (11, , B W , where A and B are two arbitrary complete lattices (except that in the proof of Theorem 10.3, extended in this way, the last inequality should be replaced by ,4).

a°

10.2 For the beginnings of the history of the usual subdifferential af (x0) and of the usual s-subdifferential a, f (x0) (of (10.36), (10.37), and, respectively, (10.39), for a locally convex space X and W = X*), see, for instance, Moreau [205], Rockafellar [235], and the references therein. For these particular cases the results of Section 10.1 reduce to well-known results of (usual) convex analysis. For arbitrary sets X, W, a coupling function ça : X xW ---> Rand f e R x , xo e X with f(x0) G R, the ço-subdifferential aça f (x0) (see Remark 10.7(a)) has been introduced by Balder [19], and Martfnez-Legaz has observed ([174], p. 632) that the assumption yo (X x W) c R of Balder can be replaced by go(X x W) c R and yo(xo, wo) E R. The usefulness of this latter observation is shown, for example, by the subdifferentials of Section 10.5. For W c R x satisfying W-ER c W and yo = çOnat, the W-subdifferential af (4) of (10.36) has been considered by Dolecki and Kurcyusz [63], and for arbitrary W c R x and yo = (N at in [261], p. 323. For the case where c(yo) is replaced by Lindberg conjugation with respect to go : X x W —> R (see the Notes and Remarks to Section 8.2), corresponding subdifferentials have been studied by Lindberg ([159], [160]). For ço (X x W) c R, the e-subdifferentials 0 se(0 f (x0) (of (10.22), with A = c(ç)) have been introduced by Balder [19], as "e-yo-subdifferentials" af f (x 0 ). In the particular case where X is a locally convex space, W = X* and go = (Pnat 1 Theorem 10.4 can be found, for instance, in Moreau [205], §10. In the general case, formula (10.26) for acm f (x 0 ), Remark 10.6(a), Theorem 10.4, and Remarks 10.7(a), (b), and 10.8(b) were given in [188], formula (2.1), remark 2.1, thm. 2.1 and remarks 2.2a, b, and 2.2c, respectively.

10.3 Theorems 10.5, 10.6, and Corollary 10.2 were given in [1811, thms. 5.4, 5.5, and cor. 5.2a, b. Theorem 10.7 and Corollary 10.3 are due to Martfnez-Legaz ([173], prop. 2.7 and cor. 2.8. Remark 10.9 and Theorem 10.8 were given by Martfnez-Legaz ([173], p. 206 and prop. 3.8).

Notes and Remarks

455

For some results on subdifferentiability with respect to (pc,e ,N of (8.102), with 1 a < d-oo, and other coupling functions go, see also Balder [19], Lindberg [159], Dolecki and Kurcyusz [63], Dolecki [56], Dietrich [55], Rolewicz [239], Flores-Bazdn [89], and the references therein. There are also other concepts of "subdifferentials" that can be studied with the aid of subdifferentials with respect to conjugations associated to suitable coupling functions. For example, in Sections 10.5 and 10.6, various concepts of "subdifferentials" (quasi-subdifferentials, etc.) are studied with the aid of conjugations associated to coupling functions of type {0, —Do} (which, by Theorem 8.14, are nothing else than the conjugations of type Lau associated to arbitrary coupling functions). As another example, let us recall the following notion introduced by Plastria [225]: a function f : R" —> Ti is said to be lower subdifferentiable at xo if there exists called a lower subg radient of f at x o , such that (Do(x) — cpo(xo)

f (x) — (xo)

(x E R", f (x) < f (xo))•

(10.114)

The set of all such (1)0, denoted by a- f (x0), can be studied with the aid of the R" x R subdifferential acm f (xo) with respect to the conjugation c(go) : RRn __> associated to the coupling function go : R" x ((R")* x R) —> R defined by go(x, (41), r)) = min {(1)(x),

(x E R", (1)

E (R)*, r E

R);

(10.115)

one can prove, for example, some results similar to those of Section 10.6b (see Martffiez-Legaz [173], Penot and Voile 1218]). Alternatively, a- f (x 0) can be also studied with the aid of I-dualities (see [182], §6). Some other examples, that can be written as acm f (x0), for suitable ç o , are the "boundedly lower subdifferentiable functions" in the sense of Plastria [225] (see Martfnez-Legaz [173]) and the "a-lower subdifferentiable functions," where a E (0, 1], of Martfnez-Legaz and Romano-Rodriguez [177].

10.4 In the particular case where X is a locally convex space and W = X*, Proposition 10.3 can be found, for example, in Laurent ([156], prop. (6.6.1)). Proposition 10.4, combined with various conditions ensuring (8.143) can be found in the references given in Remark 8.21(d) (e.g., see Laurent [156], thms. (6.5.8) and (6.6.7), and Attouch and Brézis [7], Thm. (1.1) and cor. (2.1)). Propositions 10.3 and 10.4 for the general case, given here, can be found in Martfnez-Legaz [176], prop. 3.1 and Thm. 3.1, together with some related properties. The theory of results such as Proposition 10.4 in which, for a function f constructed from other functions fi, one calculates af (x0) in terms of the afi (x 0 )' s, is called subdifferential calculus; more generally, there also exists a so-called s-subdifferential calculus. For example, Martfnez-Legaz and Seeger have proved ([178], thm. 1 and remark 2) that if X is a locally convex space and f, h : X --> R U {-Foo} are two proper functions finite at xo E X and such that h is lower semicontinuous and convex, then for every s 0 we have ae(f_:,_0(x0), n oe±,f(x0)adi(xo)),

(10.116)

456

Notes and Remarks

where, for any two subsets A and B of X*, A the sense of Pontryagin [226], defined by A -±- B = {(1) c X*1(1)

B denotes their "star-difference," in

B

A)

(10.117)

(for some other applications of star-differences of sets see, for instance, [230], [116], and [180]). Hence, in particular, for E = 0, (10.116) yields (f

—h)(x0) = n fax f (xo) axh(x()))• A.?,0

(10.118)

Formula (10.118) has been extended, in [190], thm. 3.2, to the case of oconjugations (8.458), where X is any set, W c fl x WoW c W, go : X x W --> Fl is the "natural coupling function" (2.184), f, h c (11 U1-1-oop x \ H-ool, h coincides with its biconjugate, xo c X, and f (x o ), h(x0) E 11, with a suitable extension of the notion of "e-subdifferential" to this case (for all E c FI with e e, where e is the o)). unit element of (FI , Let us also mention the following result of Hiriart-Urruty and Phelps ([119], thm. 2.1): If X is a locally convex space, f, —h E Te are proper lowersemicontinuous convex functions and x0 c (dom f) n (dom(—h)), then a(f

—h)(x0) =

n Las E

f (x0)

86.(—h)(4)],

(10.119)

>0

where the bar denotes the closure in the weak* topology of X*.

We will not consider here further calculus rules for subdifferentials nor various other results of (usual) convex analysis on subdifferentials, such as those on dense subdifferentiability (i.e., conditions to ensure that the set {x0 e X I af(xo) 0) is dense in X), properties of the (set-valued) subdifferential map xo E X —> 3f(x0) E 2 x* (semicontinuity, monotonicity, etc.); for W = X* see, for example, Phelps [223] and the references therein. Concerning dense subdifferentiability for some more general sets W c RA', see, for example, Dolecki and Kurcyusz [63], §6; for some results on the set of points of single-valuedness and continuity of the set-valued subdifferential map xo c X —> af (x0) E 2 w and, more generally, of a "monotone" multifunction F : X —> 2 w for linear sets W of Lipschitz functions on a metric space X, see Rolewicz [240].

10.5 The historical development of subdifferentials with respect to conjugations of type Lau has followed a different way, namely from some particular cases and some generalizations thereof (see the Notes and Remarks to Section 10.6) to the general cases presented in Section 10.5. For an arbitrary subset Q of X x W, the subdifferential aL( 2) f (x 0 ) with respect to the conjugation of type Lau L (Q) : R x —> Te (of (8.172)) has been considered by Martfnez-Legaz ([174], pp. 632-633), who has defined it, for any f c k x and xo c X, by formula (10.58); thus, when f(x0) = —oc, that concept is different from 8 L(2) f (x0) of Section 10.5 (i.e., from the particular case c(go) = L (Q) of (10.26)) because of formula (10.33) of Remark 10.7 (b). For 8L( 2) f (x0), with some particular subsets Q of X x W, see the Notes and Remarks to Section 10.6.

Notes and Remarks

457

For an arbitrary duality A : 2 x —> 2, the subdifferential a L(A) f (X0) with respect to the conjugation of type Lau L (A) : R x —> R w (of (8.254)) has been considered by Voile ([302], p. 289), who has defined it for any f E R x and xo E X, by formula (10.59); thus, when f(x0) = —oc, that concept is different from of Section 10.5 (i.e., from the particular case c(yo) = L (A) of (10.26)), because of formula (10.33) of Remark 10.7 (b). For a L(A) f (x0), with some particular dualities A : 2 x —> 2 w , see the Notes and Remarks to Section 10.6. Remark 10.10(b) can be found in Voile ([302], prop. 1.3.1iv). Lemma 10.1 is a particular case of [258], thm. 1.1. Corollaries 10.6, 10.7, Remark 10.11(a), and Proposition 10.5 are due to Voile ([302], thm. 1.3.2, a remark on p.290, and prop. 1.3.4 and its proof).

10.6 Proposition 10.6 was given by Voile ([302], thm. II.1.1iii)). The quasi-subdifferential a)/ f (x0) of (10.72) has been introduced independently by Greenberg and Pierskalla [105] and Zabotin, Korablev, and Khabibullin [3151; this has suggested further new concepts in abstract subdifferential theory. Thus the pseudosubdifferential 3 f(x o ) of (10.104) and the semisubdifferential as f (x 0 ) of (10.88) have been introduced in [257] and [260], respectively; the version of "semisubdifferential" as° f (x0) given in (10.100) has been considered by Lindberg [160] (see also Voile [302] and Martfnez-Legaz [174]). A unified approach to quasi-, pseudo-, and semisubdifferentials has been given in [261] by introducing the following concept: Given a locally convex space X, .A4 of (2.98) satisfying (2.100) and the parametrization (2.101) of Ar of (2.89), the 0-subdifferential of f E R x at xo E X is the subset 30 f(x 0 ) of X* defined by

a° f (x0) =

{ (1) 0 E

= { 00 E

X * I Pxo) = min f (M 4) ) ,4)(00 0} X * I f(x0) = min f (X \ 9(00, (1)o(xo))),

0, the s-9-subdifferential of f and, more generally, for any s the subset 8,6 f (x0) of X* defined by

ae° f (xo) =

{ (1) 0 E

X * I f(x 0 )

E — Rx

min f (14 0 .4) ) (x0)) + s).

(10.120)

at .X0 E X is

(10.121)

Among other results, in [261] it has been shown that

0 9 f (x0) = and that for any s

{ 4) 0 E

X * I 114 4,0,4)(00) n A f(x0)( f ) = 0)

(10.122)

0,

as° f (x0) — {(1)() E X * I

111 430 ,43,00 ) n A f (x o )-F. (f)

= 0);

(10.123).

In particular, E-quasi-, E-pseudo-, and E-semisubdifferentials have been introduced in [262], and among other results, formulas (10.123) for them have been given in [262], Proposition 3.1. Let us also mention the following approach of [274] to abstract subdifferential theory: Let X be a fixed set. For any (nonempty) subset G of X and any h E R X , let

458

Notes and Remarks

us consider the primal infimization problem (PG . i)

inf h(G),

aG,h =

(10.124)

and a dual supremization problem to (P), that is, a problem (QG,h)

fi G ' h

=

(10.125)

sup A( W),

where W is a fixed (nonempty) set and X = X G ' h E T?I'v • Then for any f E Te and X0 E X the subdifferential of f at x o with respect to the primal-dual family of pairs of optimization problems {(P), (Q)) =- 1{(PG,h), (Q")) I G E 2 X \ 0, h E R X ) is the subset 8 {(P) ' ( Q)} f (x0) of W defined [274] by 8l(PMQ)} f(X0) =

{WO

E WI f (xo) =

(wo)}

(10.126)

For some results on these subdifferentials, see [274]. As has been observed in [274], if W c R x and (Q) is the family of Lagrangian dual problems to (P), G, h

)

= sup inf {[h(x) — w (x)} ± inf w (G)) wEw

xEx

(G E 2X \ 0, h E R x ),

(10.127)

then ""Q)) f (x0) coincides with the usual subdifferential (x0) of (10.36), (10.37); while if (Q) is the family of "surrogate dual" problems to (P), in the sense of Section 0.8b, then al (13).(Q)) f(x0) coincides with the surrogate subdifferential f (x0) of (10.120). In particular, for example, if X is any set, W c R x , and G, h)

fiG,h

sup wew

inf

x€X w(x)?.inf w(G)

(10.128)

then al(P ' 1 f (x0) reduces to the quasi-subdifferential aY f (x0) of (10.72), with X* replaced by any W c Proposition 10.8, on subdifferentials with respect to Greenberg-PierskallaCrouzeix quasi-conjugations (8.200), is due to Martfnez-Legaz ([174], remark 4.4). Proposition 10.9, Remark 10.15, and Corollary 10.8 have been given by Lindberg in [160], p. 10, where it has been also observed that au") f (xo) of (10.71) is a convex cone with vertex 0. Part of Proposition 10.10(a), namely, that aY f(x0) is a convex cone with vertex 0, has been shown independently by Greenberg and Pierskalla ([105], thm. 6iv, ii) and Zabotin, Korablev, and Khabibullin ([315], p. 65). Proposition 10.10(b) has been given by Zabotin, Korablev, and Khabibullin ([315 1, p. 66), and independently the equivalence 1° . 2° has been obtained also by Greenberg and Pierskalla ([105], thm. 6v). Remark 10.16(b) has been noted by Greenberg and Pierskalla ([105], thm. 6ii). Formula (10.79) of Remark 10.16(c) has been given, independently, by Greenberg and Pierskalla (in [105], thm. 6iii, which contains also the example (10.80)) and Zabotin, Korablev, and Khabibullin ([315], p. 65). Formula (10.82) of Remark 10.16(d) can be found in Greenberg and Pierskalla ([105], §6),

Notes and Remarks

459

where it has been taken as the definition of aY f (x0), while formula (10.72) has been deduced from it (as [105], thm. 6i). Definition 10.4 of the semisubdifferential as f (x0) has been given in [260], formula (24). Remark 10.19(a) and formula (10.98) of Remark 10.19(b) have been observed in [260], formulas (21) and (24). Formula (10.99) for aL( ") ) f (x0) has been given, implicitly, by Lindberg [160], who has noted that "it is hard to get" such subgradients, "except in exceptional cases." The results on a L(w , 00) f (x 0 ) corresponding to Propositions 10.13(b) and 10.14, mentioned in Remark 10.19(c), can be found in Voile prop. 111.3.3 and thm. 111.3.5), where it has been also shown that L(w.()()) f (x0) ([302], is a weak* evenly convex cone ([302], thm. 111.3.4). Definition 10.5 of the pseudosubdifferential f (x0) has been given in [257], formulas (4.1), (4.2). Remark 10.20 (b) can be found in Voile [302], prop. 111.4.3 (see also Martfnez-Legaz [174], cor. 2.14). Remark 10.20(d) has been made in [257], prop. 4.1. Formula (10.110) of Remark 10.20(e) can be found in [257], where it has been taken as the definition of az f (x 0 ), while formula (10.104) has been deduced from it (as [257], formula (4.2)). For other results on ay (x 0 ) and its relations to aY f (x 0 ) of (10.72) and af (x 0 ) of (10.36), see [257]. For the subdifferential aL( "f (x0) with respect to the conjugation of type Lau L(W, 0) of (8.242), some results corresponding to those of Section 10.6b on subdifferentials with respect to quasi-conjugations and on quasi-subdifferentials (with X, X* and in X replaced by I?", U(Rn) and L in R", respectively) have been given by Martfnez-Legaz ([171], [174]) and Voile ([302], p. 303 ). 10.7 The remarks of Section 10.7 were given in [188], §3 and §4. Some applications of A (v, 1)-subdifferentials (10.113) to lower subgradients (10.114) have been mentioned in [182], §6.

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Notation Index

aff G, 62 Ad(f), 72 A(W x R), 84 A(x), 141 A(x), 141

dom f, 22, 72, 111, 229 dom u, 391

D = D(E, F), 186 D, 59

Diepi go, 441 D(M), 64

B, 6, 13, 44, 53 e, e(x, w, a), 230 eo, eo(x, w, a), 232 el, ei(x, w, a), 232 e', 233

Y, 395

c, 16,242 c(gp), 248 é- (go), 277

CA, 231

e,, 448 e (y, ) , 407 e*, 448

[c], 275 [c]A, 296 co(F), 12 co G, 7, 57

eco G, 61 econe G, 61 E, (E, 34 (E, 0), 447 Ex, 427 Epif, 95, 225 Exts G, 394 95 E(W), 97

G, 59

COB, 8 co B

G, 8

COB x, 44

com, 8, 38 com G, 8,51 coivt x, 37 co(W), 11 COW G, 8, 72 cow, y, G, 88 cone G, 60 c(W, go), 255 C, 167 Ccc , 167

f, 123,145, 446 fa , 446 lc, 17, 242 fcc' , 245

—x —w C( R , R ), 250

fc0,

l

CL(R x , ), 280 C(B), 44 C(JVI), 8, 34, 51 C(u), 9, 56 C(W), 93 C(W; x0), 102 C(W; go), 127

feo(T), 11 fc0(w), 11,93 fco(W,y9), 128 f c(w4) , 255 fc () , 248 f`' (0 , 277 fc;, 290

fc--6, 23

475

476

f', 288 47, 292 ( , 437 438 feci , 144 fc,h, 20 f" , 435, 436 f L( ") , 285 f L(M) , 284 f L(A) , 298 f L((P ) , 276 f L(ç2) , 280 fm((P) , 17 fq , 139 fzi, 142 fq (13), 166 Aca 145 fq (m), 131 fq(u), 166 fy (w), 148 fq( w, (p) , 162 f,-, 434 fir ], 434 (f, r)(x , d), 99 f", 291 g) , 240 11, 93 fYY, 289 f A , 16,219 f A(e) , 231, 241 f °(" ) , 232, 241 .f A( e 1) , 232 f A(r) , 420 f A(r1) , 425 f AA ', 222 f , 289 f", 292 q) f , 420 F, 437 F- , 437 1F1, 390 146, 307 f>., 307 f*, 16, 248, 444 f**, 255 f # , 435 f", 435 f 0 , 446 f u(0 , 431 f n(9') , 431 , 243 g` ic , 245 gds , 291 gdY , 288

Notation Index

edr, 292

E s 438 g A' , 221 g A' A , 222 g', 289 g* , 252 Graphf, 118, 420 6w, 8 G(Y), G(Y1), 198 GO, 79 • 88 G (a^ ' 401 ,• 416 417 • G v , 416 G vv , 417 G () , 14,416

Gm , 417 Hypof, 95

Ha , 121 Ha, N , 121

inf E , 34 81 J.A4, 81

K, 162, 167 K 90 , 167 1C(W), 8, 72 1C(W, (09), 86

lirn infAf, 408 lirn supA f , 408 lin G, 62 Lac1), 279 LG.17 , 20, 24 L r , 27 L(W , 6 ) , 285 L(M), 284 L(6,), 298 L(cp), 276 Lac,* , 278 L ((p), 281 L(S2), 280 L(R"), 141 L(I?", R r 404 ),

max, 34 min, 34 Ma , 65

ku , 66

Ma , 67 if/a, 68

Notation Index

M,4, 13, 34, 51 M w, 84 MA, 193 MA,T, 184, 193 M 99 , 287

Ar,

U ( f), 93 U (G), 51 U(x), 36 U(R"), 141 U(Y), 41 /4( ko),

69 Afx, 408 N ^ 401

vw.,p , 86

0, 398 0(x), 143

Vw. so , 86 Va , 121 Va .A1 , 119

P = PG,h, 20, 23 mud, 132 , 14 (I) ) = (PG,h), 20 (PA) = (Pi`; h ), 20 Prx., 377 q (xo, ; cl)), 291 Q(B), 165 Q(B; xo), 166 Q(M), 129 Q(M; xo), 139 Q(u), 165 Q(u; x0), 166 Q(W), 146 Q(W; x0), 150 Q(W, y9), 162 Q((W, y9); xo), 163 (Q) = (Q"), 20, 24 (Qs) = (VP? ), 28 (QE) = (Q h ), 29 (R" ,T), 398 R +, 17,445 R, 10 R +, 17 xv, 11,72 , ), 13,92 sup L , 34 Sd CO, 72 S(W x R), S(W x R), 72 Sy (h), 228 T, 181, 192, 245 T f (x0), 382 T. A , 183 To, 222 T, 398

u, 8, 13, 47, 52 u,,, uin+ 1 , 392

v, 11, 93

W, 8, 70, 72 ÜV, 90 (W, 0), 70 x, 188 xs, 111 (x, y), 2, 3 X#, 57 X', 58 X- = (x, ?,), 67 (X, B), 391 (X, 0), 398 Y, 40, 53, 94, 225 Yo, 95, 225 k, 95 ko, 95 Z, 398 ceG,h, 20, 458

p, 79 fi G ' h ,

21, 458

G ' /1 ,28 fq,h ,

y (x , a), 226 Yul,h , 350 V vr,h' i 355 F (e), 420 F(X, W),407 PA(e ), 423 FA(e, ), 426 449 441 v , 448

A, 15, 172 A', 15, 176 A", 178 --A",

189

477

478

Notation Index

Ac , 296 A(e), 231, 241 A(e0), 232, 241 A(ei), 232

a, 79

183, 193 amax, 186 amin, 186 A w , 223

EG,w, 29 [(09], 216

A .A4,

w , 223 A(F), 420 A(F1), 425 15, 176 Ao, 195 A(v, 1), 345 4, 210 Ai ço i, 216 • 357 A(*, V), 340

A -

(p', 210 (pc , 248 (Pnat, 85 (pa , 266 (Pcy,N, 263 211 cp[Ai, 216 PA,-j-' 357

(

(py, 89 (ion, 212 (PI, 215 (491)[(pi, 216 XG, 21, 81, 443, 449 X(A} ± a, 224

(f), 237

Vfm,,„ 421 Vfm,,,, 426

137

0, 70 0A, 196 Oço , 89 0° , 90 e e , 421 e el , 426, 428 K,

E,29

208

,

11G,

aG, aG.v, 88

Kw, 77

*A .1,, 340 *OW, 90 co, 41 co, 96

S2,., 280 S2A, 208 go , 212

k, 79 K

49'

+00 E ,

261

,

, 30, 31, 458

• 21,24

00 E , 34

2 x , (2 x , D), xv,, 13,50 32 0, 5, 34

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