Duality in Optimization and Variational Inequalities (Optimization Theory & Applications)

DUALITY IN OPTIMIZATION AND VARIATIONAL INEQUALITIES

Optimization Theory and Applications series of books and monographs on the theory and applications of optimization. Edited by K.-H. Elster t and F. Giannessi, University of Pisa, Italy Volume l Stochastic Linear Programming Algorithms: A Comparison Based on a Model Management System J. Mayer

Volume 2 Duality in Optimization and Variational Inequalities C.J. Goh and X.Q. Yang

DUALITY IN OPTIMIZATION AND VARIATIONAL INEQUALITIES

Department of Mathematics, Auckland University and Department of Mathematics and Statistics, University Of Western Australia and

X.Q. Yang Department of Applied Mathematics, The Hong Kong Polytechnic University

London and New York

First published 2002 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneouslypublished in the USA and Canada by Taylor & Francis Inc, 29 West 35th Street, New York, NY 10001 Taylor & Francis is an imprint of the Taylor & Francis Group O 2002 Taylor and Francis

Publisher's Note This book has been prepared from camera-ready copy provided by the authors. Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Every effort has been made to ensure that the advice and information in this book is true and accurate at the time of going to press. However, neither the publisher nor the authors can accept any legal responsibility or liability for any errors or omissions that may be made. In the case of drug administration, any medical procedure or the use of technical equipment mentioned within this book, you are strongly advised to consult the manufacturer's guidelines. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record has been applied for.

ISBN 0-415-27479-6

Dedicated to

April and Linus

Zoe and Lucy

CONTENTS

xi xiii

Figure Captions Preface

Chapter 1 Mathematical Prelimimaries 1.1 Basic Notations and a List of Symbols 1.2 Elements of Convex Analysis 1.3 Fenchel Transform and Conjugate Duality 1.4 Elements of Graph Theory 1.5 Elements of Optimization and Variational Inequalities

1.6 WeakIStrong Duality, Duality Gap and Gap Function 1.7 Motivations for Studying Duality

Chapter 2 Duality in Network Optimization 2.1 Duality of Paths and Cuts and Minimum Spanning Trees 2.2 Duality in a Painted Network 2.3 Flow, Divergence, Potential and Differential 2.4 Duality of Max Flow Min Cut 2.5 Duality of Max Match Min Block vii

2.6 Duality of Max Tension Min Path 2.7 Duality of Feasible How and Feasible Potential 2.8 Duality of Linear Optimal Flow and Optimal Potential 2.9 Duality of Convex Optimal Flow and Optimal Potential

Chapter 3 Duality in Linear Systems 3.1 A Crash Course in Linear Programming 3.2 Duality in Linear Programming 3.3 Duality and Complementary Slackness 3.4 Duality and Theorems of Alternatives 3.5 Painted Index Theory 3.6 Duality in Monotropic Optimization

Chapter 4 Duality in Convex Nonliwar Systems 4.1 Conjugate Duality in Unconstrained Optimization 4.2 Optirnality Conditions for Constrained Optimization 4.3 Lagrangian Duality 4.4 Lagrangian Duality Specialized to LP, QP and MP 4.5 Wolfe Duality and Gap Functions for Convex Optimization

Chapter 5 Duality in Nonconvex Systems 5.1 Examples of Nonzero Duality Gaps

5.2 Zero Duality Gap via a Nonlinear Lagrangian Function 5.3 Optimality Conditions via a Monotone Function 5.4 Optimality Conditions via a Monotone Composition Formulation

5.5 Zero Duality Gap via an Alternate Lagrangian Function 5.6 Zero Duality Gap for a Discrete Optimization Problem 5.7 Zero Duality Gap via an Extended Lagrangian Function

viii

Chapter 6 Duality in Variational Inequalities 6.1 6.2 6.3 6.4

Duality in Variational Inequalities Gap Functions for Variational Inequalities Primal and Dual Gap Functions for Extended Variational Inequalities Gap Functions and Dual Fenchel Optimization

Chapter 7 Elements of Multicriteria Optimization 7.1 7.2 7.3 7.4

Elements of Multicriteria Convex Analysis Vector Conjugate Duality Scalarization of Multicriteria Optimization Optimality Conditions for Multicriteria Optimization

Chapter 8 Duality in Multicriteria Optimization 8.1 8.2 8.3 8.4 8.5 8.6

Duality in Linear Multicriteria Optimization Conjugate Duality in Convex Multicriteria Optimization Lagrangian Duality in Convex Multicriteria Optimization A Case of Symmetric and Self Duality Gap Functions for Convex Multicriteria Optimization Duality in Convex Composite Multicriteria Optimization

Chapter 9 Duality in Vector Variational Inequalities 9.1 9.2 9.3 9.4 9.5 9.6

Introduction to Vector Variational Inequalities Scalarization of Vector Variational Inequalities Duality in Extended Vector Variational Inequalities Gap Functions for Extended Vector Variational Inequalities Generalized Gap Functions A Solution Method and a Related Gap Function

References Index

193

FIGURE CAPTIONS 2.1.1 2.5.1 2.5.2 2.8.1 2.8.2 2.8.3 2.9.1 2.9.2 4.3.1 4.3.2 5.1.1 5.1.2 5.1.3 5.2.1 5.4.1

A planar graph and its dual A graph representation of the matching problem Proof of Konig-Egervafy Theorem Kilter curve for arc j in optimal flow Kilter curve for arc j in optimal potential Kilter curve for arc j in optimal flow/optimal potential A convex function and its kilter curve Graphs and kilter curves for Example 2.9.1 Geometry of Lagrangian duality Zero duality gap for convex optimisation Supporting hyperplane to a convex perturbation function Example of a nonzero duality gap Supporting cone to a nonconvex perturbation function Dual function for Example 5.2.1 Monotone composition function for Example 5A.2

PREFACE

The concept of duality is ubiquitous in mathematics. It appears in group theory, number theory, mathematical physics, control theory, stochastic processes, financial engineering, the list goes on. The primary objective of this book is to focus on duality in optimization, and its generalization in variational inequalities. This is a domain that can sometimes be considered totally esoteric and suitable only to the purest of pure mathematicians, but at other times can be used in practical ways for solving engineering and business problems. Needless to say, the concept of optimization is paramount in just about every area in physical/natural/social/financial science and engineering. Companies use it to save money, engineers use it to design superior products, governments use it to plan the economy, biologists use it to study brain functions, and even mathematicians have used it to win the Nobel prize in economics! While there is an abundance of books that specialize in optimization, the great majority of them are concerned with furnishing methods, algorithms and applications. Some advanced monographs dwell more in the theory, but there are very few books in the literature that are entirely devoted t o discussing duality. Two notable exceptions are found in [R21 and [Wl], which both are devoted to a generalized Lagrangian model based on conjugate duality. Some justifications for the existence of this book are as follows. There are, in the main, two approaches to the study of duality in optimization models. Classically, the primal optimization problem is first defined and then the duality theory is constructed and developed from there. At a more abstract level, the primal and dual optimization problems are derived from an abstract Lagrangian function L : X X y -+ IR in the following manner: (primal ~roblem) (dual problem)

min f (X)

where f (X) = sup L(x, y ) ,

y..; g(y)

where g(y) = inf L(x, y),

YEY

xEX

xEX

where X and Y are appropriate domains defined in some primal and dual spaces. This approach to duality is founded on conjugate duality and invariably a convexity assumption is made. This approach also places strong emphasis on the saddle point property and minimax theorems (see Section 1.6). There is much merit in pursuing duality in this abstract way. Apart from its complete symmetry, it can be reduced to, as special cases, many other duality results in optimization. Nevertheless, we have chosen not to pursue duality in the abstract Lagrangian way, the main reason being that the two books [R21 and [Wl] have already done an excellent job with such an approach. S..

Xlll

Furthermore, this book is intended to cover specialized duality results ranging from network programming, to vector optimization, to vector variational inequalities, and some of these are yet to be considered as special cases of the abstract Lagrangian approach. Lastly, convexity is imperative in the abstract approach. The conventional wisdom is that when the convexity assumption is not satisfied, then it is difficult t o derive necessary and sufficient optimality conditions apart from those trivial ones based on definitions. In Chapter Five, we discuss strong duality results and equivalent optimality conditions in the absence of convexity. This is made possible only if we are prepared to depart from the Lagrangian model. The first exposure to duality that most undergraduates encounter is likely to be the duality in linear programming (LP). From our experience in teaching this subject to undergraduates, the probability of the following question, or its variant, recurring each year is close to one: "OK, this duality stuff is all very nice, but so what? Does it help me to solve the problem?" Our answer to this question is not usually well accepted by the average undergraduate who is typically obsessed with doing sums. In fact, it would usually take several years of work in optimization theory before one would appreciate the assertion that many methods and ideas in optimization are practically driven by duality. To help users of optimization to understand and appreciate this assertion is thus the main aim of this book. Most readers would understand duality as:

A pair of optimization problems, the primal and the dual, both share the same optimal cost, and the solution of one yields the solution to the other. Examples of this are found in Fenchel Duality Theorem, the Linear Programming Duality, the Lagrangian Duality Theorem, and vector optimization. In a broader sense, duality relates two systems, or geometric entities, or sets, or functions, or objects, in one of the following ways: Theorems of alternatives: Given two systems or sets, if one is non-empty, the other one must be empty, and vice-versa. Examples of this are found in the Farkas Lemma, the Painted networklindex theorem, the feasible flow and the feasible potential theorems. Duality via transformation: A set of transformation rules that allow one object/set/function t o be transformed into another. A further transformation under the same rules returns the original object. For example, planar graph duality, cone duality, and conjugate functions. Properties of one system have a one-to-one correspondence with the properties of another. For example, cycles and cuts, match and block, flow and potential, controllability and observability in control theory, and the inverse of an one-to-one map. Readers will no doubt discover other ways in which duality can occur. It is clear that all these manifestations are centered around some kind of symmetry. In fact duality is effectively the study of a form of symmetry. xiv

By and large, this book is intended for readers with a prior knowledge of optimization, and who have already gained some exposure to computational algorithms for solving optimization problems. Our contention is to emphasize and explain duality as clearly as possible from a theoretical perspective. To avoid getting entangled in unnecessarily abstract notions, just about all results in this book are presented for finite dimensional vector spaces such as R". Most results in this book can be easily extended to abstract spaces, albeit with added notational complexity. This book is organized in nine chapters. Loosely speaking the first four chapters are of relatively light weight while the last five are of relatively heavy weight. The light weight chapters are used to remind the readers of some foundational material and to facilitate the understanding of duality in a clear and concise manner. Chapter One contains the standard mathematical preliminaries that advanced readers may skip. Chapter Two is a collection of duality results in network optimization, which in our opinion is probably the simplest way of understanding duality as there is usually some kind of visualizable geometry involved. Chapter Three is concerned with duality in linear programming and associated linear systems. Linear programming is likely to be the first encounter of most readers with duality. The reason why we choose to put this chapter after that of network optimization is due to several reasons. First, it is easier to understand duality in the network context. Second, the topics on painted index theory and monotropic optimization are generalizations of the network problem, yet these contain significant overlaps with linear programming (LP) and readers would have been better off understanding LP duality first. Chapter Four is concerned with traditional duality results in nonlinear programming. Chapters Five to Nine contain fairly recent research materials, many of which have yet to appear in book form. Chapter Five attempts to break away from the traditional Lagrangian duality theory and proposes new approaches to the duality of nonconvex nonlinear systems. Chapter Six is about duality in variational inequalities, a subject that includes optimization as a special case. From Chapter Seven onward, multicriteria consideration dominates, and we shall begin with an introduction to multicriteria optimization and vector variational inequalities in Chapter Seven. Chapter Eight is devoted to the study of duality in multicriteria optimization, a topic of great theoretical interest. The last chapter contains fairly recent research results in the duality of vector variational inequalities. We are indebted to Bruce Craven, Gaven Martin, Alistair Mees, Elijah Polak, Alex Rubinov and Kok Lay Teo for their valuable advice, comments, and feedback on earlier drafts of this book. The final draft has been improved by the watchful eyes of Xuexiang Huang, Shengjie Li and Xinmin Yang who assisted in the proofreading. Les Jennings' help in our struggle with !Q$ has made it possible for us to meet the publication deadline. We thank Frances Horrocks of Taylor and Francis for coordinating the final effort in getting this book to press. This book could not have been written without the earlier inspiring work of Terry Rockafellar, who has also provided us with his encouragement, and his kind permission in allowing us t o extract materials from his book. We are in particular grateful to Franco Giannessi for his continuous encouragement and detailed comments during the preparation of this book. We acknowledge that our research activities have been continuously

supported by several grants from the Australian Research Council and the Research Grants Council of Hong Kong.

xvi

CHAPTER I

MATHEMATICAL PRELIMINARIES

The material in this book spans a fairly wide range of mathematics. It would be foolhardy to attempt to cover all the requisite knowledge and theories and hence only that relatively small subset of fundamental material is presented here so as to make the book as self-contained as possible. The purpose of this chapter is not to facilitate comprehensive understanding by a novice but rather to ensure the notations and basic concepts introduced are consistent and to remind the reader of certain fundamental results. The reader is therefore assumed to have an elementary grounding in linear algebra, real analysis, calculus, and have done at least a first course in linear and nonlinear programming. Some exposure to graph theory, vector optimization and variational inequalities would certainly be an added advantage. Nothing in this chapter is new of course. It is inevitable that materials are often imported, in an abridged form or otherwise, from other sources, but with notation suitably modified to conform to the standard notation of this book. We have tried to present the results complete with proof most of the time, although occasionally some results of a more involved nature are presented without proof. Whenever in doubt, the readers should refer to the following classics. Much of the materials in this chapter are also extracted from these references. Real Analysis: [R4], [R5]. Convex Analysis: [B3], [ETl], [Rl], [HLl], [vTl]. Linear and Nonlinear Programming: [Ll], [BSI], [Cl], [C2]. Graph Theory and Network Optimization: [R3], [KHl], [AMOl], [H3]. Vector Optimization: [SNTl], [L2], [Jl]. Variational Inequalities: [NI].

MATHEMATICAL PRELIMINARIES

2

1.1 Basic Notation and a List of Symbols For the benefit of the reader, we have compiled a (possibly incomplete) list of notations and symbols used throughout the book, so that one may refer to this section for a vaguely-familiar-but-can't-remember symbol. This section is not intended to be carefully read and digested at the first sitting, but t o be browsed through and its existence noted. In general, a vector is a column vector, unless otherwise specified, and is typically represented by a lower case Roman or Greek alphabet in bold face, e.g., X or 5. The superscript of a vector represents its ordering in a sequence of vectors, e.g., xi is the ith vector in a sequence of vectors. The transpose of a vector X is denoted by xT. The ith component of the vector X is typically denoted by xi or [x]~.The unit vector ei has entries given by [ei] = 6,, where 6, is the Kronecker delta tensor with 6ij = 1 if i = j and bij = 0 otherwise. A matrix is typically represented by an upper case Roman or Greek alphabet in bold face, e.g., A or A; the superscript of a matrix represents its ordering in a sequence of matrices, e.g., Ai is the ith matrix in a sequence of matrices. The (i, j ) entry of a matrix A is typically represented by Aii or [AIij. The transpose, rank, determinant, trace, kernel of a matrix A are denoted respectively by A ~rank , (A), det (A), tr(A) , ker( A). The identity matrix is denoted by I. The kth column of A is denoted by a k , and the kth row of A is denoted by ak. A scalar is typically represented by a lower case Roman or Greek alphabet; A superscripted scalar represents its ordering in a sequence of scalars. A set is typically represented by an upper case Roman or Greek alphabet in calligraphic font, e.g., A or N;if a set A is discrete, IAl denotes its cardinality. An abstract space or subspace, is typically represented by an upper case Roman or Greek alphabet in script font, e.g., S or B ; dim(S) denotes the dimension of the spasce S.

A scalar-valued function is represented by a lower case Roman or Greek alphabet, e.g., f (X) is a scalar-valued function of the vector variable X, a vector-valued function is represented by a lower case Roman or Greek alphabet in bold face, e.g., f( 4 -

IR is the set of real numbers;

R=IRU{CU);

IR" = R X R X -

X

G>

R is the Euclidean vector space of dimension n;

= { F E IRnl 0 , j = 1,2, - - - , n ) is the closed positive orthant of R"; and int R: = {E E RnI Ej > 0, j = 1 7 2 , . - - , n is ) the interior of R : .

Rn= IRn U {m) where m

nent .

is the imaginary point which has m for every cornpo-

Given a reallscalar-valued function f , dom(f) is the effective domain of the function f ;

MATHEMATICAL PRELIMINARIES

3

Range(f ) is the range of f ;

V f (X') or a row vector;

g I x o is the gradient of f with respect to

a

df (xo) is the convex subdifferential of f at xO;

a

dof (xo) is the Clarke subdifferential of f at xO;

X

evaluated at xO,and is

a f;(J) and fL(

By default, 0, and

+

Consequently yy (1 - 7)z E r ( X ) and hence r ( X ) is convex. Furthermore X G M r ( X ) since every X E X can be written as lx.

Definition 1.2.7 A set X C IR? is called an a f i n e set (or a linear variety) if given X', x2 E X, all affine combinations of X' and x2 also belong to X, i.e. axl (1 - a)x2 E X.

+

Theorem 1.2.5 Every affine set X

C IRn is a translate of a subspace.

Proof: We need only to show that the set y = X - X for some X is a subspace. Choose any point X E X, then clearly y contains the origin. Pick any y l ,y2 E y ,

MATHEMATICAL PRELIMINARIES

9

then by definition, there exist X', x2 E X such that y1 = X' - X and y2 = x2 - X. But y1 y2 = X' - X x2 - X = (X' x2 - X) - X E y, since (xl x2 - X) E X.

+

+

+

+

D

Definition 1.2.8 Let X C R". The afine hull of X, denoted aff(X), is the intersection of all affine sets containing X (i.e., all affine supersets of X). Remark 1.2.2 Clearly X 0

aff(X);

aff(X) = X if and only if X is affine; aff (X) is the smallest affine set containing X.

Theorem 1.2.6 aff (X) is the set of all affine combinations of points in X. Proof: Let the set r ( X ) of all affine combinations of points in X be defined by:

If X

X E r ( X ) , then x = C:=, ~ i x h i t X% h X C aff (X). But aff (X) is affine, hence E aff(X). This implies that r ( X ) C aff(X).

To show that aff(X) C r ( X ) , we need only to show that r ( X ) is an affine set that contains X. Take

and y E IR. Then

where

Consequently yy+ (l-y)z E l? (X) and hence r ( X ) is affine. Furthermore X since every X E X can be written as l x .

c r(X) I

Definition 1.2.9 The dimension dim(X) of an affine set X is the dimension of the unique subspace that is translated from X. The dimension of an arbitrary set X is defined to be dim(aff (X)), i.e., the dimension of the affine hull of X.

10

MATHEMATICAL PRELIMINARIES

Definition 1.2.10 The relative interior of a set X is defined by

ri(X) = {X E X

I B(x, b) n aff(X) G X

for some 6 > 0).

The relative boundary of X is defined by rebd(X) = X

\ ri(X).

Definition 1.2.11 The points X', x2, - ,xk are said to be afinely independent if dim(aff {X', x2, - - . x k ) ) = k - l. Alternatively, this is equivalent to the k - l vectors xZ- X', i # l being linearly independent for any given 1. Definition 1.2.12 A hyperplane in Rn is an n - l dimensional (or CO-dimensionl ) affine set. Theorem 1.2.7 Let a E R", and b E R . The set 'H = {X E Rn : aTx = b ) is a hyperplane in R n . Conversely, let H be a hyperplane in R n . Then there exists a vector a E Rn and a scalar b E R such that 'H = {X E Rn 1 a T x = 6 ) . Proof: It is easy to check that 'H is an affine set. Let x1 be any vector in 3-1. Translating 'H by -X' gives the set 'H' = 'H -X'. 'H' contains the origin and consists of all vectors in IR" orthogonal t o a , hence it is an n - 1 dimensional subspace.

Conversely, let x1 E 'H and translating by -xl gives the set 'H' = 'H - x1 which is an n - 1 dimensional subspace. Let a be a nontrivial vector orthogonal to this subspace, i.e., a E ('H')', so that 'H' = {X : a T x = 0 ) . Let b = a T x l . clearly x2 E 'H + x2 - x1 E W, and hence aTxl - aTx2= 0 which implies that aTx2= b. {X : aTx = b ) . Since H is, by definition, of dimension n - 1 and Thus 'H {X : a T x = b ) is of dimension n - 1 from above, by uniqueness these two sets must be equal. I Definition 1.2.13 Let a E R n , and b E R . The hyperplane 'H = {X E Rn : a T x = b ) divides Rn into the positive closed half space 'H+ and negative closed half space 'H-, where 'H+ ={X E Rn : aTx b)

>

'H- ={XE R " : a T x _ b . For all X' E X' and x 2 E X 2 , we have x 2 - X' E x2 - X ' , hence a T ( x 2 - X ' ) > b. Consequently, a T x 2 > aTx' b VX' E x ' , x 2 E x2,

+

or ~

inf a T x 2 2 sup a T x l 2

~

Let r be such that

x

2

xl€X1

+b >

sup a T x l

x l EX1

since b

> O.

inf a T x 2 > y > sup aTx' .

x2€X2

xl € X 1

Then the hyperplane 7-t = {y I a T y- y = 0) separates X' and X 2 in a strict sense. B

Note that Theorem 1.2.10 is not true in general without the compactness assumption.

MATHEMATICAL PRELIMINARIES

13

Theorem 1.2.11 (Supporting Hyperplane Theorem) If X C IRn is closed and convex, then there exists a supporting hyperplane at each boundary point in dX. Proof:Let d be a direction pointing to the exterior of X from some boundary point x0 E aX. Consider the sequence of points C" xO Eid X as Ei 5 0 through a decreasing sequence {Ei, i = 1,2,. For each point C', there exists a set of separating hyperplane {y E Rn I ( P ~ -)72 ~= 0) ~ that separates Cqrom X in a non-strict sense, i.e.,

+

.)a

Let Li be the set of normalized P" i.e., JIP211 = 1, then (i) the set Li is bounded for each i, since IIPyI = 1; (ii) the set Li is closed for each i, since equality is included in (1.2.2); (iii) the sequence Li , i = 1,2, - - - is monotone, since €2 < Ej Ci C Lj . Since the sequence {Li) is closed, bounded and monotone, by the Bolzano Weierstrass Theorem [R5], there exists a non-empty limit L,, i.e., there exists (at least) I one supporting hyperplane a t xO.

Lemma 1.2.12 Let X C IRn be convex, 7-t be a supporting hyperplane of X, and = X n 7-t. Then every extreme point of M is also an extreme point of X.

M

Proof: Assume that 7-t takes the form 7-t = {y I aTy = b) where aTx2 b, Vx E X. Suppose x0 E M is not an extreme point of X. Since x0 E X , there exist X' E X and x2 E X , X E ( 0 , l ) such that x0 = Xxl + (1 - X)x2. Since x0 E M, we have = XaTxl+( l - X)aTx2.This, and the fact that aTxl 2 b and aTx22 b, b = aTxO imply that aTxl = aTx2= b and therefore X' E M and x2 E M. Hence x0 is not I an extreme point of M. Theorem 1.2.13 A compact convex set X C Rn is the convex hull of all its extreme points. Proof: We proceed by induction on n. Clearly the assertion holds in IRn for n = 1. Next assume it holds in R"-'. Let y be the convex hull of all the extreme points of X. Clearly y C X. We need to establish the converse X c y. Suppose there exists X E X with X 4 Y, then Theorem 1.2.8 asserts that there exists a # 0 such that aTx < infyE aTy. Let a = minXExaTx < m. Since X is compact and aTx is a continuous function, the Weierstrass Theorem (see Remark 1.5.1) asserts that there exists x0 E X such that aTxO= a, and consequently 7 i = {y E Rn 1 aTy= a) is a supporting hyperplane for X. Let M = X n 3-1, then M is a compact convex set in Rn-l and is non-empty since it contains the point xO. By the induction hypothesis, M is the convex hull of all its extreme points, and there exists at least one such extreme point. By Lemma 1.2.12, the extreme points of M are also the extreme points of X. Thus we have found extreme points I of X that do not belong to Y, a contradiction.

14

MATHEMATICAL PRELIMINAWES

Corollary 1.2.14 A polytope is the convex hull of all its extreme points. We shall now turn our attention to functions. Definition 1.2.18 Let f : Rn + (eflective) domain of f is defined as

be an extended real-valued finetion. The

The range of f is f (X) C R. Remark 1.2.3 If f : X + IR defined on a set X G IRn associates with each point of X a real number f (X), then define

Then dom(F) = X. So the set X is called the domain of f . Consequently the study of the constrained optimization problem min( f (X) I X E X ) = min f (X) is equivalent to the study of the unconstrained problem min{F(x) I X E IRn). The min, or strictly speaking, infimum is oo if the feasible set X is empty. Definition 1.2.19 (Continuous and semicontinuous functions)

A real-valued function f : X -,IR is upper semicontinuous ( u s .C.) at for any given E > 0, there exists a S > 0 such that

X

E

X if,

y E X and lly - xII < S + f (y) - f (X) < E. f is an upper semicontinuous function if f is upper semicontinuous at every point E X. A vector-valued function f : X 4 IRm is upper semicontinuous at X E X if every component of f is upper semicontinuous at X. f is an upper semicontinuous (vector-valued) function if every component of f is an upper semicontinuous function.

X

A real-valued function f : X -, R is lower semicontinuous (1.s.c.) at for any given E > 0, there exists a S > 0 such that y E X and ((y- xll

X E

X if,

< S + f (y) - f (X) > -E.

f is a lower semicontinuous function if f is lower semicontinuous at every point E X. A vector-valued function f : X -, Etm is lower semiwntinuous at X E X if every component of f is lower semicontinuous at X. f is a lower semicontinuous (vector-valued) function if every component of f is a lower semicontinuous function.

X

0 A real-valued function f : X -+IR is continuous at there exists a 6 > 0 such that

X

E X if, for any given E

> 0,

MATHEMATICAL PRELIMINARIES

15

f is a continuous function if f is continuous at every point X E X. A vectorvalued function f : X -, Etm is continuous at X E X if every component of f is continuous at X. f is a continuous (vector-valued) function if every component of f is a continuous function. Note that a function that is simultaneously upper and lower semicontinuous is continuous. A function f is U.S.C. (resp., 1.s.c.) if and only if -f is 1.s.c. (resp. u.s.c.).

Definition 1.2.20 Let X 2 Rn be a nonempty set, and f : X -,R . f is said to be diflerentiable at some X E int(X) if there exists a (row) vector V f (X) such that

where g is a function satisfying lim g(y - X) = 0.

Y+X

V f (X), or sometimes written as

gI

it has components [Vf(x)Jj =

or f '(X) is called the gradient of f at

g/, ,

X

X,

and

j =, 2 1 - , n .

X

f is said to be twice differentiable at some X E int(X) if there exists a matrix H(x) such that

e,

where g satisfies (1.2.3). H(x) is called the Hessian of f evaluated at X, and has components [H(x)lU=

Xi X j

i,j = 1,2, m--,n.

f is said to be differentiableltwice differentiable if f is differentiableltwice differentiable at every interior point of its domain.

A vector-valued function f is said to be differentiableltwice differentiable if every component of f is differentiableltwice differentiable. Definition 1.2.21 The epigraph of a function f : Rn 4 {(X,Y) E R?+' I X E Etn, y E R l f ( 4 5 Y).

is the set epi(f ) =

Definition 1.2.22 A function f : Rn -+ R is said to be convex if its epigraph epi(f) is a convex subset of Etn+'. If, furthermore, dom(f ) # 0 and f (X) > -00 V x E IR", then f is a proper convex function. f is said to be concave if -f is convex. Alternatively, a more conventional definition is as follows. Let the set X be convex and f : X -,R . f is convex on X if given any x , y E X and X E [O,l], then

16

MATHEMATICAL PRELIMINARIES

This is equivalent to saying that the function F as defined in Remark 1.2.3 is convex. Moreover, f is said to be strictly convex if given any X, y E X, X # y and X E (0,l):

Remark 1.2.4 If f : X [R11):

+ IR

is convex, then it is well known that (see, for example,

f is continuous in ri(X); if f is differentiable on an open set containing X, then f is convex on X if and only if f(Y) 2 f ( x ) + V f ( x ) ( y - 4 V ~ , YE Rn; if f is twice differentiable on an open set containing X, then f is convex on X if and only if the Hessian of f is positive semidefinite on X. If the Hessian of f is positive definite, then f is strictly convex on X. In general, the converse is not true. Definition 1.2.23 A function f : X -+ IR is said to be quasiconvex if for X E [0, l] andx,y~ X, f (Ax + (1 - X)Y) 5 max{f ( 4 , f (Y)h

and strictly quasiconvex if

and strongly quasiconvez if

Definition 1.2.24 A function f : X

-+

IR is said to be pseudoconvex if for

and strictly pseudoconvex if

Definition 1.2.25 A proper convex function f : IRn epi(f ) is a closed set in Etnf l. Definition 1.2.26 The closure

f of a function f

-t

: IRn -+

R is said to

be closed if

is defined to be

J(x) = sup F(x) V x E IRn , FEA(f

where A( f ) is the family of all affine functions F defined on IRn such that F ( x ) 5 f (X) Vx E Rn.A(f ) is called the ABne Mznorant of f .

MATHEMATICAL PRELIMINARIES

17

Remark 1.2.5 It can be shown (see [vTl]) that, a proper convex function f is closed if and only if it is lower semicontinuous, if and only if f = f. Lemma 1.2.15 Let

F be a family of convex functions on IRn, then,

is a convex function.

Proof: epi(f) = ngEFepi(g) is convex by Corollary 1.2.2.

B

Definition 1.2.27 Let f : IRn -+ R be a convex function, and let X E dom( f ) . The vector z E IRn is said to be a subgradient of f at X if

is called the subd2flerential of f at X. If

X

dom( f ) , then d f (X)= 8 by convention.

Remark 1.2.6 It can be easily established that for convex functions, d f (X) is closed and convex;

af (X) is a singleton if and only if f is differentiable at X. In this case d f (X) = {Of ( 4 1 Definition 1.2.28 Let f : IRn -+ R, X E dom(f) and d E IRn. The directional derivative of f at X in the direction of d, if it exists, is defined by

Theorem 1.2.16 Let f : R" -P R be a convex function, and let IRn. Then y E df(x) if and only i f y T d 5 f r ( x , d ) V d E R n .

X E

dom(f), y E

Proof: (Necessity) If y E df (X), let t > 0. Then from Definition 1.2.27,

The conclusion follows by letting t j, 0.

18

MATHEMATICAL PRELIMINARIES

(Sufficiency) If yT d

< f '(X,d )

y T d 5 lirn tl0

;

then

f (X+ td) - f (X) vx E R n t

hence y E af (X).

D

Definition 1.2.29 Let f : IRn lR be a locally Lipschitz function. The Clarke subgradient of f at X E lRn is defined by

is the Clarlce generalized directional derivative of f at

X

in the direction y.

f is said to be subdifle~entiallgregular a t X if the directional derivative ff(x; y) exists and f f ( x ; y )= f O ( x ; y ) , ' f y E R n . The following properties of f "(X;y ) are stated without proof, see [C5]. Theorem 1.2.17 T (i) f O ( x ; y )= m a x , ~ a o f ( ~ )Y. z (ii) Let f (X) = G(g(x)) where g : lRn -+ I F , G : lRm -+ IR. Let

MATHEMATICAL PRELIMINAFUES

1.3 Fenchel Transform and Conjugate Duality The theory of conjugate duality [F21 plays a very important role in convex analysis. It is aesthetically appealing as a nice symmetry prevails in many of the results. An entire book [R21has been devoted to the subject. This contains many more results than what this section can afford to discuss, together with an extensive range of useful applications. Again, while the results in this section are presented for finite dimensional vector spaces, they can be generalized to infinite dimensional versions in abstract spaces.

Definition 1.3.1 (Fenchel Transform) Given f : Rn + R, the Fenchel transform (or conjugate, or polar) of f is the function f * : Etn + R defined as:

The biconjugate (or bipolar) f ** : IRn + R of f is the conjugate of f *.

Remark 1.3.1 One can interpret the meaning of the Fenchel transform in the following way. Given a y E IRn and an a E R , one defines a linear function t : Rn + R by t(x) = y T x - a. One desires that the linear function t belongs to A( f ), the family of affine minorant (see Definition 1.2.26) of f . This means that !(X) = y T x - a 5 f (X) V x E Rn7or a y T x - f (X). This property is ~ = ~ f *(y). { ~ Another ~ X economic accomplished by choosing a 2 S U ~ ~ -~f (X)) interpretation goes as follows: if xi represents the quantity held of each of the ith of n commodities, f (X) is the cost of producing them, and y+ is the price of selling the ith commodity, then y T x - f (X) is the net profit fiom producing and selling the commodities, and f *(y) is the maximum possible profit for the given price y.

>

Remark 1.3.2 Note that the function f does not need to be convex for its Fenchel transform to exist. Although more often than not we talk only about the Fenchel transform of convex functions, and f * is often called the convex conjugate of f . Lemma 1.3.1 (Fenchel's or Young's Inequality)

Proof: Follows directly from Definition 1.3.1.

I

Example 1.3.1 Let X C Rn. The indicator function dx : Rn + (0,oo) of X is defined bv 0 ifx€X oo otherwise. Then the Fenchel transform of 6x is given by the support function of X defined by

:6 (X) = sup y T X. Y'=

20

MATHEMATICAL PRELIMINARIES

Theorem 1.3.2 Let f : Rn -+R. Then the Fenchel transform f * : Rn -+ convex on R n .

is

Proof: For each X E Rn7the mapping y -+y T x - f (X) is linear (and convex) in y. Let F = {yTx - f (X) I X E ]Rn} be the family of functions which are linear in y. Then, by Lemma 1.2.15,

is convex.

Theorem 1.3.3 (Properties of Fenchel transform) Let f : Rn -+ R. ~ ~ f;(y) = f * ( y ) - y T x O V ~ E I R " . (i) If fl(x) = f ( x + x O ) V X E R then (ii) I f 0 E R and f2(x) = f(x) + p b'x E Etn, then G ( y ) = f*(y) --P Vy E Rn. (iii) If # 0 and f3(x) = ~ ( P x ) Vx E R n , then f,*(y) = f*(y/P) Vy E Rn. (iv) I f p > O a n d f4(x) = P f ( x ) V X E I R ~then , fz(y) =Pf*(y/P) Vy E R n . Proof: 6)

(ii)

MATHEMATICAL PRELIMINARIES Theorem 1.3.4 If f is a proper convex function, then f ** = f. Proof: Young's inequality (Lemma 1.3.1) asserts that, for a given

X,

From Definition 1.2.26, f(x) = up,^,(^) F(x) where A( f ) is the affine minorant, it follows that yTx- f*(y)5 f(x) Vy,xE ]Rn. In particular, f**(x)= sup {yTx- f*(Y)}5 f(x) Vxc IRn. yERn

On the other hand, if for some y E IRn and b E IR such that

then

Consequently, f*(y)5 sup {xTy- f*(y)}= f**(x) Vxc Ktny€Rn

By Definition 1.2.26, f(x) is the supremum over all y and b such that xTy -b 5 f (X), D therefore, f(x) f ** (X).


2 and IAl 2 1. If an arc is associated with a direction (as indicated by an arrow on that arc), it is called a directed arc. If all arcs of the graph are directed, then the graph is called a directed graph (or a digraph in short). Each arc of A is incident to two end points, and if an arc j E A is also directed, then it is sometimes represented as an ordered pair of nodes ( a ( j ) ,w(j)), where a ( j ) is the source node of j and w(j) is the sink node of j . In this manner, the arc set A can be perceived as a subset of Af X N . Two nodes are said t o be adjacent to each other if they are joined by some arc. The digraph is complete if A = n/ X N . Definition 1.4.2 The node-arc incidence matrix E of a digraph G = ( N , A) is X IAl integer-valued matrix with entries given by: defined as an 1 -1 0

if a ( j ) = i , if w(j) = i , otherwise.

Lemma 1.4.1 l T= oT ~ where lT= (1,1, - -

a ,

1).

Proof: Each column of E has a 1 and a -1.

Definition 1.4.3 A path P in a digraph G = ( N , A) is an alternating finite sequence ofnodesandarcs{io,jl,il,jz,---,jK,iK)whereK> l , i k ~ N a n d a n a r jr c EA

MATHEMATICAL PRELIMINARIES

25

is either jk = (ik,i k - 1 ) (whence it is traversed negatively) or jk = ( i k - 1 , ik) (whence it is traversed positively). The source node and sink node of P are io and i~ respectively, and the notation P : io 4 i~ is used to denote that the path joins . length of the path is the number of the source node io to the sink node i ~ The arcs in it. If some arc appears more than once in the path, either positively or negatively, the path is said to be one with multiplicities. An elementary path is a path without multiplicities and where no node occurs more than once in the path. In a path without multiplicities, the set of all positively traversed arcs is denoted by P+ while the set of all negatively traversed arcs is denoted by P- . The arc-path incidence vector ep E RId1is defined by

0

otherwise.

Definition 1.4.4 A cycle is a special path P = ( 2 0 , jl,il,j2, jK,iK) where io = i ~ .An elementary cycle is one without multiplicity and that, apart from io = zk, each node of P occurs only once. The orientation of the cycle is identified with the order of appearance of the arcs and nodes. - - a ,

Theorem 1.4.2 If P = { i ~ , j lil, , j1,- - - ,jK,i ~ is } an elementary path or cycle of a proper digraph G = (N,A), then

where ei E RIN1is a vector with [ei]j= Sij and E is the node-arc incidence matrix of G.

Proof: Let Ej be the j t h column of E, then if jk = (ik-i,ik), then [ep]jk= 1, hence, [ep]jkEjk = ezk-1 - eik, otherwise;

j k = (&,&I),

then [ep]jk= -1, hence, [epIjkEjk= ezk-1 - ezk.

Thus in all cases

Corollary 1.4.3 If P is a cycle, then Eep = 0, and therefore the columns of E corresponding to this cycle are linearly dependent. Definition 1.4.5 A graph g' = (hf',d')is a subgraph of the graph G = (N,d) if N' C_ N and A' A. G' = (NIld')is a spanning subgraph of G = ( N , A )if

N'

=N.

Definition 1.4.6 A graph G = (N,d)is said to be connected if given any pair of nodes (2, z'), a path P : i +'i connecting the nodes can be found. A component of

MATHEMATICAL PRELIMINARIES

G is a connected subgraph G1 of G such that there is no other connected subgraph of G which contains G1 strictly. Definition 1.4.7 A proper graph G = (N, A) is acyclic if no cycle can be found in 9. An acyclic graph is called a forest. A connected forest is called a tree. A spanning tree of 7 is a spanning subgraph of G which is also a tree. The cotree T* of a spanning tree 7 is another spanning subgraph containing exactly those arcs which are not in 7. Theorem 1.4.4 Let G = (N, A) be a proper graph and 7 = (NI, A') be a subgraph of G. Then the following are equivalent: (i) 7 is a tree, i.e., acyclic and connected. (ii) Given any pair of nodes (2, it) in NI, there exists an unique path in 7joining i to i t . (iii) 7 is connected and IN1 ' = 1-4'1 + 1. (iv) 7 is acyclic and IN'J= IA'I + 1. Proof: (i)+(ii): If not, then a cycle can be formed in 7 by concatenating any two paths joining i and it, contradicting the fact that 7 is acyclic. (ii)+(iii): If there exists a unique path in 7joining any pair of nodes, then clearly = Id']+ 1 by induction. This is clearly true for a proper graph with two nodes and one arc. Assume that it is true for any tree with less than m nodes. If 7 has m nodes then removal of an arc will disconnect the graph into two components, ?; = (NI, A1) and 3 = (N2,A2). Since each of 5 and 5 has less than m nodes,

7 is connected. We prove ln/'l

Thus

lNtI = IJvll+IN21,

(the last one being that of the removed arc). The conclusion follows from combining (1.4.1) to (1.4.3). (iii)+(iv): If 7 is not acyclic, then a cycle P with m nodes and m arcs exists. For any other node not lying on this cycle, it must be part of a path that joins to one of the nodes on the cycle by the connectedness assumption. Call this a tentacle. Each tentacle will have the same number of nodes (not counting the last one in the cycle) and arcs. Adding up, 7 will have IN'I = IA'I, a contradiction. (iv)+(i): The acyclic part is directly implied. If 7 is not connected, then there exist K > 1 components, each being a tree with one less arcs than nodes. Adding up, 7 will have lNtl = IA'I K nodes, a contradiction.

+

I

MATHEMATICAL PRELIMINARIES

27

Remark 1.4.1 The concept of path has a dual concept called cut. It will become clear in Chapter Two that the duality between path and cut plays a central role behind many network optimization problems. Definition 1.4.8 A cut Q in a graph G = ( N , A) is a set of arcs whose removal results in a disconnected graph. The cut is said t o be elementary if the removal of arcs in this cut increases the number of components by exactly one. If G is a digraph, a cut is conveniently expressed as Q = [ S , N \ S] where S G N . This represents the set of arcs which straddle the node set S and its complement N \ S . The set of arcs in Q can be partitioned into two disjoint subsets: Q+ = { j E A I a(j)E S,w(j) E N \ S) (the set of forward arcs), Q - = { ~ E A I ~ ( ~ ) E S , ~ ( ~ ) E (thesetofreversearcs). A~\S) If S = 0 or N, then the resulting cut Q = [S:nf incidence vector ep E IRId1is defined as:

0

\ S] is an

empty cut.

The cut

otherwise.

The set indicator vector es E IRINI for the node set S that generates the cut is defined as: 1 i f i ~ S , [esli = 0 otherwise.

{

Definition 1.4.9 A node is an end point if it has only one arc incident on it. Remark 1-42The following facts are intuitively obvious, and rigorous proofs can be easily established following the proof of Theorem 1.4.4.

A tree has at least two end points. The deletion of an end point of a tree and the arc incident on it results in another tree. Every connected graph has a spanning tree. There exists a unique path in a tree that joins the two end points of a cotree arc. The following result establishes the correspondence of a spanning tree to a basis of the node arc incidence matrix E.

Theorem 1.4.5 Let G = ( N , A ) be a proper connected digraph with a node arc incidence matrix E. A set of columns { E j 1j E A') with Id'[ = IN1 - 1 is a basis for the column space of E if and only if 7 = ( N , A') is a spanning tree for G. Proof: (sufficiency) We first prove that the set of columns { E j ( j E A') correspond-

ing to a spanning tree 7 of G is (i)linearly independent and (ii)spans the column space of E. This establishes that { E j 1j E A') is a basis.

28

MATHEMATICAL PRELIMINARIES

(i) Since 7 is a spanning tree, !A'\ = JNJ - 1. If the set {E', j E A') is linearly dependent, then there exists X j ; j E A', not all zero, such that

Let t be an end point of 7 and let p be the arc in 7 incident on t. Since t is an end point, [Elt, = 1, or -1; but [Eltj = 0 V j # p. (1.4.4) thus implies that X, = 0. The deletion of the node t and the arc p results in another tree. If we apply the above argument recursively, then eventually X j = 0 V j E A', a contradiction. (ii) Any arc k in G but not in 7 is a cotree arc. Let P be the unique path in 7 joining the two end points of k, then k together with P forms a cycle. By Corollary 1.4.3, the columns of E corresponding to all the arcs in the cycle are linearly dependent, implying that can be expressed as a linear combination of the columns of E corresponding to the tree arcs. Consequently {Ej Ij E A') spans the column space of E . (necessity) If {E' lj E A') with /A'[ = IN1 - 1 is a basis for E , it must be linearly independent. By Corollary 1.4.3, no cycle can be found in 7 = (N,A'). By part (iv) of Theorem 1.4.4, 7 is acyclic and has one less arcs than nodes, hence it must be a tree. Since it contains all the nodes in G, it is also a spanning tree. I

Corollary 1.4.6 Let E be the incidence matrix of a connected graph. Then rank(E) = Ih/l - 1. Proof: Follows from Lemma 1.4.1 and Theorem 1.4.5. Note that the property rank(E) = E.

-

m

1 is known as the rank one deficiency of

1.5 Elements of Optimization and Variational Inequalities The reader of this book should already be familiar with the concept of optimization. We now present a very general overview of the subject and a brief introduction to variational inequalities in order to lay the foundation for subsequent duality analyses. The need to order vectors arises frequently in this book. A formal discussion of this topic will be given in Chapter Seven. For the purpose of the first six chapters,

MATHEMATICAL PRELIMINARIES we adopt a set of informal vector ordering relationships: two vectors and q,

c

c

0 > means that every component of component of r ) ;

29

, 2,2 . Given

is strictly greater than the corresponding

6 2 r ) means that every component of ( is greater than or equal to the corresponding component of v,with at least one component strictly greater;

c

0 2 r ) means that every component of sponding component of r ) .

c is greater than or equal to the corre-

The most general optimization problem is succinctly stated as follows: (Prototype Optimization Problem)

min f (X) xES

subject to

X

E X. (1.5.1)

Sometimes the above is succinctly stated as: min f (X). Some elaboration of the various terminology is warranted here. X is called the variable of the optimization problem, and is usually a member of some space S which could be continuous or discrete, finite or infinite dimensional. We are mainly concerned with duality in continuous spaces in this book. Duality in discrete spaces is quite a lot more subtle, and is perhaps not as clean as the continuous case. Nevertheless some discussion of discrete optimization is included in Chapter Five. The space S can be an infinite dimensional Banach space or Hausdorff space, although more often it is just the finite dimensional vector space IRn. To quote from a well-known applied mathematician, one can indulge i n any fancy space one likes, but i f one has to compute a solution, then everything is back to IRn. To avoid an unnecessarily abstract notation that may obscure the underlying ideas, we have chosen t o discuss duality mainly in the context of Etn, although it should be noted once again that most results are generalizable to abstract spaces with only little modification required.

The function f : S +-range( f) is usually called the cost function, or functional if S is infinite dimensional, of the optimization problem. Optimization is mostly concerned with real-valued functions. Usually, the cost function f is real-valued, and hence range ( f ) C IR, or possibly R. Scalar optimization is often referred to as just optimization by default. In the more general context, f could be a vector-valued function, in which case, range f C IRm with m > 1, and the optimization problem is called a multicriteria optimization, or vector optimization, or multiobjective optimization. Multicriteria optimization will be dealt with in the latter half of this book, as it entails a more complex interpretation of optimality. By convention, optimization usually means minimization by default, hence min is used. In the event when certain closedness or compactness assumptions cannot be satisfied, we often replace min by inf to denote finding the greatest lower bound (or infimum) instead of the minimum, which by convention the latter is attainable.

MATHEMATICAL PRELIMINARIES In the case where the minimum is attainable, inf and min are used interchangeably. Any optimization problem concerns with maximization can be easily converted to a minimizing one by simply changing the sign of the objective function. The set X is a subset of the space S, and is called the feasible set (or constraint set) of the optimization problem. If X = S, then the problem is unconstrained, and is constrained otherwise. Usually, X is nonempty, otherwise the problem is called infeasible and the minimum cost is assigned oo by convention. Often, X is assumed t o be connected with a nonempty interior. Sometimes, the set X may have additional properties such as compactness or convexity, in which cases, there are many stronger results pertaining to the solution when f is nice. Often the feasible set X is expressed as a set of points that satisfy a number of constraints, e.g.,

Here the functions g and h are defined on the same domain as the cost function f and can either be scalar valued or vector-valued, or non-existent. Each component of g is called an equality constraint and each component of h is called an inequality constraint. The special set of inequality constraints given as A? = < X 5 U are called simple bounds. If the infimum in (1.5.1) is attainable by some feasible point, then the minimum cost is denoted by min f (X) = f (X*) where X* is understood to be the optimal solution to the problem. Strictly speaking, this should be a globally optimal solution, although often X* is expected t o be only locally optimal. Some special cases deserve special attention. If f , g and h are all linear, then the problem is called a linear programming problem. This will be dealt with in some length in Chapter Three. Otherwise if one or more of f , g or h is nonlinear, the problem is called a nonlinear programming problem. If g and h are linear and f is convex and separable, i.e., f (X) = C .f .(xj), then the problem is called 3 a monotropic programming problem, and this will also be dealt with in Chapter Three. If f is convex and X is convex, we have a convex optimization problem. Much stronger results can be obtained for this convex case than otherwise, and these will be presented in Chapter Four. If the problem is defined on some special structure, e.g., a graph or network, then ad-hoc titles like network optimization are ascribed to the problem. Many beautiful and geometrically meaningful duality results are found in network optimization, and the whole of Chapter Two is devoted to this special case. Optimization problems are sometimes made complicated by other considerations. If S is an infinite dimensional space, then the problem is called infinite dimensional. Optimal control problems [C21 and calculus of variations [GFl], [El] are special classes of infinite dimensional optimization problem. If the cost and constraint functions are perturbed by noise or random factors, then the problem is called a stochastic optimization problem [KWl]. Interesting duality results prevail in both optimal control problems as well as stochastic programming problems. Unfortunately, these are not within the scope of this book, and the readers are referred t o [HI] and [R81 for details.

Definition 1.5.1 Let X C IRn and f : X

--+ R

be a real-valued function.

MATHEMATICAL PRELIMINARIES A point

X*

E X is said to be a local minimum of (1.5.1) if there exists

suchthat X E X, and Ilx-x*II

A point that

X

31

< E * f(x)

> f(x*).

X* E X is said to be a local maximum of (1.5.1) if there exists c E X , and Ilx - x*II < E f (X) f (X*).


0

> 0 such

< f (X) V x E X.

A point

X*

E X is said t o be a global minimum of (1.5.1) if f (X*)

A point

X*

E X is said to be a global maximum of (1.5.1) if f (X*)2 f (X) V x E X.

Remark 1.5.1 Let X C Rn be a nonempty compact set, and f following facts are well known (Weierstrass Theorem, see [R5]):

:

X

-+

R. The

If f is lower semicontinuous, then f attains a global minimum over X. If f is upper semicontinuous, then f attains a global maximum over X . If f is continuous, then f attains both a global minimum and a global maximum over X. The following properties for the minima of convex functions can be established.

Theorem 1.5.1 Let f be a closed and convex function on lRn. Then (i) a local minimum of f is also a global minimum, (ii) the set of all global minima is convex, and (iii) if f is strictly convex, then the global minimum is unique. Proof: (i) If f has a local minimum at X*, then f (X*) 5 f (X)V x E B(x*, c ) for some c > 0. For any X E IRn and a sufficiently small X > 0, Xx (1 - X)x* E B(x*, E), hence f (Xx + (1 - X)x*) 2 f (X*).Since f is convex, this means that

+

From which we have X f (X) - X f (X*)2 0 and since X > 0, the conclusion that a global minimum follows. (ii) If

But

X'

X*

is

and x2 are (global) minima, f (X') = f (x2), then, for 0 < X < 1,

f (Xxl

+ (1

-

X)x2) 1 f (X2)

since x2 is a global minimum. (1.5.2) and (1.5.3) together imply that Xx' is a global minimum. (iii) If f is strictly convex, and X'

# x2 are global minima with f (X')

This contradicts (1.5.3), so X' = x2.

+(1

- X)x2

= f (x2), then

B

In particular, we have the following characterization of a global minimum for a convex function in terms of the subdifferential.

MATHEMATICAL PRELIMINARIES

32

T h e o r e m 1.5.2 Let f : IRn + R be a convex function, and let is the (global) minimum of f if and only if 0 E d f (X*).

X*

E dom(f ). Then

X*

Proof: (Necessity) If 0 E af (X*),then by Definition 1.2.30, f (X) 2 f (X*)tr'x E ELn, and hence X* is a (global) minimum. (Sufficiency) If f is globally minimized at X*, then f (X)- f (X*)2 0 = o T ( x - X*), implying that 0 E d f (X*). m

We now turn our attention to variational inequalities, henceforth abbreviated as VI. In some sense, V1 can often be perceived to be a generalization of optimization. The topic was first studied in the context of solving partial differential equations in infinite dimensional spaces [KSl], but it has since found many applications in optimization and general equilibrium models in finite dimensional spaces.

C IR"

be a convex

Vx E X.

(1.5.4)

P r o t o t y p e Variational Inequality P r o b l e m (VI) Let X set and f : R" --+ Rn be a vector-valued function. Find a point

X*

E X such that f ( ~ * ) ~-(X*) x 20

The problem VI as stated in (1.5.4) reduces to several well-known problems under further assumptions on the function f and the set X. These are discussed briefly in turn as follows, readers are referred to [Nl] for details. 0 Solution of simultaneous equations. Let X = R", then if and only if X* solves the V1 (1.5.4). 0

y

X*

solves f(x) = 0

Fixed point problems. Let X be a closed and convex set in IRn, then for any

> 0, X* satisfies the fixed point relationship

if and only if as

X*

solves the V1 (1.5.4), where the projection operator Prx is defined Prx(x) = argminYExllx- yll.

Complementarity problems. Let X = R:, then X* solves the complementarity problem: f(x*) 2 0 , X* 2 0 , ~ ( x * ) ~ x=*0 if and only if

X*

solves the V1 (1.5.4).

Our primary interest, however, lies in the context of V1 as a generalization of optimization problems. Consider the prototype optimization as stated in (1.5.1). To avoid confusing f with f , we shall restate the optimization problem using a different symbol for the cost function as follows: Optimization problem

min g(x)

xE X

(1.5.5)

MATHEMATICAL PRELIMINARIES

33

where g is assumed to be a differentiable function. The following result is wellknown, and supplement the results to be discussed in Chapter Four. Theorem 1.5.3 (Optimality conditions for optimization) Assuming that the feasible set X is closed and convex. A necessary condition for X* to be a local minimum of the optimization problem (1.5.5) is that X* solves the following VI,

If in addition, g is convex, then X* is a global minimum of the optimization problem (1.5.5) if and only if X* solves the V1 (1.5.6). Proof: Let 0 < t -m.

Definition 2.6.5 (Max Tension Problem) Given two disjoint node sets Nfand N- of 8, find a potential U where ui is constant on n/+ and (a different) constant on N-, such that the spread of U from N+ to N- is maximized, subject to the

constraint that the resulting differential v = Au is feasible with respect to spans.

Theorem 2.6.3 (Weak Duality) If U is a potential satisfying the conditions of the max tension problem, and P : Nf -+N- is a path joining some node in N+ to some node in N-, then:

spread of

U

from N f to N- 5 d;.

Proof: Follows from the second inequality of Theorem 2.6.1 and the proof of TheB orem 2.6.2. Theorem 2.6.4 (Strong Duality - Max Tension Min Path Theorem) Assuming that there exists at least one potential satisfying the conditions of the max

DUALITY IN NETWORK OPTIMIZATION tension problem, and that there does not exist a cut Q : N+ span, then

65

N-

of unlimited

max in max tension problem = rnin in rnin path problem.

Proof: This is done by constructing a path P : N+ -+ N - using the following rnin path algorithm. If a path can be found such that d; is equal to the spread of a U from nf+ to N-, then by the weak duality theorem 2.6.3, P must be the solution t o the rnin path problem and U must be the solution to the max tension problem. m

It remains to present the rnin path algorithm. This has the same flavor as many network algorithms in that a tree (or forest if N+ is a set) is grown from its root in N+ until it reaches some node in N - . At each stage of the growth, an arc that stands for the shortest extension away from the tree is added t o the tree. If a feasible potential exists, it can be assumed, without loss of generality, that d y 5 0 5 d l for all arc j , so that an initial feasible potential is given by U = 0.

Algorithm 2.6.1 (Min Path Algorithm) Initialization. Let U = 0 be the initial feasible potential. Let S = N+,A' = 0 , 7 = ( S , A 1 ) , Q = [S,N\S],W i = O V i E S . Do until S nn/- # Q): Compute = rnin

+

W,(,, d: ww(j)-d;

if j E Q+; ifjEQ-.

If p = oo, Q is a cut of unlimited span, stop. Else take the minimizing arc j in (2.6. l), A' + A ' U ( j ) ; itw(j)ifj~Q~,i+a(j)ifj~Q-;

S + S U {i}; Wi

End

+

P

End Upon exit, let wk = P V k # S, and let t E S n N - . The potential U = W solves the max tension problem, and the path P : N+ -+ t solves the rnin path problem, with d; = W,. D

Remark 2.6.4 Note that the above rnin path algorithm is intended to facilitate understanding and is not designed for computational efficiency. A much more efficient version of the rnin path algorithm due to Dijkstra can be found in standard Operations Research texts such as [T2].

DUALITY IN NETWORK OPTIMIZATION

66

2.7 Duality of Feasible Flow and Feasible Potential The ultimate goal of this chapter is t o establish the duality between the optimal flow problem and the optimal potential problem as first defined in Section 2.3. However, before the optirnality issue is discussed, it is necessary to establish if a feasible solution exists at all. This section is entirely devoted to studying the feasibility issue. This effort is particularly rewarding for it demonstrates many elegant applications of the painted network theory. As before, the driving force behind the theory is the duality between path/flow and cut/potential. Consider a digraph G = ( N , A ) with a node arc incidence matrix E, a flow requirement vector b E IRINI,a flow capacity interval Cj = [ C ; , C:] and a span interval Dj= [d;, d:] for each arc j E A. The two feasibility problems are defined as follows:

Definition 2.7.1 (Feasible Flow) A feasible flow x E Xj E

and

Cj V j

E

is such that

A

Ex = b.

Definition 2.7.2 (Feasible Potential) A feasible potential u E lRINI is such that

Dj V j E A v = -E T U.

uj E

where

Trivially, the feasible flow problem seeks to find a feasible flow, and the feasible potential problem seeks to find a feasible potential. These two problems can be considered as dual to each other. We shall devote the first half of this section to studying the feasible flow problem.

Definition 2.7.3 Given a set S C_ N , the net supply of the set S is given by b~ =

xiEs

Theorem 2.7.1 (Feasible Flow Theorem) The feasible flow problem has a solution X if and only if b~ = 0 and bs < c; for all cuts Q = [S,Af\ S]. Proof: (Necessity) If b~ = l T b # 0, then (2.7.2) can never be satisfied, since 1 E = oT. Furthermore, bs is the same as the flux of X across the cut Q = [S,N\S] which must be less than or equal to c; by the second inequality of Theorem 2.4.2. (Sufficiency) The proof of sufficiency is by construction through the following fea1 sible flow algorithm.

DUALITY IN NETWORK OPTIMIZATION Algorithm 2.7.1 (Feasible Flow Algorithm) Initialization. Find a flow X such that divergence y = Vx = Ex. Do until N+ = JZ/- = 0, return

X

xj

E

CjV j (this is easy!) and compute the

as the feasible flow, stop.

Let N+ = {i E N I bi > yi} (set of nodes with surplus) and N- = {i E N I bi < y i ) (set of nodes with deficit), and apply the painted network Algorithm 2.2.1 after painting all the arcs according to the scheme: Green if c;

< xj < ,c:

Red if

C;

= Xj =,:C

White if

C;

=Xj

Black if

C;

< Xj = C.:

< C,:

If a painted cut Q = [ S , N \ S] is found, then bs > c&, no feasible flow can be found, stop. Else a painted path P is found. Compute

(C

-j

( yi

- bi

Change the flow by:

End

for arc j E P+, forarcj~p-, for the starting node of P in N+, for the ending node of P in N - . X +- X

+ qep.

End

Remark 2.7.1 One can think of the feasible flow Theorem 2.7.1 as a theorem of alternative: either there exists a feasible flow, or there exists a cut Q = [S,N \ S] such that bs > C;, but not both. Remark 2.7.2 Assuming that the flow capacity intervals are all finite, then at each step the flow increment 7 must be positive and finite. If we further assume that c,: c;, and bi are all commensurable, then 7 must be integral, and hence the non-zero difference of lbi - yil for each node in N+ and N- decreases to zero by non-diminishing amounts. The algorithm either exits with a feasible flow when Nf = JZ/- = 0 or returns a cut that violates the condition of the feasible flow Theorem 2.7.1. In either case, the number of steps needed t o exit from Algorithm 2.7.1 must be finite. There are two conditions for X to be a feasible flow. In the feasible flow algorithm, one starts with a flow that satisfies the capacity constraint (2.7.1), and then works towards satisfying the conservation constraint (2.7.2). In a symmetrical manner, one can also start with a flow that satisfies the conservation constraint

DUALITY IN NETWORK OPTIMIZATION

68

(2.7.2), and then works towards satisfying the capacity constraint (2.7.1). To do so, we change the flow by a circulation so that the conservation constraint is preserved.

Algorithm 2.7.2 (Flow Rectification Algorithm) Initialization. Find a flow X such that Ex = b. (This is easy!) Do until A+ = A- = 8, return

X

as the feasible flow, stop.

Let A+ = { j E A I X j > }c: (set of arcs with too much flow) and A- = { j E A I x j < c;} (set of arcs with too little flow). Choose some arc k E A+ U A-.

If k E A+ let N+ +- {a(k)},N- + { ~ ( k ) } ; Else k E A- let M- + { ~ ( k ) N+ } , + {~(k)}. End Leave out the arc k from the digraph and apply the painted network Algorithm 2.2.1 after painting all the arcs according to the following scheme: Green if

C;

< X j < c,:

Red if

C;

= X j = C:,

< ,C:

White if x j

5 C;,

Black if c;

< x j , c: 5 xj.

Xj

If a painted cut Q = [S,N \ S] is found, then bs > c&, no feasible flow can be found, stop. Else take the painted path P and compute:

q = min

Xj-c; C-; -xk

Change the flow by: xk +xk

-v

forarcjEP-, if k E A-,

if k

E

A+, xk + x k + v if k E A-, and

X+-xfqep. End End Remark 2.7.3 The justification of finite termination here is similar to that of Remark 2.7.2. The flow increment must be positive and finite in each iteration under the assumption that c,: c;, and bi are all commensurable. In which case v must be integral, and hence the non-zero difference of x j - c: for each arc in A+ and c; - x j for each arc in A- will be reduced to zero by non-diminishing amounts.

DUALITY IN NETWORK OPTIMIZATION

69

The algorithm either exits with a feasible flow when A+ = A- = 8 or returns a cut that violates the condition of the feasible flow Theorem 2.7.1. We now turn to the dual problem of finding a feasible potential. As in the case of feasible flow, a necessary and sufficient condition for a feasible potential to exist is as follows: Theorem 2.7.2 (Feasible Potential Theorem) The feasible potential problem has a solution U if and only if dg 0 for all elementary cycles P.

>

Proof: (Necessity) The spread of v = Au is 0 around any circuit P, and this (i.e., 0) must be less than or equal to d$ by Theorem 2.6.1. (Sufficiency) The proof of sufficiency is by construction through the following feasible potential algorithm. I Algorithm 2.7.3 (Feasible Potential Algorithm) Initialization. Assign an arbitrary potential U and compute its induced differential = Ax = - E ~ u .

V

Do: until pi = 0 Vz E N,return U as the feasible flow, stop.

; = djf - vj and 6; = d; - vj. For all i E N, compute Let 6

Let k E N be such that pk < 0,N f + {k), N- + {k). Except when traversing the first arc out of k, let 6 ; c max{O, 6;) and 6; + min{O, 67). Apply the (modified) min path Algorithm 2.6.1 using the span interval 6;] V j to find a cycle from k back to itself.

[g,

If k is reached a second time, thus completing a cycle P with P < 0, then d& < 0 and no feasible potential can be found, stop. Else stop (the min path algorithm) as soon as a value P 2 0 has been reached. Set wi = 0 for all nodes not in the tree. U t- U W. End

+

End

I

Remark 2.7.4 The justification of finite termination here again requires the comand the starting potential. When the potential is changed, mensurability of d: the new potential will be such that pk = 0 and that for all other nodes, the corresponding pi is no less than the old value. Thus feasibility is improved by a finite number of quanta each time. The algorithm either exits with a feasible potential when pi 2 0 Vi, or returns a cycle that violates the condition of the feasible potential Theorem 2.7.2.

,p;,

70

DUALITY IN NETWORK OPTIMIZATION

Remark 2.7.5 Just as the case of feasible flow, the feasible potential theorem can be regarded as a theorem of alternatives: either there exists a cycle P such that d$ < 0, or there exists a feasible potential, but not both. Another algorithm for finding a feasible potential that can be thought of as a dual to the flow rectification Algorithm 2.7.2 is as follows:

Algorithm 2.7.4 (Tension Rectification Algorithm) Initialization. Assign an arbitrary potential v = Ax = - E ~ u . Do until A+ = A- = 8, return Let

U

U

and compute its induced differential

as the feasible potential, stop.

A- = { jE A

I uj > d:}

A+ = {j E A

I vj < d; 1 (set of arcs with too little tension).

and

(set of arcs with too much tension)

Choose some arc k E A+ U A-, and paint the network according to the scheme: Green if d; = uj = d,:

< ~j < d,: White if U j 2 d,: V j > d;, Black if U j 5 d;, Vj < d.: Red if d;

Apply the painted network Algorithm 2.2.1 to find a cycle containing k. If a painted cycle P is found, then d$ < 0, no feasible potential can be found, stop. Else take the painted cut Q (which contains k) and compute

q = min

d: - uj uj - d; d; - v k uk - d l

Change the potential by:

for arc j E Q', for arc j E Q-, if k E A+, if k E A-. U +- U

+ qeN,s.

End End Remark 2.7.6 In each iteration of the tension rectification Algorithm 2.7.4, the new potential improves by a finite number of quanta over the previous one since the new differential uj is no further away from the span interval D j than the old differential. In particular, uk is closer to its span interval by the positive amount q.

DUALITY IN NETWORK OPTIMIZATION

71

2.8 Duality of Linear Optimal Flow and Optimal

Potential

We now turn to a fairly important class of network optimization models which includes all the network models, i.e., max flow, shortest path and assignment problems that we have discussed so far as special cases. Some of the better known special cases of network optimization models that are not mentioned yet include the transportation problem, the warehousing problem, critical path analysis, facility location planning, just to name a few. This class of problems apparently accounts for 70% of all linear programming applications in the real world (see [AMOl]). Under some appropriate assumptions on their cost and capacity/span intervals, the linear cost optimal flow problem and the linear cost optimal potential problem are also dual to each other.

A) with a node arc incidence matrix E, a flow Consider a digraph G = (N, requirement vector b E lR.INI, a flow capacity interval Cj = [C;, c:] and a span interval Dj = [d;, d:] for each arc j E A. Let d, c E lRIdI and b E l T b = 0. Unless otherwise stated specifically, all cuts and paths/cycles are assumed to be elementary. The two optimization problems are defined as follows:

Definition 2.8.1 (Linear Optimal Flow Problem) min X

dTx

subject to xj E Cj = [C;, and E x = b .

C],:

Vj E A

Definition 2.8.2 (Linear Optimal Potential Problem) max

-bTu-cTv

",V

subject to and

E Dj= [d;, d:], V j v = -E T U.

vj

E

A

(2.8.3) (2.8.4)

Clearly both problems are special cases of linear programming. What makes them important and interesting to study is that because of the special network structure, these problems can be solved much more efficiently than the usual linear programs, usually by several orders of magnitude in computation time. One may be tempted to think that these two problems are dual to each other, but as they stand this is not quite true yet. At the end of this section, special assumptions on the cost and capacity/span intervals will be imposed to establish the duality. We have addressed the issue of feasibility in the preceding sections. For optimization, an intimately related issue is whether the optimal solution is finite. As the reader will soon discover, these issues are dual to each other in the sense that

72

DUALITY IN NETWORK OPTIMIZATION

feasibility of the primal corresponds t o finiteness of the dual optimal, and that finiteness of the primal optimal corresponds to feasibility of the dual. Definition 2.8.3 In the linear optimal flow problem, we say that a cycle P is unbalanced if

Note that the qualification dTep < 0 looks rather similar to the upper span of the cycle in the feasible potential problem, although "span" has no meaning in a flow problem. Recall that the feasible potential problem has a feasible solution if and only if there is no elementary cycle with the upper span d: < 0. The striking resemblance here is no coincidence. In due course it will become clear that feasibility of the dual and finiteness of the primal are closely related. This observation is also applicable to the notion of an unbalanced cut to be defined as follows: Definition 2.8.4 In the linear optimal potential problem, we say that an elementary cut Q = [S,N \ S] is unbalanced if

+

In the above, note that 0 = b~ = bs b ~ \ SO~ that , bNjs = -bs. We shall study the optimal flow problem and the optimal potential problem in turn before proving an important duality result. Theorem 2.8.1 (Existence for linear optimal flow) Assume that the linear optimal flow problem has at least one feasible solution. A finite optimal solution exists if and only if no elementary cycle is unbalanced. Proof: (Necessity) If an unbalanced cycle exists, an infinite amount of flow can be sent around this cycle without violating the conservation constraint (2.8.2). This incurs an infinitely negative cost, inf = -00. (Sufficiency) A specialized proof based on the theory of extreme flow can be found in [R3]. We shall defer the proof of sufficiency to the next section where the existence of the more general problem of convex optimization is proven in a unified framework. D

Definition 2.8.5 (Kilter curve for optimal flow) For each arc j, the Kilter Curve for arc j is the set


0, less goods should be sent. If ( d j - v j ) = 0, then it doesn't matter as long as the amount sent is feasible with respect to capacity.

Algorithm 2.8.1 (Optimal Flow Algorithm) Initialization. Find a feasible flow X. Do: Let the (flow dependent) span interval be defined by:

Apply either Algorithm 2.7.3 or Algorithm 2.7.4 to find a cycle P with d; feasible potential.

< 0 or a

If a feasible potential to the above span interval is found, return optimal solution, stop.

X

Else no feasible potential is found, but a cycle P with d; instead. Let c+ - x j for arc j E P + , 7 = min Xj-c; forarcjEP-.

as the

< 0 is found

If 7 = 00, P is an unbalanced cycle, the solution is unbounded, stop. Else q > 0 is finite. Change the flow by: X +- X + qep . End End

End

W

Remark 2.8.2 If a feasible potential is found, the kilter condition is satisfied and by Theorem 2.8.2, the solution is optimal. If a cycle P with d$ < 0 is found, then by definition of the span interval, the flow on this cycle must be such that Xj < c: V j E P+, xj > c; V j E P-, and consequently 7 > 0. The new solution has a cost dT(X qep) = dTx 7dTep. Since dTep = d; < 0, the new solution has a strictly lower cost. If = oo, by Theorem 2.8.1, the optimal solution is unbounded. Otherwise 7 is finite and if the usual commensurability assumption is satisfied, 7 will not be ever-diminishing, this process must terminate in a finite solution that gives the minimum flow.

+

+

We now turn to the problem of optimal potential. Because of duality, the analysis is very similar. The following result can be considered to be a dual counterpart to Theorem 2.8.1.

DUALITY IN NETWORK OPTIMIZATION

75

Theorem 2.8.3 (Existence for linear optimal potential) Assuming that the linear optimal potential problem has at least one feasible solution. A finite optimal solution exists if and only if no cut is unbalanced. ProoE (Necessity) If an unbalanced cut Q = [S,N \ S] exists, then we can change a feasible potential U to U + te,vjs which will remain feasible for any large t ; since the differential in S and in N \ S are unchanged and the span interval across the cut can accommodate any arbitrarily large increase in differential. Furthermore, the new potential will have a higher cost for any positive t. This thus incurs an infinitely large increase in cost, sup = m. (Sufficiency) As in the case of flow, we shall defer the proof of sufficiency to Section D 2.9 as a special case of the unified proof. Definition 2.8.6 (Kilter curve for optimal potential) For each arc j, the Kzlter Curve for arc j is the set Note that the above conditions imply that x j = cj if d; < vj < d,: x j 5 C j if d; = dj, and x j 2 cj if d: = vj. This curve is depicted in Figure 2.8.2 in the case where both d; and d: are finite. Note the symmetry with the Kilter Curve for optimal flow in Figure 2.8.1.

FIGURE 2.8.2

Kilter curve for arc j in optimal potential

Theorem 2.8.4 (Optimality condition for linear optimal potential) U solves the linear optimal potential problem if and only if there exists a flow X such that V X = E X = b and (xj,vj) E rj V j E A, where V = A u . ProoE (Sufficiency) Assume that X,U satisfy the optimality conditions. Given any other feasible potential U' (with Au' = v'), then either d;

< v j , ~ ;< d: then x j - c j = 0;

DUALITY IN NETWORK OPTIMIZATION

76

In all three cases (cj - ~ j (v; ) - vj) 2 0. Summing over all j and using the identity (2.3.3) give: cT(V' - V) 2 xT(v1 - V) = -bT(U' - U)

Hence U must be maximal. (Necessity) Application of the following optimal potential algorithm either finds a better solution, i.e., one with a higher cost, or finds an optimal U and an optimal X that satisfies the kilter condition. I The following algorithm for solving the optimal potential problem can be considered as a dual counterpart to the optimal flow Algorithm 2.8.1.

Algorithm 2.8.2 (Optimal Potential Algorithm) Initialization. Find a feasible potential U, and compute its differential v = Au. Do: Let the (differential dependent) capacity interval be defined by:

Apply either Algorithm 2.7.1 or Algorithm 2.7.2 to find a feasible flow X such that E x = b a n d x j € C j Vj,oracutQ=[S,N\S]withc& 0 is finite. Change the potential by: U t U + qehr\s. End End End

m

Remark 2.8.3 If a feasible flow is found, kilter condition is satisfied, and by Theorem 2.8.4 the potential must be optimal. If a cut Q with c$ < bs < oo is found,

DUALITY IN NETWORK OPTIMIZATION

77

then by definition of the capacity interval, the differential across the cut must be such that vj < d: 'J j E Q+, vj > d; 'J j E Q-, and consequently 7 > 0. The new solution has a cost

Since

the new solution has a higher cost than the previous solution. If 7 = oo, the cut is unbalanced, and by Theorem 2.8.3, the optimal solution is unbounded. Otherwise 7 is finite and if the usual commensurability assumption is satisfied, rl will not be ever-diminishing, this process must terminate in a finite solution that gives the optimal potential. The optimal flow algorithm and optimal potential algorithm are intended to accentuate duality as the driving force, but are not designed to be efficient. In practice, the network simplex (or simplex-on-a-graph) algorithm with appropriate scaling strategies is acknowledged to be the most efficient method for solving linear cost network optimization problems. Because of duality, there are two versions of the network simplex method: the simplex algorithm for flow, and the simplex algorithm for potential. As the names imply, these are specialization of the simplex algorithm to network flow and potential problems. Like the more general method, the primal algorithm for flow maintains primal feasibility (which corresponds t o dual optimality) and works towards primal optimality (which corresponds to dual feasibility). As this book is not intended to be a cookbook for solving problems, the readers may like to refer to specialized texts such as [KHl] or [AM011 for details of the network simplex algorithm. Some underlying ideas that require some familiarity of linear programming (LP) (see Algorithm 3.1.1 of Chapter 3) are as follows: 0 One starts from an initial basic feasible solution, where a basis in this case corresponds t o a spanning tree, the basic variables correspond t o flow in the tree arcs, and the nonbasic variables correspond t o flow in the cotree arcs. The Tucker representation Theorem 2.3.2 asserts that the flows in the tree arcs are uniquely determined by the flows in the cotree arcs. The latter nonbasic variables will take values either on the lower or upper bound of the capacity interval of the cotree arcs.

In the pricing step, the potentials of all nodes are uniquely determined by setting the cost and the differential of each tree arc to be the same. The relative or shadow cost of each cotree arc turns out to be the difference of its cost and differential. If there exists a cotree arc such that its relative cost is negative and its flow is at its lower bound, or if the relative cost is positive and the flow is at its upper bound, then the total cost of flow can be reduced by adjusting the flow of this cotree arc (increase flow if relative cost is negative, decrease flow if the relative cost is positive).

DUALITY IN NETWORK OPTIMIZATION

78

In the pivoting step, the flow change is implemented around a cycle so as to maintain conservation. This cycle includes the cotree arc concerned, and the unique path in the basis tree joining the two end points of the cotree arc. The maximal amount of flow change around this cycle is determined by the smallest slack from the bounds amongst all the arcs in this cycle. In the changing of basis step, the cotree arc concerned will enter the basis tree, replacing one of the previous tree arc that has its flow hitting one of its bounds. The new spanning tree will thus define a new basis, and the procedure is repeated again from the pricing step until the shadow cost of each cotree arc is such that no reduction in cost is possible, hence the optimal solution. The simplex algorithm for potential follows in a symmetrical manner. There is one other well known method for solving the optimal flow or optimal potential problem which warrants special attention. This is more for aesthetic reasons rather than for practical reasons. Duality again plays a key role behind the completely symmetrical nature of the Out-of-Kilter algorithm [F3]. Unlike the simplex method, this does not need modification for it to be applicable t o either the optimal flow or the optimal potential problem. Furthermore, it does not require the initial solution t o be feasible nor t o satisfy the optimality condition. The idea is simple: given a flow that satisfies Vx = b and a potential U, one tries to force each of the flow/differential pair (xj, vj) onto the corresponding kilter curve by either changing the flow around some cycle, or changing the potential across some cut. In what follows, we shall present the out-of-kilter algorithm as if we are solving the optimal flow problem, but it does not require much convincing to see that the algorithm can be directly applied to the optimal potential problem. Algorithm 2.8.3 ( T h e Out-of-Kilter Algorithm) Initialization. Find any flow that satisfies Vx = b and any potential U, compute the differential v = Au. DO: until all arcs are in kilter, i.e., (xj, vj) E

r j V j E A, stop.

Paint the arcs according to:

Cj = [C;, ]C:

=

I

Green if C; < x j < C: and d j = dj; Red if C; = x j and vj < dj, or if C: = x j and White if ;C = x j and vj = dj, or if x j < C;, o r i f x j d j ; Black if :C = x j and d j = dj, or if x j >C:, orifxj >C;, andvj dj;

(Essentially, this amounts to painting the interior of the lR2 plane to the left, and the left corner, of the kilter curve as White, the interior of the lR2 plane to the right, and the right corner, of the kilter curve as Black, the relative interior of both the vertical segments of the kilter curve as Red, and the relative interior of the horizontal part of the kilter curve as Green.)

DUALITY IN NETWORK OPTIMIZATION

79

Select any arc k that is out-of-kilter (this must be either Black or White).

If k is White, then let JV+e- {W (k)), N- +- {a(k)). Apply the painted network Algorithm 2.2.1 to find a compatible cycle P : w(k) -+a(k) that includes arc k as part of P + , or a compatible cut Q : w(k) 1 a(k) that includes arc k as part of Q-. Else k is Black, then let i\l+ + {a(k)), ih/- + {w(k)}. Apply the painted network Algorithm 2.2.1 to find a compatible cycle P : a(k) -+ w(k) that includes arc k as part of P - , or a compatible cut Q : a(k) I w(k) that includes arc k as part of Q+.

End

If ( P r i m a l Phase) a compatible cycle P is found, compute ' c3+ - x j c3- - x ~ c, - xk a = rnin c2 -xk xk - C; xk - C: 3

j

C

X.

-cj

for arc j € P + a n d u j > d j , forarcj~P+anduj dk, for Black arc k E P- and vk < dk, for Black arc k E P- and vk 2 dk, for arc ~ E Pa n- d u j > d j , for arc j E P- and uj < d j .

If cr = m, P is an unbalanced cycle, the solution is unbounded, stop. Else a > 0 is finite. Change the flow by: X +- X + crep. End Else(Dua1 Phase) a compatible cut Q is found, compute

q = min

vj-dj forarcj~Q-withzj>c;, for White arc k E Q- with xk 2 cl,, uk - dk for Black arc k E Q+ with z k 5 c:. ( dk -

If q = 00, Q is an unbalanced cut, there is no feasible solution, stop. Else q > 0 is finite. Change the potential by: U +-u qeN\s. End

+

End End

D

The usual justification of the out-of-kilter algorithm is based on the concept of a kilter number. This is a measure of the out-of-kilterness of the flow-differential pair for each arc. If the flow-differential pair for an arc is in-kilter, the corresponding kilter number is zero. Otherwise, the kilter number is strictly positive. There are two possible ways to define this kilter number. If the starting flow is feasible with respect to capacity, then the area of the rectangle subtended by the point (xj, ~ j )

80

DUALITY IN NETWORK OPTIMIZATION

with the kilter curve can be used as one definition of kilter number. If the flow is infeasible with respect to capacity, the horizontal distance from the out-of-kilter point (xj: vj) from the kilter curve can also be used as the kilter number. It remains to show that each iteration of the out-of-kilter algorithm strictly reduces the total kilter number, and if the usual commensurability assumption applies, this reduction is a multiple of some positive quantum, and therefore will not be ever-diminishing. In the primal phase, the change of flow around a cycle will be such that the flowdifferential pair for each arc in the cycle remains on, or moves horizontally closer to the kilter curve, and therefore the kilter number, defined in either way, will be non-increasing for each arc on the cycle, with at least one strictly decreasing. In particular, the arc (or arcs) in the cycle that achieves the minimum in the calculation of q will enter into the kilter curve. In the dual phase, the flow-differential pair for each arc on the cut remains on, or moves vertically closer to the kilter curve. In particular, the arc (or arcs) in the cut that achieves the minimum in the calculation of q, will enter into the kilter curve. Thus if the kilter number is measured as the area of rectangle, the kilter number for each arc in the cut will be non-increasing, with at least one strictly decreasing. Otherwise if the kilter number is measured as the horizontal distance, then only the arc (or arcs) in the cut that achieves the minimum q will have its kilter number reduced from some positive value to zero. Since the total kilter number decreases strictly as a multiple of a quantum, eventually the algorithm must terminate. If there is some unbalanced cycle or unbalanced cut discovered by the algorithm, then by Theorem 2.8.1 and Theorem 2.8.3, either the optimal solution is unbounded, or the problem is infeasible. Otherwise the finite I optimal solution will be obtained after a finite number of iterations. The linear cost optimal flow problem and the linear cost optimal potential problem are dual to each other under a further assumption. Bearing in mind that adding or subtracting a constant to the objective function of an optimization problem does not change the optimal solution, let cj, dj, p j , and qj be numbers satisfying the following constraint:

Definition 2.8.7 (Primal Optimal Flow Problem) min subject to and

Cdjxj+pj xj

> cj V j E A

E x = b.

Definition 2.8.8 (Dual Optimal Potential Problem) max U,"

subject to and

-C(cjvj+qj)-Cbi~i jEA i€N vj 5 dj V j E A v = -E T U.

DUALITY IN NETWORK OPTIMIZATION

81

It is easy to check that the two problems share the same kilter curves (see Figure 2.8.3):

and therefore they also share the same optimality condition. Thus solving for the optimal flow to the primal will yield a potential which is the optimal solution to the dual, and symmetrically, solving for the optimal potential to the dual will yield a flow which is the optimal solution to the primal.

FIGURE 2.8.3

Kilter curve for arc j in optimal flow/optimal potential

The above conclusion can be deduced through the usual weaklstrong duality routine.

Theorem 2.8.5 (Weak Duality) If X is a feasible solution to the primal, and U is a feasible solution to the dual then the primal cost is always greater or equal to the dual cost. Proof: Let X and then from (2.2.3)

U

be feasible solutions to the primal and the dual respectively, T

X V =

-b T U.

The difference between the primal cost and the dual cost is then:

(2.8.12)

DUALITY IN NETWORK OPTIMIZATION

T h e o r e m 2.8.6 (Strong Duality) If both the primal problem and the dual problem have feasible solutions, then the optimal primal cost and the optimal dual cost are finite and share the same value. If only the primal has a feasible solution, it has an unbounded infimum. If only the dual has a feasible solution, it has an unbounded supremum. Proof: Note that the primal is feasible, and that the primal has a finite minimum as the dual is feasible, by the weak duality. An application of the out-of-kilter algorithm will end up with all arcs satisfying the kilter condition, and hence the inequality in (2.8.13) holds as equality for all arcs. By weak duality, the minimum of the primal and the maximum of the dual must be equal. If the dual is infeasible, it has a supremum of -00,so has the infimum of the primal by applying the outof-kilter algorithm. If the primal is infeasible, it has an infimum of CO, so has the I supremum of the dual. R e m a r k 2.8.4 Either by a suitable change in origin, or by setting the parameters = cj = q j = 0: the primal and dual problems take the form of a standard pair of asymmetric linear programs

pj

2 0,

Primal:

min dTx

s.t. Ex = b,

Dual:

max bTu

s.t. - u T E 5 d T .

The kilter condition of (xj, vj) E tary slackness condition:

rj

X

V j translates into the well-known complemen-

The results in this section are all special cases of a much more general class of convex optimization problems, which will be taken up in the next section.

DUALITY IN NETWORK OPTIMIZATION

83

2.9 Duality of Convex Optimal Flow and Optimal Potential We now return t o the general optimal flow and optimal potential problems first addressed in Definitions 2.3.4 and 2.3.5. Here we impose the additional assumption that all the underlying functions are convex. This general class of problems include all the problems we have discussed so far as special cases. The resulting duality theory can thus be considered as a generalization of all the duality results discussed in this chapter. Consider a digraph G = ( N , A ) with a node-arc incidence matrix E. Each arc j E d is associated with a capacity interval C j and a convex cost function f j : Cj -t R. To extend the domain of f j to R , we adopt the convention:

In which case we can rewrite C j as

Unlike in Section 2.8, here the capacity intervals Cj's need not be closed, although the function fj's are assumed to be closed. Let the end points of C j be c; and c:, where c; can be -m and c: can be m. In general f j can be assumed to be continuous on Cj but may not be continuously differentiable. Its left and right derivatives are defined as follows.

Definition 2.9.1 (Left and right derivatives) Let derivatives of f j : Cj + IR are defined by

5E

Cj. The left and right

Let

Remark 2.9.1 Note that, since f j is convex, then (i) For all E Cj, fi- (E) 5 f;+ (EL

c

with equality if and only if f j is differentiable at 5. (ii) Both the left and right derivatives are finite in Cj with

fi+(E') 5 f;- (Q) (iii) (iv)

whenever

C < E".

fi- and fi+are both monotone non-decreasing on C,.

4-and f:+

V:-cc), f;+ cr)].

form the end points of the subdifferential of fj, i.e.,

(2.9.5)

afj(O

=

84

DUALITY IN NETWORK OPTIMIZATION We extend the domains of

fi+and fi- from Cj to R by defining

fi-(5) = f;+ (E) =

+W

f;-(O = fi+( 0, then clearly the complementary slackness condition of Theorem 3.3.1 is satisfied, and as such X is optimal for the primal and ( & p ) is optimal for the dual. To find a feasible solution to the primal problem under these conditions, we set up the artijkiat primal problem : (Artificial Primal Problem)

min

subject to:

lTy

Ax

+ y = b,

xi = 0 if

~

~- CIa- pi '

< 0,

DUALITY IN LINEAR SYSTEMS

109

Theorem 3.3.3 (Primal-Dual Optimality Theorem) If (X*,p*) is feasible for the dual, and the optimal solution of the artificial primal problem is y = 0 and X * , then X* is optimal for the primal and (X*, p * ) is optimal for the dual. Proof: If y = 0 is optimal for the artificial primal, then the corresponding X* is feasible for the primal, and the conclusion follows from the complementary slackness condition. D Remark 3.3.2 Of course there is no guarantee that given any feasible (X, p ) for the dual, the artificial primal will always yield y = 0 as the optimal solution. In which case, the dual solution (X, p) will he adjusted accordingly. This is the basis of the so-called primal-dual algorithm. Although the detail is not provided here, some of the network flow algorithms (e.g., the out-of-kilter algorithm) discussed in the previous chapter would have given the reader a flavor of the underlying mechanism.

3.4 Duality and Theorems of Alternatives Theorems of Alternatives were first studied by Minkowski, and these are sometimes called transposition theorems. These results are not only useful in proving many results in optimization theory, they are also intimately linked to the duality of optimization and variational inequalities [G3]. The Painted Network Theorem discussed in Chapter Two is an example, and the reader should be convinced by now that this theorem is instrumental in proving many of the duality results in Chapter Two. Other network related examples are found in the Feasible Flow Theorem 2.7.1 and the Feasible Potential Theorem 2.7.2. While these results are valid in nonlinear systems too, our interest is only confined to those of linear systems in this chapter. Typically, one has two linear systems, or polyhedral sets, structured in such a way such that exactly one of them is nonempty. The basic tool used in proving these results is usually the Separating Hyperplane Theorem 1.2.8. Before we go into a formal analysis of theorems of alternatives, more background material in convex analysis is needed. Definition 3.4.1 (i) X E IRn is said t o be a cone if X E X + Xx E X VX > 0. (ii) If in addition, X is convex, then X is said to be a convex cone. Alternatively, a cone X is convex if it is closed under addition, i.e., X', x2 E X + x1 x2 E X. (iii) A cone X E Rn is pointed if it contains no subspace of Rn other than the origin.

+

Remark 3.4.1 Note that: (i) A convex cone needs not be pointed, e.g., a line through the origin.

DUALITY IN LINEAR SYSTEMS (ii) A pointed cone needs not be convex, e.g., two disjoint half rays through the origin. (iii) A cone X needs not be closed. But if it is closed and non-empty, then it is necessary that 0 E X. (iv) Usually the term "cone" means a convex cone by convention. Definition 3.4.2 Let X E IR" (not necessarily a cone). The polar cone (sometimes called a dual cone, and in fact more appropriately so) X* of X is defined by X* = {y E R." I xTY5 0 VX E X). If X = R", then X* = {0) and vice versa. If X = 0 then X* is taken t o be Rn by convention. The bipolar cone X** of the set X is the polar cone of X * . R e m a r k 3.4.2 Note that: (i) A polar cone is a cone. (ii) If the cone X is a subspace, X* is its orthogonal complement {y E ]Rn I x T y = 0 v x E X). L e m m a 3.4.1 If X E IRn is nonempty, then X* is a closed convex cone, and X X**.

c

Proof: If

X E

X, then x T y 5 0 Vy E X*, which implies that

X E

X**.

m

L e m m a 3.4.2 If X', X2 E R" are nonempty, then X' C X2 + (X2)* C (X1)* Proof: If y E (X2)* then xTy 5 0 V x E X2. Since X' C x 2 , this means that m xTY5 0 Vx E X 1 ,or y E (X1)*. T h e o r e m 3.4.3 If X E Rn is a nonempty closed convex cone, then X** = X. Proof: X E. X** by Lemma 3.4.1. We need only to show that X** S X. Let X E X**. If X # X , then the Separating Hyperplane Theorem 1.2.8 asserts that there exists a hyperplane 'F1 = {y I aTy = b) (with b 0 without loss of generality) that separates X and X , i.e., aTy 5 b V y E X and a T x > b. If a T z > 0 for some z E X, then, by virtue of X being a cone, aT(Xz) > 0 can be arbitrarily large for X > 0 which contradicts that aTy 5 b Vy E X. Hence a T z 0 Vz E X and therefore a E X*. Now X E X** + aTx 5 0 which contradicts aTx > 0. Therefore B X E X , and thus X** S X.

>


O .

Alternatively, this may be stated in an equivalent form: exactly one of the following systems is non-empty. System I System I1

l X 2 0 , A x = b) E IRm I X T b < 0 , X ~ 2oT) A

{X E IRn {X

DUALITY IN LINEAR SYSTEMS Proof: (Necessity) If X E lRm and X ~ 2AoT then since XTb 0.

>

111

X

>0, we have X T ~ =x

(Sufficiency) If not, let X = {y E RmI y = A x = C:=l xaa$ xi 2 0 Vz.) where ai is the ith column of A. Clearly X is a closed convex cone (in particular, a polyhedral cone). Thus if b @ X , i.e, b # A x for X 2 0, then by Theorem 1.2.8 there exists a separating hyperplane that separates b from X , i.e., 3X such that X T b < 0 and I X T a 2 0 V2 (or, equivalently, X T ~oT).

=>

R e m a r k 3.4.3 Note that if X is as defined in the proof above, then its polar cone is given by X* = {y E lRm I y T 5~oT). Since X is a closed and convex cone, by Theorem 3.4.3, X = X**. Thus a given vector b either belongs to X = X** or it does not. System I above says that b belongs to X , while system I1 says otherwise. Corollary 3.4.5 Let A E lRmX is non-empty.

and b E lRm. Exactly one of the following systems

Proof: System I is equivalent to

The conclusion follows by replacing the matrix A in Farkas Lemma by [A, I].

I

Corollary 3.4.6 Let A E IRmXn,B E lRmXPand b E lRm. Exactly one of the following systems is non-empty. System I System I1

{[;l

dEtn+pI A x + B y = b , x ~ O

{X E Rm 1 XTb < 0, X ~ = B 0, X T >gT] ~

Proof: If we replace y in System I by equivalent to

U -v

where

U

2 0,

2 0, then System I is v -

The conclusion follows by replacing the matrix A in Farkas Lemma by [A, B , -B]. I

Theorems of alternatives are intimately related to LP duality. For easy reference, we restate the asymmetric primal and dual LP problems as follows:

DUALITY IN LINEAR SYSTEMS

(Primal) (Dual)

min cTx such that Ax = b, X 20. rnax bTX such that XT A 5 cT.

(3.4.1) (3.4.2)

We restate Corollary 3.2.3 (to the weak duality Theorem 3.2.2) as follows:

Corollary 3.4.7 If the primal is feasible and unbounded below, then the dual is infeasible. If the dual is infeasible and the primal is feasible, then the primal is unbounded below. Alternatively, if the primal is feasible, then the primal is unbounded below if and only if the dual is infeasible. Corollary 3.4.8 If the dual is feasible and unbounded above, then the primal is infeasible. If the primal is infeasible and the dual is feasible, then the dual is unbounded above. Alternatively, if the dual is feasible, then the dual is unbounded above if and only if the primal is infeasible. A simpler proof of Farkas Lemma (Theorem 3.4.4) can be obtained using these two corollaries for LP weak duality.

Proof: (Farkas Lemma by LP duality) In the primal problem in (3.4.1), let c = 0, so that the primal feasible set is given by System I of Farkas Lemma. Note that X = 0 is feasible for the dual problem in (3.4.2). Hence the dual is feasible. By Corollary 3.4.8, the primal is infeasible if and only if the dual is unbounded, if and only if there exists X such that bTX < 0 and XTA >oT,or there exists X' such that bTX' > 0 and ATX' s o T , since the solution tX' gives an unbounded dual cost as t W. This means that the dual is unbounded, or system I is infeasible, if and I only if the system I1 is feasible. Two other well-known theorems of alternatives can also be established in a similar manner by using Corollary 3.4.7 and 3.4.8.

Theorem 3.4.9 (Gale's Transposition Theorem) Let A E Etmxnand b E Rm. Exactly one of the following systems is non-empty. System I System11

(XE Rn ( x 2 0, cTx < 0, Ax = 0) { X E R ~ I X ~ A & ~ }

Proof: In the primal and dual problems, let b = 0, so that the primal and dual become (Primal) (Dual)

min cTx such that Ax = 0 , X 20. max c T such ~ that XTA 5 cT.

(3.4.3) (3.4.4)

Since X = 0 is primal feasible, by Corollary 3.4.7, the dual (3.4.4) is infeasible if and only the primal is unbounded below. That is, System I1 is infeasible if and only if the primal is unbounded below. But the primal is unbounded if and only if there exists X such that cTx < 0, Ax = 0, and X 2 0 since tx gives an unbounded I solution as t W.

DUALITY IN LINEAR SYSTEMS

113

Theorem 3.4.10 (Gordan's Transposition Theorem) Let A E IRmXnand b E IRm. Exactly one of the following systems is non-empty. (Note that writing X 2 0 is the same as X 2 0 and X # 0.) System I

{X E

Rn I

X

2 0,

A x = 0)

{X E IRm I XTA < oT}

System I1

Proof: In the primal and dual LP, choose c = -1 (a vector of all ones) and b = 0. Then, System I1 is feasible if and only if the system {X E RmI X ~ 5A-lT}is feasible if and only if the system{x

/

- l T x < 0, X

2 0,

A x = 0) by Theorem 3.4.9

if and only if the system{x I l T x > 0, X if and only if the system I is infeasible.

2 0,

A x = 0)

is infeasible is infesible

We shall end this section with two other well known theorems of alternatives due to Motzkin and Tucker.

Theorem 3.4.11 (Motzkin's Theorem of Alternative) Let A E IRmXn, B IRmX P and C E IRmXq. Exactly one of the following systems is non-empty.

E

{X E Rm( XTA > oT, X T 2~0 , X T c = oT}

System I1

Proof: We first show that if System I1 is feasible, then System I is infeasible. If System I1 is feasible, and for X 2 0, y 2 0 we have

or equivalently, (AX

+ BY + C Z ) > o

contradicting the first equality of System I. Next, we show that if System I1 is infeasible, then System I must be feasible. If System I1 is infeasible, then the system XTA 2 t l T for some t

> 0, XTB ? o T ,

is infeasible, or equivalently, the system

XTc

=0

114

DUALITY IN LINEAR SYSTEMS

is infeasible, or equivalently, by Corollary 3.4.6, the system

is feasible, or equivalently, the system

is feasible, or equivalently (after letting v = -z) the system (System I) Ax+By+Cz=O,

x>O, yzO,

is feasible.

Theorem 3.4.12 (Tucker's Theorem of Alternative) Let A E IRmXn, B and C E ]RmX Exactly one of the following systems is non-empty.

E

IRm

System I1

{X E lRm I X T 2~oT7X T 2~0 , X T c = oT}

Proof: The proof follows closely that of Theorem 3.4.11 and is omitted.

W

In subsequent chapters, theorems of alternatives will be used over and over again to establish duality results for linear and nonlinear systems.

3.5 Painted Index Theory This section is concerned with a very powerful generalization of the painted network theory discussed in Chapter Two. While the theoretical foundation is based on the theory of oriented matroids (see, for example, [L4]), it is possible to get away with a simpler version without the need t o master all the background material required for a full understanding of matroid theory. The development here follows closely that of Rockafellar [R3], [R7]. The reason why this treatment is separated out from Chapter Two where all the network results are collected is because the current duality theory overlaps significantly with concepts in linear

DUALITY IN LINEAR SYSTEMS programming presented in the earlier part of this chapter. Naturally, a firm grasp of LP concepts enhances the understanding of the duality of painted index theory. Under fairly general conditions, almost all the formal properties that have been said about a specialized node-arc incidence matrix E in Chapter Two also hold true be a real matrix, with n > m. We for an arbitrary real matrix. Let A E RmXn assume that A is of full rank, i.e., rank(A) = m. Given real vectors x E Rn and U E Ktm, let the vectors y and v be defined as:

We refer to the variables in (x,y) as the primal variables, and the variables in (U,v ) as the dual variables. Thus if A is specialized to a node-arc incidence matrix for a digraph, then the vectors X,y ,U,v represents the flow, divergence, potential, and differential vectors respectively. In the current context, these vectors have no special physical significance and should be considered in their abstract framework. As in the network case, we have the fundamental identity:

so that Let

e = (X E Rn I A x = 0)

and

D = {v E Rn I v = - A ~ Ufor some U E Rm}. be two spaces representing the kernel of the linear mapping represented by A and the row space of A respectively. Since given any X E e, Ax = y = 0, so that x T v = 0 by (3.5.1) and therefore e and 'D are orthogonally complementary to each other, i.e., e={xcIRn I V ~ X = O V V E V ) = B ~ and

'D = {v E Rn 1 v T x = O V X E C) = eL.

As in the case of linear programming, we can construct a basis matrix B from A by choosing m linearly independent columns of A to form a non-singular basis matrix B. The remaining n - m columns make up another matrix N E Etm By an appropriate reordering, we may express the matrix A as A = [B, NI. Correspondingly, we partition the vector xT = [X;, X&], into the basic variable x p E Rm and the nonbasic variable XN E such that

+

Given any X E C!, A x = BxB N x N = 0. Consequently, we may express the basic variable explicitly as a linear function of the nonbasic variable,

DUALITY IN LINEAR SYSTEMS

116

where T = -B-lN E R.mx(n-m) is referred to as the Tucker Tableau [Tl]. Similarly, let v E D and partition v in correspondence with B and N so that by definition, there exists some U E IRm with

Thus U

and consequently, VN

= (-B~)-'V~,

= - N ~ ( - B ~ ) - ~=v -~ T ~ v ~ .

(3.5.5)

(3.5.3) and (3.5.5) are referred to as the Tucker representations of the linear systems of variables [Tl]. Note that the Tucker representation given in (2.3.6)-(2.3.8) is the network specialization of the general case here. Since there exists a maximum of

(z)

ways of choosing the basis matrix B (not all of them necessarily give rise to

a non-singular B), the number of possible Tucker representations is therefore finite.

When one (or more) column of B is exchanged with another column (or more) from N, the Tucker tableau T switches from one form to another. For computational purposes, we restrict the number of changed columns at any time to just one, say B and N are changed to B' and N' where B and B' differs by exactly one column. In which case, the new Tucker tableau T' = -(Bf)-'N' can be easily computed from the old Tucker tableau T = -B-'N by a simple rank one update procedure commonly known in LP as the pivoting step. This is a well-known procedure for readers who are familiar with the simplex algorithm for solving linear programs.

Primal and Dual Support Let the column index set of A be 3 = {l,2, ,n). The theory of oriented matroid is concerned with combinatorial structures related to subsets of the index sets 3 induced by the elements of C! and D. We will not go into a full discussion of matroid theory, but will instead present a slightly more compact version due to Rockafellar [R2], [R8].

Definition 3.5.1 (Primal Support) A subset P = P+ U Psupport of C! if there exists X E C such that

c 3 is a primal

Clearly only the trivial vector X = 0 has an empty primal support. A primal support P is said to be elementary if there does not exist a PO= P$ UPc # 0 such that POC P (as a strict subset). A vector in C? is said to be elementary if its support is elementary. Given a primal support of C!, the associated vector X E C may be non-unique, although it is not hard to show that two vectors corresponding to the

DUALITY IN LINEAR SYSTEMS

117

same primal support are a scalar multiple of each other. This minor complication can be overcome by identifying a unique associated vector ep (generalized incidence vector for the primal support) under the following normalization scheme:

IlepIJ1= lPl, the cardinality of P.

(3.5.6)

Definition 3.5.2 (Dual Support) A subset Q = Q+ U Qof C! if there exists v E D such that

3 is a dual support

Clearly only the trivial vector v = 0 has an empty dual support. A dual support Q is said to be elementary if there does not exist an Q. = Q: U Q; # 0 such that Q. C Q (as a strict subset). A vector in 'D is said to be elementary if its support is elementary. Given a dual support of 'D, the associated vector v E D may be non-unique. This again can be overcome by identifying a unique associated vector e~ (generalized incidence vector for the dual support) under the following normalization scheme:

JJeQJI1 = IQ(, the cardinalityof Q.

(3.5.7)

The fact that we are using symbols that are almost identical to that used in Chapter Two is no coincidence. In the event that A is the node arc incidence matrix of a digraph, then P, Q, e p , e~ reduce to a cycle, a cut, a cycle incidence vector, and a cut incidence vector respectively. It is possible to show that any X E C! with a corresponding support P can be expressed as a linear combination of generalized incidence vectors (for primal support), each of which corresponds t o an elementary primal support. Similarly, any vED ' can be expressed as a linear combination of generalized incidence vectors (for dual support) as well. Although the restriction t o elementary supports is not necessary for the subsequent analysis, we shall nevertheless make such an assumption for convenience in view of the above.

Definition 3.5.3 (Index painting) A painting of the index set 3 is a partition of 3 into four disjoint subsets, some possibly empty, such that the index in each of the subsets are labelled Green, Red, White, and Black respectively. The following is a generalization of the painted network Theorem 2.2.1, and can be thought of as another theorem of alternative similar to those discussed in the previous sections. This result, however, has a strong combinatorial flavor to it. The qualification of "elementary" as appearing in square parentheses below means that its appearance is optional.

Theorem 3.5.1 (Painted index theorem - combinatorial form) Given any painting of 3 and any C E J that is Black or White, exactly one of the following is true:

DUALITY IN LINEAR SYSTEMS

118

I. There exists an [elementary]primal support P containing l such that Vj E Pp+, j is painted Green or White, and Vj E P - , j is painted Green or Black. 11. There exists an [elementary] dual support Q containing l such that Vj E Q+, j is painted Red or Black, and Vj E Q-, j is painted Red or White. (Partial) proof: If there exists an X E

(2 that

satisfies the conditions in Alternative

I: and there exists an v E B that satisfies the conditions in Alternative 11, then xe and ve must be of opposite sign and neither is 0; if j # .l is Green or Red then

xjvj = 0 since either X j = 0 ( j is Red) or vj = 0 ( j is Green); if j # l is White, then 2 0 and vj 5 0; and if j # l is Black, then x j 5 0 and vj 2 0. Consequently,

Xj

T

X V = XeVe

+

C

XjVj f

C

XjVj

0,

min V f (x)y.

YEX

We may then rewrite the restricted Wolfe dual as d(x) = f (X) - V f (x)x

+ Ymin V f (x)y EX

Gap functions have several interesting properties, which we shall study with respect t o bounds, convexity and differentiability in turn. The first two properties of the following theorem can also be regarded as the defining properties of a gap function.

Theorem 4.5.4 (Bounds on y) (i) y(x) 2 O, (ii) y (X*)= 0, where X* is the optimal solution to the primal, (iii) The gap function is minorized by the convex function f (X) - f (X*),i.e.,

and, in particular y(x)

2 Vf (x)(x- X) = 0.

(ii) Since X* is the primal optimal solution, it is necessary that

and hence, Vf(x*)(y -X*)

2 0 vy

This together with (i) imply that y (X*) = 0. (iii) Since f is convex, we have, for a fixed X,

E

X.

DUALITY IN CONVEX NONLINEAR SYSTEMS Taking the minimum on both sides with respect to y E X gives

Theorem 4.5.5 (Convexity of gap function) Assume that each component of V f (X) is concave in X, and that V f (x)x is convex. Then y (X) is convex. Proof: For convenience, define the auxiliary functions

Then clearly y (X)= h(x) - s(x). Note that: (i) By its first definition, W is a monotone function since, given a larger z, the underlying linear program has a larger feasible set. (ii) Furthermore W is concave since its second definition asserts that majorant of linear functions.

W

is the affine

h is assumed convex, so it remains to show that s is concave. Given xl,x2 E Rn and X E [0, l],we have,

+ (1 - x)x2) = w ( 0f (xxl + (1 - x)x2)

s(Xxl

2 w(XVf (X1) + (l - X ) V (X2)) ~ by the monotonicity of

> Xw(Vf (X')) + (1 - X)W(Vf (x2)) = Xs(xl) + ( l - X)s(x2).

W

and concavity of V f

by the concavity of

W

Theorem 4.5.6 (Subdifferential of y) Assume that each component of V f (X) is concave in X, the subdifferential of y is given by:

where

y(x) = { y E X I y E argminZExVf (x)z) , and V2f is the Hessian of f .

Proof: Since the components of V f (X) are concave, we have

150

DUALITY IN CONVEX NONLINEAR SYSTEMS

Multiplying by y

=> 0, we have

Since y E y(x), V f (x)y = w(Vf (X)) and similarly V f (z)y = w(Vf (z)). Therefore

and hence

as(x) 2 {yTv2f(X) I y

E

Y(x)}.

Consequently,

Remark 4.5.2 From the above results, an alternative way of solving the convex optimization problem CP is to solve the unconstrained problem min y (X). X

In fact Polyak [PI] has proposed an algorithm for minimizing a non-differentiable function where the optimal cost value (0 in this case) is known a priori. see [H21 for detail.

CHAPTER 5

DUALITY IN NONCONVEX SYSTEMS In Chapter Four, the conventional Lagrangian function used for solving constrained optimization problems is a linear combination of the cost and constraint functions. Optimality conditions based on the linear Lagrangian theory typically are either necessary, or sufficient, but not both unless the underlying cost and constraint functions are also convex. Recently a more general Lagrangian function is defined in [S4]. This is a unified approach of the conventional Lagrangian duality and the surrogate duality [SZ], 1531. In this chapter, we propose a somewhat different approach to solving nonconvex inequality constrained optimization problems based on a nonlinear Lagrangian function. This is a nonlinear combination of the cost and constraint functions, which leads t o optimality conditions that are both sufficient and necessary, without any convexity assumption. Further more, by appropriately defining the dual optimization problem, we show that a zero duality gap will always hold regardless of convexity, contrary to the case of linear Lagrangian duality. This has a clear geometric meaning from the fact that the perturbation function to be defined in (5.1. l ) is monotone non-increasing. Thus even though there may not be a supporting hyperplane at the optimal point, there will always be a supporting cone. This is related to the concept of nonlinear penalty functions as will be shown in Section 5.7. The supporting cone concept has motivated a series of research (see, [GYl], [GY2], [HYl], [HY2], [YG2], [YLl], [RGYl], [RGY2], [YHl], and [YTl]) on which this chapter is based.

5.1 Examples of Nonzero Duality Gaps As an introduction to this chapter, we shall provide some intuitive ideas before a formal analysis. The property of zero duality is important in designing algorithms

DUALITY IN NONCONVEX SYSTEMS

152

for nonlinear optimization problems. It is well known that there may be a nonzero duality gap between the primal optimization problem and its (linear) Lagrangian dual problem unless the primal problem is convex or satisfies some generalized convexity conditions, see [L6],[JWl],[R2]. Consider the following inequality constrained optimization problem: (Problem P,) ~.t.

inf fo(x)

xEX

fi(x) 5 Oi, i = 1 , 2 ; - - , m ,

where fi : ]Rn -,R,i = 0,1,2, - - ,m are continuous functions, and X is a subset of lRn. Define the set of all feasible solutions to be

Let g(x) = [f 1(X),f 2 (X), - - - , fm (X)] and B = [01,- - - ,Q,]. problems associated with problem P, is defined by: (Problem P,) S-t.

The family of perturbed

inf fo(x)

xEX

g(x) s y

where the vector y = [yl, yz, - - - ,T y ], is a perturbation t o the parameter vector 0 of the original problem PO. When y = 6,the perturbed problem reduces to the original problem P,. Let the perturbation function W : Rm -,lR be defined by

Using the conventional notion that inf 0 = +oo, then W has an effective domain dom(w) = {y I 3x E X s.t. g(x) 5 y). Clearly the perturbation function W is a monotone non-increasing function of y. Define the epigraph of w(y) as the set:

At the optimal solution X*, the conventional Lagrangian (being linear in fo and fi) can be geometrically interpreted as a supporting hyperplane to the set epi(w), see Figure 5.1. l .

Theorem 5.1.1 If the functions fi, i = 0 , 1 , - - -,m are convex, and X is convex then the perturbation function w(y) is convex. Proof: Consider the unconstrained optimization problem:

where @(X'

{

f o ( ~ ) if g(x) 2 ~ 7 E X Y) ' oo otherwise.

By Lemma 4.1.5, we need only to show that +(X, y ) is convex in (X,y).

DUALITY IN NONCONVEX SYSTEMS

153

Consider two points ( x l , y l ) and (x2,y2)for $. If for some i E {1,2, - - ,m}, fi(xl) > [ylIi or fi(x2) > [y21i, then + ( x l , y l ) = CO or +(x2,y2) = CO, and the result holds trivially. So we need only to show the case when g(xl)

syl and

g(x2) s y 2 .

(5.1.2)

Since fi, i = 0,1, - - - ,m are all convex and X is convex, we have, for a E (0, l ) ,

Then,

and the proof is complete.

FIGURE 5.l.l

Supporting hyperplane t o a convex perturbation function

Consequently, when the functions fi, i = 0,1, - - ,m are convex and X is convex, the epigraph of W is also convex, and a supporting hyperplane exists at every point of the perturbation function w(y) or the lower boundary of epi(w). When any one of fi, i = 0,1, - - ,m is not convex, then epi(w) may not be convex, and a supporting hyperplane may not exist at some point of the perturbation function, see Figure 5.1.2.

DUALITY IN NONCONVEX SYSTEMS

FIGURE 5.1.2

Example of a nonzero duality gap

Example 5.1.1 The following example is a modified version of a problem taken from [L6]. min fo(x) = 1 - 21x2 X

20

fi(x) = X I + 4x2 5 1 It is easy to see geometrically that, at optimality, the inequality constraint is active, 15 The and therefore the optimal solution occurs at X* = [ 1 2 , 1 and fo(x*) = E. perturbation function can be easily shown t o be subject to

which is clearly nonconvex. Thus if a linear Lagrangian function L(x, X) = fo(x) X f l (X) is used, then the dual function

will always have a value of $L (X) = -m since L is minimized at X 2 0. As a result the duality gap fo(x*) - $L (X*) = m .

X

+

= (m,m ) for all

Furthermore, one can also interpret the (Lagrangian) dual function as the vertical intercept of the supporting hyperplane to the perturbation function with normal A when y = 8 (see Chapter Four). The duality gap is thus the vertical difference between this vertical intercept and fo(x*). If the perturbation function is nonconvex, then it is easy to see that the duality gap cannot be zero if the supporting hyperplane is not supporting the perturbation function at y = 8.

DUALITY IN NONCONVEX SYSTEMS

155

Consider the problem Pe again, but X is a discrete set. Now the feasible set for Problem Py may not change as y changes in Rm.Thus w(y) is a step function. Thus w(y) is a monotone non-increasing step function of y. The epigraph epi(w) is always nonconvex. In general: a supporting hyperplane does not exist at y = 8. However, it is easy to see that there is a shifted supporting cone a - ' : R at any boundary point a of epi(w). This shifted cone is represented by the weighted Tchebyshev norm.

Example 5.1.2 Consider the example given in page 181 of [BSI]: min subject to

fo(x) fi(x)=0, X=

(XI,22) E X ,

) where X = ((0, O), (0,4), (4,4), (4, O), ( l , % (2, l ) ) , f o b ) = - 2 ~ 1+ X2 + 4, f l ( ~ = xl +x2 - 3. It is clear that minx,, fo(x) = 1. If the linear Lagrangian dual is applied, then there is a nonzero duality gap of magnitude 3, see [BSI] since

1"

\

supporting cone

FIGURE 5.1.3

no supporting hyperplane at this point

perturbation function

Supporting cone to a nonconvex perturbation function

Our study in this chapter is motivated by the above observation. Specifically, the perturbation function is a non-increasing function, thus there will always be a supporting cone to the epigraph of the perturbation function, see Figure 5.1.3. The supporting cone is represented by the weighted Tchebyshev norm. This characterization has also been used to find efficient solutions for multicriteria optimization

problems, see [GY3].

156

DUALITY IN NONCONVEX SYSTEMS

5.2 Zero Duality Gap via a Nonlinear Lagrangian Function Throughout the rest of this chapter, we assume

We begin by presenting a nonlinear Lagrangian function which relies on the assumption of the existence of an optimal solution. The main reference of this section is [GYl], [GY2]. Consider the problem Pe. Without loss of generality, we assume in this section that the parameter vector 0 = [g1, - - - , gm] E int R T . The key analytical tool used here is the weighted Tchebyshev norm (strictly speaking, this does not qualify as a norm), which is a scalar-valued function mapping IRm+' to R + . Given z E lRm+' and a weight vector e E i n t ~ y + ' ,the weighted Tchebyshev norm of z is defined as L ( z ) = 0 0 as in the case of a linear Lagrangian. In fact this approach will be adopted in Section 5.7 where the multiplier takes on a special meaning. In the present case, however, division of zi by ei affords a clearer geometrical interpretation as the vector e turns out to represent the 'direction7 of the supporting cone. Note that the weighted Tchebyshev norm is a limiting case (p = W ) of the (weighted) Minkowski metric:

{

1

[ ] p } 7

whenz tIRTi1.

2=0

There is more than one way of defining the nonlinear Lagrangian function. We begin with a simpler version defined as follows, other versions are discussed in latter sections.

Definition 5.2.1 (Nonlinear Lagrangian F'unction) The nonlinear Lagrangian function for the (primal) constrained optimization problem Pe is defined by, given

Let the function y : E

+ IR

be defined as follows:

DUALITY IN NONCONVEX SYSTEMS

157

Definition 5.2.2 (Nonlinear Lagrangian D u a l Function) Assuming that the optimal solution X* for problem Pe exists and fo(x*) > 0, then the nonlinear Lagrangian dual function is defined as:

\ -m

otherwise.

Note that there is a peculiar feature of this dual function, namely, given some e E E, +(e) cannot be computed, since fo(x*)is not known a priori. Fortunately, this does not restrict the usefulness of this duality theory, as we will demonstrate later in the section. In Section 5.5, we present another version of the duality function which is computable. For the rest of this section, the non-computable dual function 4 as defined in (5.2.6) will be used. Definition 5.2.3 (The D u a l Optimization P r o b l e m t o P r o b l e m PO,first version). (Problem D*) sup +(e) eEE

The following result shows that the maximization of 4 with respect to e occurs in a relative interior point of E. As a result, max may be used instead of sup in problem D*. T h e o r e m 5.2.1 The dual function C$is maximized at e = e*, where [e*Io= 1, [e*li= i' i = 1,2,.-.,rn. f o (X*1' Proof: Clearly the dual function cannot be maximized if it takes on the value -m. We only need to establish that y(e) is a monotone non-increasing function for all e E V i = 1 7 2 , - - . , m , .~ i v e n e ' , e ~ ~e1&<e2,itisclearthat, , E such that ei 2 f o (X*1 for a given X E X, maxo,,,

X) Vi = 0, 1, - - - ,m, hence ma~o- 4

{y}, ei

ei

and consequently 7(e1) = i n f x t , u n a q j i l m

7(e2) = infxr maxojil,

{y} {y} 2 2

{ f f } . Thus the dual function must be maximized

at the smallest possible e without taking the value

-00,which

is at e = e*.

R

T h e o r e m 5.2.2 (Weak duality) +(e) 5 fo(x) V e E E , Vx E Xo. Proof: If there exists k such that

ek

< A then 4(e)

f o (X*> '

the result holds trivially. Hence we may assume that

= -m

< fo(x) Vx, and

DUALITY IN NONCONVEX SYSTEMS

158 Then,

+(e) = ~ ( e=) min max

o

(7)

Theorem 5.2.3 (Zero duality gap) fo(x*) = q5(e*)Proof: By definition, 4(e*) = min C(x,e*) X€ X

Consider the two cases: Case(i), if X is feasible, then implies that

Case(ii), if

X

et

5 1 tli = 1,2,- - ,m and fo(x) 2 fo(x*), this

is infeasible, then there exists k such that

fk

(X) > 1, hence L(x, e*)

>

fo(x*). Thus, in both case (i) and case (ii), we have C(x, e*) 2 fo(x*) Vx E X. From (5.2.9), we conclude that (5.2.10) min L(x, e*) = +(e*) 2 fo(x*). xEX

Finally, (5.2.10) and the weak duality together imply that q5(e*) = fo(x*).

I

Example 5.2.1 (Example 5.1. l revisited.) min fo(x) = 1 - 21x2 X

20

s.t.

fl(x) = X1

+ 4x2 5 1.

An application of the proposed nonlinear Lagrangian approach yields the following dual function:

=

rninxEx max {l - ~ 1 x 2 , $4x2)

if e 2 e* = 16/15, otherwise.

These dual functions are plotted (in a brute force manner) in Figure 5.2.1. Note that for the dual function,

DUALITY IN NONCONVEX SYSTEMS 15

max 4(e) = fo(x*) = e>O 16' and therefore the duality gap is zero, even though the perturbation function is nonconvex.

,

,

,

,

dual f u n c t i y version

FIGURE 5.2.1

;

,

,

Dual function for Example 5.2.1

5.3 Optimality Conditions via a Monotone Funct ion We now present an equivalent optimality condition in terms of an unconstrained problem which is both sufficient and necessary. This optimality condition leads directly to a conceptually simple method for solving convex or nonconvex constrained optimization problems. The main references for this section are [GYl],[GY2].

Theorem 5.3.1 (A necessary and sufficient condition for opt imality without convexity) Let X* be the (global) optimal solution of problem PO (as defined in Section 5.1), and let B. = fo(x*)> 0. Then a solution xo solves problem PO if and only if xo solves the unconstrained problem: (Problem P,)

inf t ( x , 9) =

xE X

inf max

160

DUALITY IN NONCONVEX SYSTEMS

where 0 = [Bo7 01,

- ,Q,].

Proof: Let us first prove that inf

max

x c x o_ fo(x*) and fi(x)

If

X

If

X E

5 0i7i = 1, - - ,m. Thus

X is infeasible, then 3j E {l,2, . . ,m ) such that f j(X) > Oj , implying that f i (X)

% 0:

{a,} > l.

Moreover,

Thus the claim holds. (Necessity) If x0 solves problem PO,then x0 must be feasible, and fo(xO)= fo(x*). Hence

i.e., x0 solves problem PI. (Sufficiency) Assume that x0 solves problem PI. Then x0 E X. If x0 does not solve problem PO,then either x0 is infeasible, or x0 is feasible and fo(xO)> fo(x*). If x0 is infeasible, then 3j E {l,2, - - ,m) such that f j(X') > Oj , implying that

If x0 is feasible and fo(xO)> fo(x*) , then the inequality (5.3.1) still holds. Thus in I both cases, x0 does not solve problem PI. The above theorem leads to a conceptually simple method for solving the nonconvex inequality constrained optimization problem PO. Consider the following scalar function of a scalar parameter O0 E R+ \ (0) (Note that Oi, i = 1,2, - . - .m are fixed parameters of the problem PO):

Under appropriate assumptions, the function are summarized as follows:

has some nice properties which

DUALITY IN NONCONVEX SYSTEMS Theorem 5.3.2 Assume that X. is compact and if X is unbounded,

Let 90 > 0. Assume that Pe admits a solution X* with fo(x*) > 0 and that for any > fo(x*), there exists X E X such that fi(x) < Oi(i = 1 , - . - , m ) , fo(x*) < fo(x) < 90. Then (i) 0 < 90 < fo(x*) =+ $(go) > 1; (ii) 90 > fo(x*) =+ W O )< 1; (iii) 90 = fo(x*) if and only if $(go) = 1; (iv) $(g0) is a monotone non-increasing function of 90.

Proof: (i) Assume that 0 < O0 < fo(x*). It is clear that @(OO)= inf max = mini

inf

max

x t x o osiirn

{ } -

'9,

inf max xEx\xo

fi ( {

T

}

}

4

a

Thus inf max xtxoo5i5m{T}'

fi (4

'"'

o

Now we show that

i maxm

-- - {O0 fo(x*) f d x ) gl

fm(x)} 9,

> l.

(5.3.3)

fi ( 4 inf max {-}>l XEX\XO OSiSm Oi

Since lim,Ex,IIzII++m fo(x) = foo, we see that there exists N > 0 such that fo(x) 2 290,

V s E A = {X E X : llxll

>N).

So we need only t o prove that inf m,ax {-)f i ( 4 zEBO 0 satisfying

there exists 6 > 0 such that

DUALITY IN NONCONVEX SYSTEMS

162 As a result,

fi (X) ) > l inf m,ax { Bi

xECO 0 and X 0, then X E Xo.

E

X. Then @(X,60)

Proof: Let X E X, @(X, 90) 2 0. In fact, if for some 1 5 i 5 m,

If

X

@(X, go) will

is feasible, then from Property (B),

X

often be used in the

> 0. Furthermore if @(X,90) =

is infeasible, then from Property (A),

DUALITY IN NONCONVEX SYSTEMS Assume that @(X,80) = 0. If is a contradiction.

X

167

is infeasible, then from (5.4.l), @(X,00) > 0, which m

In the following, a successive solution scheme via a parametric monotone composition formulation is developed for finding the global minimum of optimization problems that could be nonconvex. Specifically, a two-level iterative scheme will be proposed. In the lower level of each iteration, an auxiliary optimization problem with a fixed parameter is solved, while in the upper level the parameter is adjusted such that the least root of the optimal function value of the parametric optimization problem is found.

Definition 5.4.2 Let 4 be a function that satisfies Properties (A), (B) and (C) and 80 > 0. Define cp(Oo) to be the global optimal value of Pd(O0), i.e.,

The following result provides a necessary and sufficient optimality condition for P.

Theorem 5.4.2 Let X* solve the problem P and B; = fo(x*). Then x0 solves P if and only if x0 solves the problem P6(e6). Proof: From Assumption 5.3.l(ii), 86

From Lemma 5.4.1, @(X,86) P4(O;) is 0.

> 0, Vx

> 0. It is clear that

X*

E X. and

E X. Thus the optimal value of problem

If x0 solves P, then from Property (B), @(xO, 06) = 0, thus x0 solves Pb(B;). If x0 does not solve P, then there are two cases: Case 1. x0 is infeasible, then from Property (A) for some 1 5 i 5 m,

Case 2. x0 is feasible, then fo(xO)> fo(x*) = 06. Thus

Then x0 does not solve Pd(06).

Lemma 5.4.3 Assume that X. is compact and if X is unbounded,

DUALITY IN NONCONVEX SYSTEMS

168

Let

X*

solve the problem P. If 0 < 80 < fo(x*), then (p(OO)> 0.

Zo = X \ Xo. Assume that 0 < B. < fo(x*). It is clear that

Proof: Let

V'(e0) =

,':L +(F(%00))

= min{ inf +(F(x, Oo)), inf +(F(x, OO))). x€Xo

xEX0

From Property (B), it is easy to see that inf +(F(x, 00))= inf max{O,

X€ X0

x€Xo

f o (X) - l)>--f o b * ) $0

60

1 > 0.

(5.4.2)

Now we need only to prove that inf +(F(x, 80)) > 0. xExo As l i m x E x , I I x I IfO(x) ~ + m = +M, we deduce that there exists N

> 0 such that

fo(x) 2 2e01 vx E A = {X E X : llxll > N I . It follows from Property (A) that

Consequently, we need only to show that inf -+(F(x, 60)) > 0,

XEXOr) A

where A = {X E X : l l ~ l l5 N). Since xo is compact and fo is continuous and 0 < Oo < fo(x*), we see that for some E > 0 satisfying

Thus, inf +(F(x,00))

X€B

2 inf {-)f o ( 4 > 0. XEB

eO

So we need only to prove that inf +(F(x, 00)) > 01

xEC

where C =

{X

EX O ~ A d(x,xo) :

> 6 ) = {X E xo : d(x,xo) > 6,11x/15 N).

DUALITY IN NONCONVEX SYSTEMS

169

Note that C is compact and the function m a x { v - 1, fl(x), - - , fm(x)) is continuous. Hence, the infimum of infZEcm a x { y tainable. Moreover,

-

1,f l (X). ,- -

., fm(X)} is at-

So we have inf m(F(x,00))

xEC

fo ( 4 - 1,fl(x), - - - ,&(X)} > xinf max{EC 00 fo(4 - 1,fl(x), 80

= minmax{--xEC

(5.4.4)

- , fm(x)) > 0.

From (5.4.3) and (5.4.4), the proof is complete.

I

Theorem 5.4.4 Assume that X. is compact and if X is unbounded,

Let

X*

solve the problem P. The following hold:

(i) If 0 < 80 < fo(x*),then ~ ( 8>~0.) (ii) If 80 2 fo(x*), then ~ ( 8=~0.) (iii) ~ ( 8 is~a )non-increasing and nonnegative function of go. (iv) 80 = fo(x*) if and only if 80 is the solution of the following least root problem

Proof: (i) Follows from Lemma 5.4.1.

(ii) Let

If

X

2 fo(x*). If X is feasible, then from Property (B),

is infeasible, then there exists 1 5 i 5 m such that fi(X) > 0, and so from

Thus, @(X, 80) 2 0. Note that

Then cp(Bo) = 0.

170

DUALITY IN NONCONVEX SYSTEMS

(iii) The non-increasing property of (p(00) is assured by Property (C). Results (i) and (ii) together show that (p(OO) >_ 0, 'dBo > 0. (iv) The necessity part follows from Theorem 5.4.1 and the sufficiency part follows I from (i) and (ii).

Remark 5.4.1 Although it is intuitively straightforward to find the least root of a nonincreasing function in problem (5.4.5), finding the global optimum of the auxiliary problem P4(Oo) is, in general, not an easy task, if the objective function of P4(Bo) is not convex. The proposed method can be considered as a two-level scheme. The lower level is to solve, for a given parameter, an optimization problem with simple constraints or without constraint. The upper level is to check if the parameter 80 is the least root of the equation 4 8 0 ) = 0. Corollary 5.4.5 Assume X. is compact and if X is unbounded, lim

llxll--oo,xEX

fo(x) = m.

Let X* solve the problem P. Then 00 = fo (X*)if and only if O0 is the least root of the equation cpoo(e0) = 0, where ip,(B0)

is the optimal value of the auxiliary optimization problem Pr(Oo): f i (X) inf max{O, - 11 f1(x), ' . . ,fm(x)). 00

xEX

Proof: The result follows from Theorem 5.4.4 (iv) by letting 4(y) = &(y).

I

Corollary 5.4.6 Assume X. is compact and if X is unbounded, lim

fo(x) = m.

Ilxll-t~,xEX

Let X* solve the problem P. Let 0 the least root of the equation

< p < oo. Then 00 = fo(x*) if and only if 00 is Pp(eo) = 0,

(5.4.6)

where F,(Bo) is the optimal value of the subproblem P;(BO):

Proof: From Theorem 5.4.4, 00 = fo(x*) if and only if O0 is the least root of the equation vp(00) = 0,

DUALITY IN NONCONVEX SYSTEMS where pp(BO)is the optimal value of the auxiliary problem P$(&):

This is equivalent to saying that $0 is the least root of the equation

where qP(Bo) is the optimal value of the subproblem P:(Bo). above equation is equivalent to (5.4.6).

It is clear that the

Remark 5.4.2 (i) It is much easier t o solve

U

than P$(&) since there is no root in the objective function of (ii) Assume that X* solves P. It is clear that if for B. > f ~ ( x * 0) ~< p l , p 2 < m, the solution xp, of (Bo) and the solution xp, of PT (Bo) are feasible for P, then Pp, (60) = Pp2 (80)- Thus v,, (00) = pp2 (80)

pth

P?

FIGURE 5.4.1 Monotone composition function for Example 5.4.2 Example 5.4.2 (Example 5.1.1 revisited) minfo(x) = 1 - 5 1 x 2 , subject to f l ( x ) = x l 4x2 - l

+

X1

LO,z2 2 0 .

5 0,

DUALITY IN NONCONVEX SYSTEMS

172

From Theorem 5.4.4, this problem is equivalent to the least root problem

where (pp(BO)= (pp(&)) and pp(BO)is the (global) optimal value of the following auxiliary problem P; (Bo): min

x1,x2>0

[max{O,

+

1 - XlX2 - 1)lp [max{O, X I 60

+ 4x2 - 1)lp.

Problem P:(Bo) is solved using MATLAB [ZTl, G8]. In this simple case, the whole function (pp(Bo)is plotted in a brute force manner. See Figure 5.4.1. The least root of 4,(8) = 0, 00, is found to be 15/16. The optimal solution to P4(15/16), X = (0.5,0.125), is the optimal solution to the original problem. It is verified that all cp,(Bo)'s are equal for all p. In general, the least root of 4,(8) = 0 can be found by applying the bisection method or the Newton method if the gradient is available. The direct application of the primal-dual method [Ll] would fail in the original setting of this example problem as its perturbation function is nonconvex [L6]. This example problem was solved in [L61 by a pth power Lagrangian method. In the pth power Lagrangian method of [L6], the value of p needs to be chosen sufficiently large in order to convexify the perturbation function. Although a theoretical lower bound can be derived for p [L9], [GY8], how large is large enough for p could be a thorny issue in computational implementation. This nonconvex problem is also solved in Section 5.2 using a nonlinear Lagrangian dual formulation. When adopting the nonlinear Lagrangian dual formulation in Section 5.2, the resulting auxiliary problem is of a minimax type and is generally hard to solve. Note that the successive global optimization method derived in this section works for any p 2 1 when the parametric monotone composition formulation 4, or 4; is used. In general, p is chosen in such a way that the subproblem P4(Bo) can be easily solved or its global minimum can be found. The resulting auxiliary problem P4(B0) is also a differentiable optimization problem when all fj7s are differentiable. It is worth noting that the parametric approach developed here and in [GYl] involves a single parameter, although the subproblem in [GYl], like the one in [GY2], is of the minimax type. Thus, the successive global optimization method via the parametric monotone composition formulation developed in this section has some obvious advantages over the global optimization approach proposed in Section 5.2. To conclude this section, we show that the proposed parametric monotone composition approach is also applicable to searching for strict local minima. This concern is of some practical significance, since in many situations, only a local optimality of P4(8;) can be guaranteed. Theorem 5.4.7 Consider the problem P. If x0 is a strict local minimum of P, 8;) = 0, where then x0 is a strict local minimum of the problem P m (B;) with Q?(xO, B,. = Jo(xO). Proof: Assume that x0 is a strict local minimum of P and 0; = Jo(xO). It is clear that

DUALITY IN NONCONVEX SYSTEMS

173

There is a neighborhood Nl ( x O of ) x0 such that for any X E X. n Nl(xO)and X # xO,fo(x) > fo(xO). We will show that for any X E X n Nl(xO)and X # xO; @(X, > 0.

e;)

In fact, if a point X satisfying X E X n Nl ( X ' ) and X # x0 is infeasible for P, then from Property (A) the following holds for some 1 5 i 5 m

If a point X satisfying X E X n N l ( x o )and X # x0 is feasible for P, i.e., X. nNI ( x Oand ) X # xO,then fo ( X ) > fo ( x O= ) 0;. Thus from Property (A)

X

E

Thus x0 is a strict local minimum of the problem P4(O;).

Theorem 5.4.8 Consider the problem P. Let 86 > 0. If x0 is a strict local minimum 8;) = 0, then x0 is a strict local minimum of P. of the problem Pm(@;)and @(xO,

Proof: Since @(xO, 0;) = 0, from Lemma 5.4.1,x0 is feasible. It is clear that

Assume that there is a neighborhood N2(x0)of x0 such that for Thus 0; 2 fo(xO). any X E X n N2(x0)and X # xO,

Note that @ ( X , 0;)

f o ( x )- l ) , V X / X & ~ N ~ ( X ~ ) . = max{O, 0;

By the strict inequality (5.4.8),

Thus

f o ( x )> e;

2 f o ( x O ) ,V X E x ~ ~ N ~ ( x ~ ) , x # x ~ .

Then x0 is a strict local minimum of the problem P.

m

The condition @(xO, 06) = 0 holds if x0 is feasible and 0; = fo(xO).The following result shows that a local minimum of P is actually a global minimum of P4(8,*).

174

DUALITY IN NONCONVEX SYSTEMS

Theorem 5.4.9 Consider the problem P. If x0 is a local minimum of P and 86 = fo (xO),then x0 is a global minimum of the problem Pm(B;) with @(xO,0;) = 0. Proof: Assume that x0 is a local minimum of P and 06

= fo(xO). I t

is clear that

Then from Lemma 5.4.1, x0 is a global minimum of the problem P4(Oz) with q x O ,e;) = 0. The following example shows that a local minimum for P4(0;) may not be a local minimum for P if the local minimum of P4(06) is not a strict local minimum.

Example 5.4.3 Consider the optimization problem: inf subject t o

fo(4 X E X,


0,i = 1;--,m).

The set & will be the feasible set for the dual optimization problem to be constructed in Definition 5.6.3. Definition 5.6.1 The nonlinear Lagrangian function of problem P is defined as N(u, e) = max f i b > OsiSrn { T

}

where e E £ , U E X Note that the function N (U, e) can be interpreted as a weighted Tchebyshev metric of the vector (fo(u), f i (U),- ,f r n ( ~ ) ) ~ . Definition 5.6.2 The nonlinear Lagrangian dual function for problem P is defined as follows 4(e) = inf N ( u , e ) , uE X

where e E E. T h e o r e m 5.6.1 The dual function $(e) is nonincreasing, i.e., for el,e2 E E , e1 e2 E R?+', $(el) 5 +(e2). Proof: Follows directly from the definition of 4.

B

DUALITY IN NONCONVEX SYSTEMS It is clear that computing the nonlinear dual function $(e) is equivalent to solving a minimax discrete optimization problem or a convex composite discrete optimization problem. This, in general, is not an easy task. Thus the difficulty in the following dual optimization problem is t o evaluate the dual function $(e). However, the constraint has a simple structure. Definition 5.6.3 The dual optimization problem D of problem P is defined as sup $(e) (Problem D) subject to e E E. R e m a r k 5.6.1 Observe that while the primal problem P is a discrete optimization problem, the dual optimization problem D is continuous in nature. This peculiar feature is not without precedent, one other example is found in the max flow min cut duality of Section 2.4. T h e o r e m 5.6.2 (Weak duality) Assume that fo(x) and e E E. Then f o b ) $(e).

> 0, Vx E Xo.

Let

X

>

Proof: It follows from the assumption and

Thus fo(x)

X

E X.

(5.6.1)

E X. that

2-N ( x , e ) > inf N(u,e) uEX

= $(e).

Thus, (5.6.1) holds.

I

Note that an equality constraint can be trivially converted into two inequality constraints. Thus our results are applicable t o optimization problems with both inequality and equality constraints, see Example 5.6.1. We now establish a property of zero duality gap for the present discrete optimization problem and their nonlinear Lagrangian dual problem. We consider both the cases where the discrete set X is a set of finite number of elements (Theorem 5.6.3), or a set of infinite number of elements (Theorem 5.6.4). Theorem 5.6.3 (Zero duality gap) Assume that X is a discrete set with a finite number of elements and X. # 0. If fo (X) 0, Vx E X. , then

>

min fo (X) = sup $(e) .

X € XO

eEE

Note that here 'min' is used since X is a discrete and finite set of elements. Proof: If the equality does not hold, then by weak duality, there exists that min fo(x) - E 2 q5(e), Ve E E. x€Xo

E

> 0 such (5.6.3)

DUALITY IN NONCONVEX SYSTEMS

180 For

€14 > 0, there exists X

E X. such that

fo(3

E

2& m fob) 2fo(3 - 4'

It is clear from (5.6.2) and (5.6.3) that

fo(x) - c

2 rnin f o ( x ) - E X € X0

> 4(e),Ve E E.

Then, by the assumption and (5.6.4),

2 min

{

E

f o ( x ) - -, rnin 4 xcx\xo

m.ax

O 0. Since X \ X. has only a finite number of elements, there exists ri > 0 such that f i ( x ) > ri, for any X E X \ X. and fi (X)being positive. In this case, let ei be such that

Then fi

(4>Y,

ei

Vx E X

\ X.

and f i ( x ) being positive.

Case 11. For all X E X \ Xo, f i ( x ) 5 0. Let that fk(x) > 0, i.e., Case I holds. Thus

rnin

max

xEX\Xo O 0 and 2 E X. such that

and e) >

inf

max

- xex\x0O < Z < r n

{fi(x) T ,,E}

£.

(5.6.11)

It follows from the continuity of fo and the boundedness of X. that there exists S > 0 such that if X E X \ X. and d(x, X. ) 5 S, then E

--

2

E

< fo(x) - fo (x2) < , for some x2 E Xo. 2

Note that fo(x2) 2 infXExofo(x) 2 fo(x) - €14. Then

From (5.6.9)-(5.6.11), we have

182

DUALITY IN NONCONVEX SYSTEMS

By the assumption, there exists A > 0 such that

It is clear that there exists p > 0 such that for any X E X \ Xo, llxll 5 A satisfying d(x, Xo) > S, fi(x) > p, for some i. Define

Let

It is clear that i3 = (1, el, - - - ,Grn)E E. Moreover,

fi (W>- -E 2 max o 0 , (i E I)

Thus the desired result follows. We now establish the weak duality result and the zero duality gap result for the optimization problem P and its modified Lagrangian dual problem D.

Lemma 5.7.2 (Weak duality) MP 2 MD. Proof: Let X E X. and t E R " ~ ,to = 1. It follows from Lemma 5.7.1 that p(F(x,d)) = fo(x). Therefore fo(x) 2 infxlExp(F(xf,t)) = q ( t ) . Thus inf fo(x) = MP.

xEX0

Theorem 5.7.3 (Zero duality gap) Assume that X. is a compact set. Furthermore if X is unbounded, then we assume also that the function fo is such that:

DUALITY IN NONCONVEX SYSTEMS

187

Then M P = M D .

> MD.

Proof: It follows from Lemma 5.7.2 that M P there exist E > 0 and % E X. such that

MP

-

3 ~ > q ( t ) for all t E 7

and

foF) L

(5.7.6)

IMP > fo(%) - 26.

Assume that is sufficiently small so that fo(%) - 26 and (5.7.7) that

fo(%) - 36

Assume M P > MD. Then

> 0. It follows from (5.7.6)

> M P - 3 ~ > q ( t ) for all t E 7.

(5.7.8)

Applying Lemma 5.7.1 and (5.7.7) we deduce that

q ( t ) = min{ inf p ( F ( x , t ) ) , inf xE X0

x€X\Xo

= min{Mp,

inf

p ( F ( x ,t ) ) )

P ( F ( x ,t ) ) )

x€X\Xo

2 min{ f o ( k ) - 2 ~ , inf

x€X\Xo

p ( F ( x ,t ) ) )

where X \ X. is the closure of the set X \ Xo. Combining this inequality with (5.7.8), we obtain q(t) inf p ( F ( x , t ) ) for all t E 7. (5.7.9)

>

x€X\Xo

Since fo is continuous and X. is compact it follows that we can find 6 > 0 such that for each X E X \ X. with the property p(x, Xo) = min{llx - x011 : x0 E X o ) 6 , the inequality fo(xl) < fo(x) E




It follows from (5.7.7) that Ml ( t ) fo(%) - 3 ~ SO, . by applying (5.7.9) we have for all t E 7:

DUALITY IN NONCONVEX SYSTEMS

188

Combining this inequality with (5.7.8), we have, q(t) 2 Mz(t)=

x';"J.p(F(x, t ) )for all t E 7.

It follows from Property (A) for p that there exist numbers ai > 0 , (i = 0 , l - . - ,m) with a0 = 1 such that for t E 7, we have:

Since lim X E X , f 0 ( x )= $00, it follows that there exists a number A X --+30

t ) )=

q ( t ) 2 ::$p(F(x,

inf

x € X 6 , llxll < A

> 0 such that:

p ( F ( x ,t ) )for all t E 7.

Let g ( x ) = r n a x l ~ ~f i~( x, ) . It follows directly from the definition of the set X. that X. = {X E X : g ( x ) 5 0 ) . Since the set { X E X' : llxll 5 A ) is compact and the function g is continuous we conclude that:

Thus for any

Let

X

E X 6 , llxll

< A there exist some i such that

X

Bi where

E

.& = 1 and for i E I:

where

Recall that

(to, fl

E

have been chosen sufficiently small so that f ( g ) - 2 ~> 0. So

- ,L)E R " ~ . Applying (5.7.12) and (5.7.1) we can find a vector

X', llull

< A such that -

q(t)

inf

= J C E X ~Ilxll_ 0 'v' (x,y)E lRn (ii) For all

X

X

R" and O(x,x) = 0 'dx E R";

E R", O(x, y ) is convex in the second argument;

(iii) 0 = V2Q(x,x) V x E IRn, where V2Q(x,x) is the gradient of O(x,y) with respect to the second argument, and evaluated at y = X. Consider the extended variational inequality problem EVI (6.1.2) together with another variant of the gap function defined as follows. (Note that the inclusion of

DUALITY IN VARIATIONAL INEQUALITIES

209

the term was also discussed in [ZMl] in the context of the less general variational inequality problem V17 albeit with some subtle difference.)

We shall establish that this is also a gap function in the following Theorem.

Theorem 6.4.1 Assume that f (X) is differentiable. Then y is a gap function of the problem EVI, i.e., y (X) satisfies the Definition 6.2.1.

Then,

gn[-@(x,y)] = - y€Rn min @(X,Y). R n , then minyERn@(x,y) < 0 Vx E

(6.4.2)

y(x) = Since @(x,x) = 0 Vx E y(x) 2 0 V x E E*. Next, suppose that have

X

Rn, and hence

solves the problem EVI. Then by Assumption 6.4.1 (i), we

Conversely, suppose y(x) = 0, i.e., maxyERn[-@(X,y)] = 0, which implies that -@(x,y) 5 0 Vy E IRn, or @(x,y)2 0 Vy E IR". Observe that @(x,x)= 0, which implies that X is a solution of the following optimization problem: min @(x,y).

yERn

Since by Assumption 6.4.1 (ii) and the assumption that f is convex, it is clear that @(X,y ) is convex in y. Consequently, the solution X of the convex optimization problem also solves the following variational inequality: Find

X

E

Etn such that

VQ2(x,x ) (y~- X)

2 0 Vy E R n ,

where Va2(x,X) is the gradient of @(X,y) with respect to the second argument, and evaluated at y = X. Hence we have

~ - X) VY E R n . Thus, Since f is proper convex, f (y) - f (X) 2 V f ( x ) (y ~ ( x ) ~-( X) y

+ f (y) - f (X) + v&~(x,x ) ~ ( Y- X) 2 0 Vy E E n .

DUALITY IN VARIATIONAL INEQUALITIES

210

Finally, since 0 = V2fI(x,X) by Assumption 6.4.l(iii), we have: '(x)~(Y - X) + f (Y) - f (X) 2 O Y'

E

Enl

i.e., X solves the problem EVI.

The gap function as defined in (6.4.1) has some nice smoothness properties similar to the smooth optimization problem proposed in [F4]. Furthermore, it represents an explicit primal Fenchel optimization problem, and the gap function of the corresponding dual variational inequality problem represents an explicit dual Fenchel optimization problem, where the primal and dual Fenchel optimization problems of Theorem 1.3.9 are restated as follows. (Primal Fenchel optimization problem P,) where a : IRn

+ IR

is a convex function, and ,B : IRn

(Dual Fenchel optimization problem Pd) where a* : IRn respectively.

-+ IR

and

P*

:

IRn

+

min [a(y) - P ( y ) 1,

yERn + IR

is a concave function.

max [ P* (v) - a*(v) 1,

vERn

IR are the Fenchel transforms of cx and ,B

We specialize the gap function as defined in (6.4.1) t o the following form:

where Q E IRn X IRn is positive definite, and hence the quadratic term in (6.4.3) satisfies Assumption 6.4.1. This gap function is identified with the problem EVI as defined in (6.1.2). Similarly, the following can be shown to be a gap function for the dual extended variational inequality defined in (6.1.4):

Let the primal gap function y and the dual gap function $ be identified, respectively, wit h the following pair of optimization problems: (ProblemP1) and

l

min { F ( x ) ~ ( ~ - X ) + ~ ( ~ ) - ~ ( X ) + ~ ( Y - - X ) ~ Q ( Y -}- X (6-4.5) )

yERn

DUALITY IN VARIATIONAL INEQUALITIES Theorem 6.4.2 If X solves EVI and optimization problem of P2.

Proof: Let the functions a and

U

211

solves DEVI, then P1 is the dual Fenchel

P be defined by:

Clearly a is convex and P is concave (in y). Thus the cost function of P1 is given by a(y)- P(y). We only need to show that m a x [ p (v)- a*(v)]is equivalent t o Pz.

Using the fact that the Fenchel dual of axTQx is auTQ-'U, we let get:

U

= y - X to

1 a*(v)= max {(V- F ( X ) ~ U- -uTQu1 + (v - F ( x ) ) ~+ xF ( X ) ~ X 2

uERn

Remembering that if X solves EVI and U solves DEW, then -Ft(u), and xTu = f (X) f *(U), consequently,

+

Next,

/?*(-v)= min {-vTy - f (X) yERn

=-

max IvTy - f (Y)

yERn

f

f (Y)}

1 - f (4

= -f*(v)- f (X).

Adding -a*(-v) and ,P(-v) to get

Finally, since we are maximizing over the whole of IRn,

U

= -F(x), X =

212

DUALITY IN VARIATIONAL INEQUALITIES

This completes the proof.

B

Under further assumptions, it is also possible to establish a convex lower bound to the gap function. The following result is an extension of Theorem 3.3 of [ZMl] to the problem EVI.

Definition 6.4.1 Let f : Rn+ E be a scalar-valued lower-semi-continuous proper convex function. F is said to be strongly pseudo-monotone with respect to f and with modulus p, if there exists a positive constant p such that

Theorem 6.4.3 In addition to Assumption 6.4.1, we further assume that: (i)R is continuously differentiable with respect t o the second argument; (ii)V2R(x7X) = 0 Vx; (iii) the gradient of R with respect to the second argument, V2R7 is Lipschitz continuous in the second argument with constant La; (iv)F is strongly pseudo-monotone with respect to f and with modulus p. Let x0 be a solution to the problem EVI. If (i)-(iv) hold, then there exists a positive constant a such that y(x) 2 allx - x0112, Vx E Rn.

Proof. Since x0 solves the problem EVI, we have

By Assumption (iv), this implies that

Let

xt = x + t ( x O - X ) ,

t E (0,l).

Then, by the convexity of f , we have

By the convexity of R with respect to the second argument (Assumption 6.4.l(ii)), we have,

n(x,X')

- R(X, X)

v 2 a ( x ,x ~(xt)- X) ~ = (V2n(x,xt) - V2R(x,x ) )(xt ~ - X) 5 L0 llxt - x112. (Assumption (iii))

(Assumption (ii)) (6.4.13)

DUALITY IN VARIATIONAL INEQUALITIES By definition of the gap function in (6.4.l),we have, Vt E (0, l),

> + f (X) - f (xt) a ( x , x t ) > F ( x ) ~(X - x t ) + t[f (X) - f (X')] - (o(x, x t ) - n(x, X))

~ ( x ) F ( x ) ~ ( x- x t )

-

(from Assumption (ii) and (6.4.12))

2 ~ [ F ( x ) ~ -( xX')

+ f (X) - f (X')]

- Lnllxt - x1J2 (from (6.4.11) and (6.4.13))

(from (6.4.9) and Assumption (iv)) ~tpllx-~'11~-t~~~llx-x'11~ =

(tp - t2Ln)llx - ~ ' ( 1 ~ .

Since the unconstrained maximum of (tp - t2Ln) occurs at p/(2Ln), we choose

to obtain r(x) where

2 allx - x01I2, p-Ln,

O

1 is a vector-valued function. Thus a vector optimization, or multicriteria optimization, or multiobjective optimization problem, is concerned with the optimization of a vector-valued function, possibly subject to one or more constraints. The notion of optimization essentially deals with the comparison of the magnitude of similar objects. Most of us should feel quite comfortable when there is only one thing to compare. For example, is this apple better (hence more optimal) than that one? In the real life, however, one is often faced with the problem of comparing apples with oranges, or objects that are measured in different ways or scales, and most of us would feel quite uneasy about that. As it turns out, the notion of optimality can be generalized to vector-valued functions in a meaningful and rigorous way, provided that we adopt a systematic way of comparing or ordering vectors. The subject of multicriteria optimization is not new of course and can be dated back to 1896 when Pareto introduced the notion of a Pareto-optimal solution. Unlike scalar optimization where the optimal solution is usually a point, the solution to a multicriteria optimization problem is an infinite set, and hence it is usually much harder to solve a multicriteria optimization problem than a scalar one. R o m a theoretical point of view, it turns out that many of the theoretical properties of scalar optimization can be generalized t o the case of multicriteria optimization, albeit in a non-trivial way. In this regard, duality is probably the best example of these generalizable properties. Chapter Eight and Nine are thus devoted t o the discussion of these duality results. Some prerequisite material required for Chapter Eight and Nine will be discussed in some detail in this chapter first. Section 7.1 is a generalization of Section 1.2, where the notion of convexity is extended to the case of vector and set-valued

216

ELEMENTS O F MULTICRITERIA OPTIMIZATION

functions. Section 7.2 is a generalization of Section 1.3 where the concept of conjugate duality is extended to the case of vector and functions. In Section 7.3, we lay the foundation of multicriteria optimization, and present some fundamental results pertaining to multicriteria optimization in Section 7.4. Related references for this chapter are found in [CPl],[CGY4],[G7],[G91,[GWl], [GY4],[GY5],[GY6],[Jl],[L2], [SNTl],[Y4].

7.1 Elements of Multicriteria Convex Analysis Since we will be dealing with vector-valued cost functions, it is necessary to compare or order vectors in a meaningful way. The following ordering relationships for vectors will be used for the next three chapters. These are a formalization of the ordering relationships

Definition 7.1.1 (Ordering of vectors) Let C be a closed and convex cone in R P . Given E, q E RP, the vector ordering relationships c, ilarly.

V -E

Z ~ C \ { ~Lint ) , c zint c, are defined sim-

Some basic relations for working with the above ordering relationships are given as follows. The proofs are straightforward.

Lemma 7.1.1 Let C be a closed and convex cone in R P . Then

ELEMENTS OF MULTICRITERIA OPTIMIZATION

217

Lemma 7.1.2 Let C be a closed and convex cone in W. We have (i) If y"c\{o) z,

i = 1, 2, - - and limi,,

y"

y, then y

*

C g i n t C E;

Z.

(ii) The following hold:

E z i n t C 1) l i n t C C

Definition 7.1.2 (Ordering of sets) Let C be a closed and convex cone in RP with intC # 0. Given A, B RP,the set ordering relationships ,I, 0 such that for each i ,

for some j such that

yj

> y; whenever y E y and yi < y:

Definition 7.1.5 Let C be a closed and convex cone in RPand X C RP. (i) X is said to be C-bounded if there exists b E -C such that X C_ b C. (ii) The set X is said to be strongly C-bounded if there exists b E -C such that X C b + int C. (iii) The set X is said to be strongly C-closed if X + cl(C) is closed.

+

Remark 7.1.1 Note that the set of all properly minimal points is a subset of minc\{olY, and minc\(o}Y C_ mini,, c ( y ) . Note also that if C has a non-empty interior, then C-boundedness implies strong C-boundedness . This is certainly the case if C =R:. Theorem 7.1.3 Let C = R P . If y is R:-bounded and closed (resp., RC-bounded and closed) ; then minc\{o}Y # 0 (resp., maxc\{oj Y # 0). Proof: Refer to [SNTl].

H

Lemma 7.1.4 Let y l , y2 C_ RP.

+

+

Proof: Let z E minc\{o)(Y1 Y2), then z = y1 y2 for some y1 E Y1 and y 2 . Suppose y1 minc\{o)(yl), then there exists 9 E y1 such that y 5 Y'.

Y2 E

ELEMENTS O F MULTICRITERIA OPTIMIZATION

219

The vector y + y2 E y' + y2 is such that y + y2 < y' + y2, contradicting that y1 y2 E minc\{o)(Y1 y2). Hence y1 E minc\{o)(yl). Similarly we can prove that y2 E minc\{Ol( y 2 ), and therefore y' y2 E minc\{o) (Y') minc\{o) (y2). B

+

+

+

+

Definition 7.1.6 (Convexity of vector-valued functions) Let X be a convex subset in Rn and C a closed and convex cone in IRp. A vector-valued function f : X -+ RP is said to be C-convex (quite often we simply call it convex, without the prefix C) on X if for x',x2 E X, t E (0, l ) , f (tx'

+ (1 - t)x2) C 0 , vx E X.

ELEMENTS O F MULTICRITERIA OPTIMIZATION

223

(iv) The set of all Type I weak subgradient of f at x0 is denoted by awf(xo),the set of all Type I subgradient of f at xo is denoted by af(xo) and the set of all Type I strong subgradient of f at xo is denoted by asf(xo). Furthermore, if the specification of Type I or Type I1 is absent, by default it is regarded as Type I.

Lemma 7.2.2 U E af (X) if and only if Ux - f (X)E f * (U),

U E &f(x) if and only if Ux - f(x) E fi(U). Proof: We shall do it only for the case of type I subgradient, the proof for type I weak-subgradient follows in a similar manner. By definition, U is a type I subgradient of f a t X if and only if f(y) - f(x) - U(Y- X) &\{cl) 0 VY E X, if and only if Ux - f(x) f c\(o) Uy - f(y) Vy E X, if and only if Ux - f (X)E f * (U).

Definition 7.2.3 (Type I1 Fenchel transform of a vector-valued function, vector version) Let f : Rn + RPbe a vector-valued function. Denote the vector 1= (1,1,-E RP. (i) The Type 11Fenchel Transfom of f is defined by a set-valued function: f; Bp--+ 2Rp as follows:

:

(ii) The weak Type 11Fenchel Transform of f is defined by a set-valued function: f;, : RP + 2Rp as follows:

(iii) The Type I1 Fenchel Transform f:* of f: is called the Type 11bi-Fenchel Transform or Type 11 biconjugate of f and the weak Type I1 Fenchel Transform fF1 of f;, is called theType 11weak bi-Fenchel Transform or Type I1 biconjugate of f.

Lemma 7.2.3 (Young's inequality for set-valued Fenchel transform, vector version) f ( x ) + f ~ l ( ~ ) - [ ~ ~ x ] l $ i n t ~ ( O }Q x , y € R n f(x) + f a y ) - [yTx1l~ c \ { o ){ O ) Proof: Follows directly from the definitions.

V ~ , YE Etn-

224

ELEMENTS O F MULTICRITERIA OPTIMIZATION

Definition 7.2.4 (Type I1 subgradient of vector-valued functions, vector version) Let X be a convex subset of IR", f : X -+ RPbe a convex function. and y E IRP be a vector. (i) y is said to be a Type-I1weak subgradient of f at x0 E X if f(x) - f(xO)- [yT(x- xO)]lgintc O7 VX E X (ii) y is said to be a Type-11 subgradient of f at x0 E X if f(x) - f(xO)- [yT(x- xO)]lgC\{O)0,

vx E X.

(iii) y is said to be a Type-11 strong subgradient of f at x0 E X if f(x) - f(xo)- [yT(x- xO)]l>C 0, Qx E X. (iv) The set of all type-I1 weak subgradient of f at x0 is denoted by h1f(xo),the set of all type-I1 subgradient of f at x0 is denoted by alf(xo) and the set of all type-I1 strong subgradient of f at x0 is denoted by aslf(xo). Lemma 7.2.4 y E alf(x) if and only if [yTx]l- f(x) E f;(y). y E aWlf(x)if and only if [yTx]l- f(x) E f:l(y). Proof: We shall do it only for the case of type-I1 subgradient, the proof for type-I1 weak-subgradient follows in a similar manner. By definition, y is a type-I1 subgradient of f at X if and only if

f ( z ) - f ( x ) - [ Y T ( z - ~ ) 1 1 ~ c \ { 0 ) OQ z E X if and only if [yTX]l - f (X) [yTz]1- f (z) Vz E X if and only if

[yTx]l- f(x) E f;(y). m

It is well-known that the Fenchel transform of a scalar-valued function is convex However, a similar result (see [CGYl])is in the usual definition (Theorem 1.M). not so well known in the case of vector-valued functions. Theorem 7.2.5 Let f : IRn -+RPbe convex. The Fenchel transform of f is type I convex. Proof:

+

f*(tul (1- t ) u Z ) = maq\{o) {(tul + (1 - t ) u Z ) x- f(x) I X E Etn}

+

{ ( t ~ ' (1 - t)u2)x- f(x) I X E Etn} = {t(ulx - f(x)) (l - t ) ( u 2 x- f(x)) I X E lRn} t{ulx - f(x) I X E Etn} (l - t){u2x - f(x) I X E R n } t maq\{o) {U'X - f(x) I X E Ktn} - C

+

+

+ (l - t) rnaxc\{~){ u 2 x- f(x) I X E lRn) = tf*(ul) + (l t)f*(uZ) C. -

Thus f* is type I convex.

-

-C

ELEMENTS O F MULTICRITERIA OPTIMIZATION

7.3 Scalarization of Multicriteria Optimization We first formalize the notion of optimality of a multicriteria optimization problem VO. Essentially, we consider two sets: a feasible set X 2 IRn and its image under a vector mapping f , y = f(X) C R P . In this section, let C = R;.

Definition 7.3.1 (Multicriteria Optimization Problem VO) Let f = (fi, f2, the feasible set.

- , fp)T

: IRn + RP be

a vector-valued function, and X

G Etn

be

Ideally, one would like t o find a solution X* to the problem V 0 such that f(x*) fj(x*) whenever fi(x*) < fi(x). f(x*) is called a properly minimal point if X* is a properly minimal solution. The set of all properly minimal points is called the properly minimal frontier.

Remark 7.3.1 Note that there are several other versions of proper minimality, see [SNTl]. The above definition due to Geoffrion (G51 appears t o be most popular.

226

ELEMENTS O F MULTICRITERIA OPTIMIZATION

Definition 7.3.3 In view of Definition 7.3.2 (ii), we define the following weaker version of problem VO. (Problem WVO)

mini,

R:

{f (X) I

X

E X) = minint RP,f (X).

We say that we

solve problem V 0 if we find the set of all minimal solutions argminR:,{,)(f7 X) and the minimal frontier minR?

f (X);

solve problem WVO if we find the set of all weakly minimal solutions argminin, RP (f,X) and the weakly minimal frontier mini,, R; f (X). The solution of multicriteria optimization problems is a set. This is in general a much more difficult problem than the scalar case. Only a few special cases of problem V 0 have been solved fully. Two such cases are found in the linear case [Y4] and the quadratic case [GY4]. Apart from these there are few satisfactory results in finding the complete set of minimal solutions and minimal frontier numerically for general nonlinear problems. For the rest of this section, we shall present some important theoretical results for solving the multicriteria optimization problem. In practice, multicriteria optimization problems are often reduced to scalar optimization problems by composing the vector cost function with a so-called utility or value function. There are two fundamental requirements of all utility functions used for scalarizing multicriteria optimization problems: Requirement 1. They should cover all minimal solutions for any multicriteria optimization problem, i.e., all minimal solutions can be computed by solving some scalarized optimization problem. Requirement 2. Solutions t o the scalarized optimization problem should also be minimal solutions to the multicriteria optimization problem.

Definition 7.3.4 Let R : be a closed and convex cone in RP, and y1 ,y2 E R P . A function $ : RP -+ R is said t o be: (i) monotone if y1 y2 + $(yl) 2 $(y2), (ii) strictly monotone if y1 Zint R: y2 + $(yl) > $(y2) , and (iii) strongly monotone if y1 2 R ~ \ {yo2} @(yl) > $(y2)-

*

The following function proves t o be very useful both in the scalarization of multicriteria optimization problems as well as in the scalarization of vector variational inequalities to be discussed in Chapter Nine. The original version due to Gerstewitz (see [GWl]) is published in German. Its first appearance in English is found in [L2]. This result is further studied and applied in [CGY4] and [GY3].

Definition 7.3.5 Given fixed e E int R: and a E R P , the Gerstewitz's function : IRP -+ IR is defined by:

,J ,

O F MULTICRITERIA OPTIMIZATION

max

l minf(X). The proof for (ii) is similar, using only I strict monotonicity. The converse t o Theorem 7.3.1 is not so obvious, and in fact is "not true" in general. In order for this to be true, the most common trick is to assume that the .: is a convex set in RP. Such a set f(X) is cone-convex, i.e., the set f ( X ) R condition holds when one is dealing with convex multicriteria optimization, i.e., f is convex and X is convex.

+

T h e o r e m 7.3.2 (Necessary condition for minimal solution i n convex problems) Assume that f (X) R : is convex, then (i) There exists some utility function 4 such that a properly minimal solution of V 0 is also a solution of &Opt. (ii) There exists some weak utility function 4 such that a weakly minimal solution of V 0 is also a solution of $-Opt.

+

Proof: Since the set f(X) + R : is convex, there exists a supporting hyperplane at every boundary point. In particular, let y* be a properly minimal point on minR:\{o)(y). Then there exists some X E int R; such that the supporting hyperplane is given by x ~ - y*) ( = ~ 0. By the definition of a supporting hyperplane,

and therefore XTy* is the minimum for the scalar optimization problem based on the (linear) utility function represented by X. The proof for (ii) is similar. I

Remark 7.3.2 Theorem 7.3.2 is applicable only when the set y = f (X) is coneconvex. This is a very strong condition, and is not likely t o hold, when either f is not convex, or X is not convex, or when X is a discrete set whence f(X) is also a discrete set and therefore convexity has no meaning.

ELEMENTS O F MULTICRITERIA OPTIMIZATION

229

In what follows, we discuss the reasons why the weighted Tchebyshev norm is, in some sense, the best utility function, since none of the strong convexity assumptions are required in order to establish the equivalence of the (weak) V 0 problem with the scalar optimization problem. We do need, however, the assumption that Without loss of generality, we may assume that the set Y = f (X) is R:-bounded. y C int R: by an appropriate shift of the origin.

Theorem 7.3.3 Given any set y C intR:, if a minimal point y* of y also minimizes d,,x(y, a) over y for some Minkowski metric utility function dp,x(., a), then it is necessary that p = m . In more formal terms,

Proof: We will prove it in the case of p = 2. This can be easily generalized to any dimension, albeit with added notational complexity. Assume that p < oo. Consider a special set Y c R: and a minimal point y* E minc\{ol(Y). We select the set y such that there exists a neighboring point y0 with the property that y0 = y*

Note that y0

+

&:[]

t Y, where 0

< r, 6 - (Y;-

A;

c-

y; - a2

y*. Then,

and hence dp,x(y*,a ) > minYcydp,x(y,a ) but y* E minc\(o)(y), a contradiction.

D

Theorem 7.3.4 Given any set y C intR?, if a minimum point y* of d , , ~(y, a) over Y for some Minkowski metric utility function with 1 < p < m , X E intR:, a E RP is also a minimal point of Y, then it is necessary that p < oo. In more formal terms,

Proof: Let p = m , so that d,,x(y*, a) = &,(y*) where e = (X;', X;', - ,X;'). Then there exists an X > 0 , a E RP, and a particular set y such that there exists

230

ELEMENTS OF MULTICRITERIA OPTIMIZATION

a non-minimal point (in particular, pick a weakly minimal point of minRP+\{O)(Y)) y0 of y such that

Corollary 7.3.5 Within the class of Minkowski metric utility functions, there is no utility function that satisfies Requirements 1 and 2 simultaneously, i.e., for any multicriteria optimization problem, there is no utility function that can cover all minimal solutions, and such that minimal solution to the scalarized optimization problem is also a minimal solution to the multicriteria optimization problem. Proof Follows from Theorems 7.3.3 and 7.3.4.

I

Corollary 7.3.5 thus asserts that there is no equivalent relationship between the problem of finding a minimal solution to multicriteria optimization problems and the problem of solving scalarized optimization problems, at least not in the class of Minkowski metric utility functions. It is likely that this is also true for all utility functions, although proving it is going to be difficult. Now we consider the problem of finding the weakly minimal points of a set y C R P , which is neither assumed to be cone-convex, nor is necessarily continuous, i.e., y is possibly a finite or discrete set.

Lemma 7.3.6 For any e E int R: , a E RP and b E -]R:,

Proof:

Lemma 7.3.7 Let b E RP such that y - b C int R:. Given any y E Y , and y* E minint RP ( y ) , then y - b 1. Proof: Assume the contrary, say y -b

I R 7 t(y* - b ) and t < 1,

then Y* - Y = [t(y* - b) - (y - b)]

+ [(l - t)(y* - b)].

ELEMENTS OF MULTICRITERIA OPTIMIZATION

231

Now t(y* - b ) - ( y - b) E R: by (7.3.1), and (1 - t)(y* - b) E int R: since t < 1, therefore y* - y E int R; or y* Zint R: y, contradicting that y* is weakly minimal. I

Theorem 7.3.8 (Non-convex scalarization theorem [GWl]) Let y C RP be a R:-bounded subset. Then y* E mini,, R; ( y ) if and only if, for some a E RPand some e E int R:, Cea(~*)= min Cea(Y). Proof: (Sufficiency) Assume that for some a E RPand some e E int RP+,Sea(y*) = min&.(y). If y* # mini,, R; (Y), then there exists an y E y such that y R; y*. By the strict monotonicity of the Gerstewitz's function, we have Sea(y) < &a(y*), which contradicts the assumption that Cea(y*) = min &,(y). Hence Cea(y*) > min(ea(Y)-

sin,

(Necessity) Conversely, assume that y* E mini,,

R~

( y ) . Thus

Note that int R: # 0 and y is strongly R;-bounded. Then there exists b E -int R: such that y C b+int R: or y - b C int R;. Choosing e = y* -b E int R: and a = 0, we have, by definition, &O

( y - b ) = min{t ( y - b E t(y* - b ) - R;}

This is equivalent to

teb (y) 2 &b(y*)

2 1 = (eo(y* - b ) by Lemma 7.3.7.

vy, or Ceb(y*) = min teb(y).

I

Theorem 7.3.8 is a strong result that establishes that finding a weakly minimal solution of V 0 is equivalent t o solving the scalar optimization problem using the special utility function tea, even when the set Y is non-convex or discrete. While we have established that the weighted Tchebyshev norm is in some theoretical sense ideal for solving multicriteria optimization problems, it should however be noted that this method is not without limitation. The first limitation, of course, is that it may yield weakly minimal solutions along with minimal solutions. Secondly, the scalarized optimization problem is inherently non-smooth, and thus sophisticated non-smooth optimization techniques may be required. In practice, one can possibly get around this by solving a sequence of smooth but only approximate sub-problems using a large but finite value of p. Then by increasing p it is sometimes possible to converge to some consistent solution, though for a very large value of p, numerical instability may occur.

232

ELEMENTS OF MULTICRITERIA OPTIMIZATION

7.4 Optimality Conditions for Multicriteria Optimization We now discuss some basic optimality conditions for multicriteria optimization where the feasible set is defined by an explicit set of constraints. Consider the following multicriteria optimization problem. Definition 7.4.1 (Inequality constrained multicriteria optimization problem) Let f : IR" -+R P , g : IRn -+ IRm be differentiable functions. (Problem VO)

minR; \col f (X)

subject to

X

E X = {X E

lRn I g(x)

0, a function : [O,a -+ lRn that is differentiable a t t = 0, and a scalar a > 0 such that

+

Theorem 7.4.1 (Kuhn-Tucker necessary condition for weakly minimal solutions) Let X* be such that the Kuhn-Tucker constraint qualification is satisfied. If X* is a weakly minimal solution to the problem VO, then it is necessary that there exist p E W and X E Rmsuch that

+

(i) p T v f (X*) x ~ v ~ ( x *=)oT, (ii) XT g(x*) = 0, X LR;. 0 (iii) p ~ R T \ { o 03 ) Proof: If X* is a weakly minimal solution that satisfies the Kuhn-Tucker constraint qualification, then there exists a differentiable arc @(t),t E [O, fl such that

for some y such that Vgj(x*)y

< 0,

for all j E J = {i

I gi(x*) = 0).

Then

Then the following system must be empty in order for X* t o be minimal:

for otherwise fi(+(t)) < fi(x*)Vi = 1,2, - ,p, and hence X* cannot be weakly minimal. The conclusion that the conditions (i), (ii), and (iii) hold then follows from the Motzkin's theorem of alternative (Theorem 3.4.11) and the fact that the I corresponding Xi = 0 for an inactive constraint gi(x) < 0.

ELEMENTS O F MULTICRITERIA OPTIMIZATION

233

T h e o r e m 7.4.2 (Kuhn-Tucker sufficient condition for weak VO) If the functions f and g are convex, and if the Kuhn-Tucker conditions (i), (ii) and (iii) of Theorem 7.4.1 hold, then X* is a weakly minimal solution to the problem VO. Proof: Let

X

be a feasible solution. Then

p T f ( ~ ) ~ ~ T f ( ~ ) + ~since T g X>@, ( ~ )

g ( x ) I ~ ; 0,

+ (X - X*))+ XTg(x* + (X - X*)) 2 p T f ( ~ *+ ) XTg(x*) + [/ATvf(x*)+ XTvg(x*)](X

= pTf(x*

- X*)

by the convexity of f and g

> pTf (X*),

by (i) and (ii) of the Kuhn-Tucker conditions.

Since for p \lO) 0, the scalarization pTf (X) constitutes a weak utility function. By Theorem 7.3.l(ii), we conclude that X* is a weakly minimal solution to the B problem VO. Stronger versions of Theorem 7.4.1 and 7.4.2 can also be obtained for a properly minimal solution in the sense of Kuhn-Tucker, which is defined as follows. Definition 7.4.3 (Kuhn-Tucker proper efficiency) A solution X* of the problem V 0 is said t o be properly eficient or properly minimal in the sense of KahnTucker if it is minimal and there exists no y E Rnsuch that

In the event that f and g are convex, it can be shown that (Kuhn-Tucker) proper efficiency implies the (Geoffrion) proper efficiency as defined in Definition 7.3.2 (iii) (see Theorem 3.1.5 of [SNTI]). T h e o r e m 7.4.3 (Kuhn-Tucker necessary condition for properly minimal solution) Let X* be a properly minimal ( in the sense of Kuhn-Tucker) solution to the problem VO, then it is necessary that there exist p E RP and X E IRm such that

+

XTvg(x*) = oT, (i) p T ~ (X*) f (ii) XTg(x*) = 0, 0. (iii) p Zin, pc 0, Proof: Follows very closely to that of Theorem 7.4.1, except that the Kuhn-Tucker theorem of alternative (Theorem 3.4.12) is used instead of the Motzkin theorem of alternative. I T h e o r e m 7.4.4 (Kuhn-Tucker sufficient condition for properly minimal solution) If the functions f and g are convex, and if the Kuhn-Tucker conditions (i), (ii) and (iii) of Theorem 7.4.3 hold, then X* is a weakly minimal solution to the problem VO.

234

ELEMENTS O F MULTICRITERIA OPTIMIZATION

Proof: Similar to the proof for Theorem 7.4.2.

B

Hitherto, the Lagrange multiplier used is a vector-valued one. To proceed, it is useful to look at the case of a matrix-valued Lagrange multiplier, which leads t o the concept of a vector-valued Lagrangian. Definition 7.4.4 (Set of positive matrices) Let

Definition 7.4.5 (Vector-valued Lagrangian) For the problem VO, the vectorvalued Lagrangian is a function L : RnX L -+RPdefined by:

Theorem 7.4.5 (Necessary condition for a properly minimal solution) Assume that f and g are convex functions. If X* is a properly minimal solution for the problem VO, and if the Slater constraint qualification condition is satisfied, then there exists A* E L such that

Proof: Since such that

X*

is properly minimal for problem VO, there exists p* Zint R;

0

(p*)Tf(x*)5 ( p * ) T f ( ~vx ) E X.

It is easy t o verify that (p*)Tf(X) is a convex scalar-valued function. By Theorem 4.3.8,there exists a X* 0 such that

and Given that p*

( x * ) ~ ~ (= x 0. *) RP

0 and X*

A* = [AIc, A;R;"

c

0, we choose a special matrix multiplier 0 and tTp*= 1. Consequently,

A* E L, ( ~ * ) ~=pX*, * A*g(x*)=O.

(7.4.4)yields the second necessary condition. If

ELEMENTS O F MULTICRITERIA OPTIMIZATION

235

then there exists X E IRn such that

By (7.4.3) and (7.4.4), we have

This is a contradiction to (7.4.1). So f(x*) E minRP+\{o){f(x)+A*g(x)

I X E Rn).a

Definition 7.4.6 (Vector-valued Saddle Point) For the problem VO, A pair (X*,A*) E Etn X L is said t o be a saddle point of L if L(x*, A*) E minR:\{o) {L(x, A*)

I X E IRn) n r n a ~ ~ : \ { ~ ~ { L (A) x *I , A E L ) .

L e m m a 7.4.6 (X*,A*) E IRn X L is a saddle point for problem V 0 if and only if (i) L(x*,A*) E minRP+\{O){L(~, A*) I X E W), (ii) g(x*) ~ R Y 7' (iii) A*g(x) = 0. Proof: (Necessity) The proof of (i) follows directly from the definition of a saddle point. For (ii), we note that since L(x*,A*) E maxRr \{ol {L(x*, A) I A E L ) by definition, then

This implies that (A - A*)dx*) ~R:\{o) O VA E or, for some p* >R:

0,

If (ii) does not hold, i.e., if g(x*) SR.; 0, then there exists X* >R.; ( ~ * ) ~ g ( x>*0). For any t > 0,we can select some A E L such that

0 such that

Consequently,

since t can be made arbitrarily large. This contradicts (7.4.7). So (ii) must hold.

236

ELEMENTS O F MULTICRITERIA OPTIMIZATION

By definition of L, Let A = 0 E

L

A * ~ ( x *RT oT.

Therefore A* E C. Furthermore,

where X* is the optimal solution to the scalar linear program (8.1.6), and :X is its basic component. Thus A* is a maximal solution to the dual VLP by (ii), and hence min~;\(o){Cx 1 X E X) E maxq\{o){Ab I A E L)-

DUALITY IN MULTICRITERIA OPTIMIZATION

241

Conversely, we assume that A* is a maximal solution to the dual VLP, then,

vp

0 BA E L such that pTnb > p T ~ * b .

R:

Let X = ATp. Then it follows that

fix>R?

0

Bp Zint RP, 0

S U C ~that

~~b > p T ~ *and b A ~ =Xc T p . (8.1.13)

In other words, the system {(X, p )

oT I A ~ = X cTP and XTb > pTn*b)

>R;Tm

is infeasible. By Gale's Transposition Theorem (Theorem 3.4.9), this is true if and only if the system

is non-empty. Without loss of generality, let = 1, then and hence is feasible for the primal VLP. Furthermore,

X*

satisfies A x

zR;. b

(8.1.14) together with (i) imply that Cx* = A*b. Thus by strong duality (ii), A*b E minR;\{o}{Cx

I X E X),

and hence,

For comparison, we present another variant of the primal and dual VLP. The following duality result is due to Sawaragi et. al. [SNTl], where the ordering relationships for the primal are induced by some closed pointed convex polyhedral cones C, M , Q in RP,IRn, and Rm respectively, while the ordering relationships for the dual are induced by the positive polar cones of C , M, Q respectively. The positive polar cone of a cone M is defined by:

R.7

then MO = M . In the following result, we specialize the result to a If M = case where the ordering cone is the positive orthant, and as such the ordering cone for the dual is also the positive orthant.

DUALITY IN MULTICRITERIA OPTIMIZATION

242

Definition 8.1.2 (Primal and dual vector linear programs, Sawaragi's version) (Primal VLP Problem) minRp+\{O}C s x subject to

0, ASx

X >R;

b,,

where C, E lRpxn, As E lRmXn,b, E lRm, (Dual VLP Problem)

for some p

>int

and

L, = { A s E lRpxm I 3p Zint R; 0 such that T

0.

RP

pTn >R;. oT and

T

As As P R;

= {[As, A'] E lRpx(m+n)1 311 2intR: T

T

and AS As p



where X* is the polar cone of X defined by

X* = {y E R" I yTx 2 0 vx E X}. Note that, unlike the case of Wolfe dual (see Section 4.5),the primal and the dual problems here have the same form of objective functions. This is known as MondWeir type dual problems [WM2]. We examine duality properties for the convex composite problem that include the corresponding results for convex problems and other related generalized convex problems.

Assumption 8.6.1 The Generalized Representation Condition (GRC) holds for P if for primal feasible X and for dual feasible (U, r,X), there exist cxi(x,U) > 0, Pj (X, U) > 0 and ~ ( xU), E X - U such that

Theorem 8.6.1 (Weak Duality) Let X be feasible for P and let ( u , r ,X) be feasible for D, and assume that Assumption 8.6.1 holds. Then

Proof:Since (U, T ,X) is feasible for D,there exist v" Eff,(Fi(u)), wj E Egj (Gj(U)) for i = 1 , 2 , - - - , pj, = 1,2;--,m satisfying Xjgj(Gj(u)) > 0, for j = 1,2,- - - , m , and

Suppose that

Then

X

# U and

DUALITY IN MULTICRITERIA OPTIMIZATION

264

since T ~ / c Y ~U) ( x> , 0. NOW,by the convexity of fi and by Assumption 8.6.1, we have P

C T ~ ( V ~ ) ~ F , ' ( UU)) ~

- ,m),

and cone(X - a) = {Xx

I

X

E

T h e o r e m 8.6.2 (Strong Duality) For the problem P, assume that the generalized Slater constraint qualification holds and that Assumption 8.6.1 holds at each feasible point of P and D. If X is a properly minimal solution for P then there exist r E RP7 rj > 0, X E Rm,X j 0 such that (X,r,X) is a properly maximal solution for D and the objective values of P and D at these points are equal.

>

Proof: It follows from Theorem 2.1 of [JYI] that there exist r E ]Rp,~j W,Xj 2 0 such that

Then (X,r,X) is a feasible solution for D.

> 0, A

E

DUALITY IN MULTICRITERIA OPTIMIZATION

265

From the weak duality Theorem 8.6.1, the point (X,T, X) is a maximal solution for D. We shall prove that (X,T, X) is a properly maximal solution for D by contradiction. Suppose that there exists (X*,?*,X*) feasible for D satisfying, for some 1

fi(Fi(x*))- fi(Fi(x)) > M[fj(Fj(x))- fj(Fj(x*))I,

for any M > 0 and all j satisfying f j (Fj(X)) > f j (Fj(X*)). Let A = {j E I I f3(Fj(x) > f j ( F j ( ~ * ) )where } ~ I = {1,2;.-,p}. Let l3 = I \ ( A u {i}). Choose M > 0 such that M/IdI>7j/7il jEA, where (AI denotes the cardinality of A. Then,

since fk(Fk(x))

-

fk(Fk(x*)) > 0, for k E d . So,

This contradicts the weak duality. Hence, (X,7 , X) is a properly maximal solution for D. I As a special case, the following multicriteria pseudolinear programming problem (PLP) is considered in [CC2]: (Problem PLP) minR;\{o) (ll (X),...,I , ( X ) ) ~ subject to hj(x) - b j 5 0, j = 1,2, - . - , m , x E IRn, where l j : IRn -+ IR and hj : IRn + IR are differentiable and pseudolinear, i.e., pseudoconvex and pseudoconcave [CC2], and bj E R,j = 1,2, - ,m. It should be noted that a real-valued function h : IRn -,IR is pseudolinear if and only if for each X, y E IRn, there exists a real constant a ( x , y ) > 0 such that

+

+

Moreover, any fractional linear functions of the form (aTx b) /(cTx d) on IRn is pseudolinear, where a,c E IRn, b, d E R . Define Fi, G j : IRn + by

266

DUALITY IN MULTICRITERIA OPTIMIZATION

Then, we can rewrite PLP as the convex composite multicriteria problem. The dual problem of PLP is as follows: (Problem DLP) maxR? \{O) (1I (U), - - ,I ~ ( U ) ) ~

T h e o r e m 8.6.3 (Weak duality) Let feasible for problem DLP, then

X

be feasible for PLP, and ( u , r , A ) be

Proof: Follows from Theorem 8.6.1.

B

T h e o r e m 8.6.4 (Strong duality) For the problem PLP, assume that the generalized Slater constraint qualification holds at each feasible point of P and D. If X is a 0 properly minimal solution for PLP then there exist r E ]RP, > 0, X E ]Rm,Aj such that (X,T ,X) is a properly maximal solution for DLP and the objective values of PLP and DLP at these points are equal.

>

Proof: Follows from Theorem 8.6.2.

B

CHAPTER 9

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

The concept of vector variational inequalities (VVI) was introduced by Giannessi [G61 in 1980 as a generalization of scalar variational inequalities. Since then the subject has found many applications, particularly in the areas of multicriteria optimization (VO) and vector equilibria (VE) problems (see [Gg]). This follows the idea discussed in Chapter Seven that most real world problems are concerned with not just a single criterion, but a number of criteria has to be considered simultaneously. In fact some early work in multicriteria optimization (see [Jl], [L2]) had led to some forms of vector variational inequalities, albeit being called different names. More specifically, W 1 often turns out to be a necessary optimality condition for some V 0 problems. Under certain convexity assumptions, it may also turn out to be a sufficient optimality condition. Further works relating V 0 and VVI can be found in [CY2], [CC11 and [Yl]. Applications of VVI in vector equilibria problems are studied in [CYl], [YGl] and [YG3]. Because of its immense theoretical interest, the subject of VVI has been generalized and extended in many different directions in recent years. Some notable ones are: Generalized Vector Variational Inequalities ( G W I ) , see, for example [C4]. Extended Vector Variational Inequalities (EVVI), see, for example [Yl] Quasi-vector Variational Inequalities (QW I ) , see, for example, [CLl],[CGY5]. Pre-vector Variational Inequalities (PVVI) or Vector Variational-like Inequalities, see, for example, [L5], [Y5], [Y6]. Vector Variational Inequalities for Set-valued Mappings, see, [LLKCl]. Vector Variational Inequalities for Fuzzy Mappings, see, for example, [LKLCl]. Vector Complementarity Problems, see, for example, [CY2], [Y7]. Relationships with multicriteria optimization, see, for example, [CY2], [L101, [Y3l Much of the literature on VVI is centered around existence results, which are beyond the scope of this book. There are very few results in terms of the solution

268

DUALITY I N VECTOR VARIATIONAL INEQUALITIES

of VVI as this is a highly non-trivial problem involving the computation of a set. A recent result is found in [GY7]. As the subject is fairly new and there is yet a serious book on the subject, we shall devote Section 9.1 to an introduction to the subject. Scalarization of VVI will be discussed in Section 9.2. The first duality result in VVI due to [Yl] will be the subject matter of Section 9.3. In [CGYI], the duality of VVI is further investigated and the concept of a gap function is generalized for VVI. This will be presented in Section 9.4. A generalization of the Giannessi's gap function as discussed in Chapter Six will be presented in Section 9.5. Lastly, a solution method is studied in Section 9.6 because of its relevance to gap functions. Although most of the results on VVI in the literature are derived in the framework of infinite dimensional abstract space like Banach or Hausdorff spaces, we shall present the results only in finite dimensional vector spaces, in order to be consistent with the rest of this book. The vector ordering relationships in Sections 9.3, 9.4 and 9.6 are induced by a closed and convex cone C C RPas in Definition 7.1.1. In Sections 9.1, 9.2 and 9.5 however, the ordering cone is assumed to be Et;.

9.1 Introduction to Vector Variational Inequalities Definition 9.1.1 Let F : ]Rn -+ EtnXPbe a matrix-valued continuous function and K: C Etn be a closed and convex ground set. The vector variational inequality problem (VVI) is defined as follows: (Problem VVI)

Find x0 E K such that

F(x')~(x - X')

$R;,(O)

0, vx E K:.

(9.1.1)

A weaker version of VVI, called the weak vector variational inequality (WVVI), is defined as follows. This is concerned with a closed set. Find x0 E K: such that (Problem W W I ) RP+ 0, t/x E K. F(x')~(x-x')

(9.1.2)

Obviously, a solution to VVI: is also a solution t o W I , but not the converse. Furthermore, the solution of a VVI or a WVVI is often non-unique, and a complete solution requires the computation of the set of all solutions. This is much harder to do than to compute the point solution of a variational inequality.

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

269

We shall illustrate the significance of VVI and WVVI via a couple of well-known examples. Example 9.1 . l (Multicriteria optimization) Consider the problem V 0 as introduced in Chapter Seven. (ProblemVO)

m ~ n R ; ~ ~ o ~ f ( x ) , s u b j e c t t o x ~ (9.1.3) ~.

The optimality conditions in terms of VVI was obtained by [CY2]. Let f be assumed to be Gateaux differentiable. T h e o r e m 9.1.1 If X is a weakly minimal solution of problem VO, then X solves the WVVI (9.1.2) with F ( x ) ~= Vf(x). If f is convex and X solves the WVVI (9.1.2) with F ( x ) ~= Vf (X), then X is a weakly minimal solution to the problem VO. Proof: If

X

is a weakly minimal solution of VO, then by definition,

implying that

Taking the limit t j, O, we have

Conversely, assume that X solves WVVI, and f is R;-convex. By the R:-convexity of f , we have f(y) >R: f ( x ) + V f ( x ) ( y - x ) , vy € K -

implying that

X

is a weakly minimal solution to problem VO.

I

It is also straightforward to prove that if f is R:-convex, and if X solves the VVI (9.1.1) with F ( x ) ~= Vf (X), then X is a minimal solution of VO. However, it should be noted that a minimal solution of V 0 may not be a solution of VVI. As a simple example, let n = 1 , p = 2,f(x) = (x,x2 and X = [-1,0]. It is easy t o check that every X E [-l, l] is minimal. But for X = 0 and y = -1, we have

+

implying that

X

= 0 is not a solution of the corresponding VVI.

270

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

Example 9.1.2 (Vector Equilibria) This example is based on a traffic equilibrium model with a single criterion taken from [M5]. The generalization to a multicriteria consideration by using VVI was studied in [CYl], [GY6] and [YGl]. We shall introduce the necessary notation by summarizing the (scalar) traffic equilibria problem. Consider a transportation network G = (N,A) where N denotes the set of nodes and A denotes the set of arcs. Because the interest is more in paths than arcs, we shall use a slightly different set of notations from that of Chapter Two. Let Z be a set of origin-destination (0-D) pairs and Pi, i E 1 denotes the set of available paths joining 0 - D pair i. For a given path k E Pi, let hk denote the traffic flow on this path and h = [hk]E IRN, N = CiErlPil. A (path) flow vector h induces a flow U, on each arc a E A given by:

where A = [Sak]E IRldlXNis the arc path incidence matrix with bar = 1 if arc a belongs to path k and 0 otherwise. Let v = [U,] E IRIAI be the vector of arc flow. Succinctly v = Ah. We shall assume that the demand of traffic flow is fixed for each 0 - D pair, i.e., CkEPi hk = di, where di is a given demand for each 0 - D pair i. In the scalar cost case, the demand di can be easily generalized to be a function of the minimum cost (see [M5]). Unfortunately to do this for vector costs will surely incur complicated set-valued notations. For simplicity, we shall assume 0 satisfying the demand is called a a b e d demand for the moment. A flow h feasible flow. Let 7-l = {h I h 0 , CkEPj hk = di Vi E E}. 71 is clearly a closed and convex set. Let t,(v) be a (vector) cost on arc a (usually the delay) and is in general a function of all the arc flows, and t (v) = [t, (v)]. If t, is only a function of v,, we say that the cost is separable. If dt,/au,l = dt,//du,, we say that the cost is integrable. This nomenclature comes about because if a vector-valued function is integrable, then there exists a scalar-valued function whose gradient yields the vector-valued function, see [ORl]. Note that a separable cost is a special case of an integrable cost. As it turns out, if the cost is integrable, then the subsequent equilibrium problem can be cast in the form of an optimization problem, see [M5]. The cost along a path this path, thus

rk

is assumed t o be the sum of all the arc costs along

E IRN. Succinctly r ( h ) = a T t ( v ) . Given a flow vector h , the minimum Let r = [rk] cost for an 0 - D pair i is defined by

ui = min rk(h). kE Pi The Wardrop's user principle [W31 is a behavioral principle which asserts that, at equilibrium, users only choose minimum cost paths to travel on, i.e., a flow h E 7-l is said to be in Wardrop equilibrium if

DUALITY IN VECTOR VARIATIONAL INEQUALITIES W a r d r o p equilibrium principle:

Vi E 1 , Vk E Pi, rk(h) > ui

271

* hk = 0.

Amongst several possible ways of stating the conditions for Wardrop equilibrium, the most popular one seems to be in the form of a (scalar) variational inequality (VI): (Problem VI)

Find h E

N, s.t. ~

( h ) ~ ( hh) -2 0,

Yh E X.

Existence and uniqueness for traffic equilibria can be established under easily satisfiable conditions such as the continuity and strict monotonicity of the function 7. The assumption that users choose their path based on a single criterion may be unreasonably restrictive. In reality, users may choose their path based on several criteria, for example, time delay and (monetary) cost, amongst others. In general, these costs are often conflicting. We generalize the scalar costs ta(h) and rk(h) to vector costs t a ( h ) and r k ( h ) E RP.Let T ( h ) be an p X N matrix with columns given by r k ( h ) . In [CYl], the following vector equilibrium principle is proposed. Essentially it asserts that, at equilibrium, users choose only minimal or efficient paths to travel on. Vector equilibrium principle: A flow h E 'Ft is said to be in vector equilibrium if: tli E 1,Vk, k E Pi, r k ( h ) - ~ ( h >R:\{o) ) 0 hk = 0.

*

A weaker form of vector equilibrium which turns out to be rather important is defined as follows. W e a k vector equilibrium principle: A flow h E 7-l is said to be in weak vector equilibrium if:

The latter is called weak since a flow h satisfying r k ( h ) - %(h) hk > 0 (i.e., not weak vector equilibrium) also satisfies rk(h) - Q (h) and hk > 0 (i.e., not vector equilibrium).

R:

0 and 0

The following sufficient condition for a flow h to be in vector equilibrium is due to [YGI]. T h e o r e m 9.1.2 (Sufficient condition for vector equilibria) h E X is in vector equilibrium if h solves the W I : Findhsuchthat

~ ( h ) ( h - h )$RPc\{OIO,V h ~ 7 - l .

P r o o E Let h satisfy (9.1.4), Choose h to be such that

(9.1.4)

272

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

Clearly h E 7-1 since VZ E 1,CjEp, hj = CjEp= h j = dimNOW

7k(h) - %(h) ~ R : \ { o ) 0 , then (9.1.5) and (9.1.6) together imply that hk = 0.

A similar sufficient condition can be established for a weak vector equilibrium: Theorem 9.1.3 h E 7-1 is in weak vector equilibrium if h solves the WVVI: Find h such that

~ ( h ) (h h)

gin,R:

0, V h E 7-i.

Proof: The proof follows exactly as in Theorem 9.1.2 but with $R? \{O) in (9.1.5) replaced by gin,R: and >R:\jO) in (9.1.6) replaced by >int RP+. I

9.2 Scalarizat ion of Vector Variational Inequalities In contrast to multicriteria optimization, the solution of vector variational inequalities is a much less mature subject. In Section 7.3, we discuss in detail how solutions to multicriteria optimization problems can be found by scalarization or composing the vector cost with a utility function. Unfortunately, such scalarization techniques in general do not work for VVI. There are very few results on solving W1 until recently. The first such result is found in [GY7] where the solution of WVVI is reduced to finding the level sets of some related scalar function. As this is related to the notion of gap functions for VVI, we shall defer the discussion of this result to Section 9.6. The maih reference for this section is [GY8]. The ordering cone for this section is assumed to be R : . If a certain integrability assumption can be made about the underlying (matrix valued) function, it is possible t o apply a scalarization technique t o relate the problem of solving a WVVI t o a (scalar) variational inequality or t o a multicriteria optimization problem [GY8]. These results are discussed as follows.

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

273

Definition 9.2.1 Let X E R ] : \ {O}. The scalar variational inequality for the vector variational inequality as defined in Definition 9.1.1 is defined by:

(Problem VI(X))

Find

X

E

K, such that x ~ F ( x () y~- X) 2 0, Vy E K,

where K ,F are as given in Definition 9.1.1. We first present a straightforward relationship between WVVI and VI(X). Theorem 9.2.1 (Sufficient condition for WVVI in terms of a scalar VI) If there exists X E RP, \ {0} such that X E K: is a solution of VI(X), then X is a solution of the WVVI (9.1.2). Proof: If not, there exists y E K such that

]: Then for all X E R

\ {0), we have

Hence X E K is not a solution of VI(X).

B

Unlike the above sufficient condition, a necessary condition for the solution of WVVI is much harder to obtain. Several non-trivial conditions are required which are to be discussed as follows. Definition 9.2.2 A (cost) matrix function F : ]Rn if,

--+

RnXPis said to be integrable

Thus a cost matrix is integrable if the Jacobian of each column of the matrix is symmetric. This nomenclature is a generalization of integrable field in physics. The following is a well-known extension of Green's theorem to functions in ]Rn. Theorem 9.2.2 If F ( x ) is integrable, then there exists f : IRn (fi (X), - - , fpWT , denoted by

such that

Proof: See Theorem 4.1.6 of [ORl]

-,RP,with

f (X) =

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

274

Definition 9.2.3 Let f : K -,RP be a differentiable vector-valued function. f is said to be intR:-pseudoconvex (or weakly pseudoconvex) on K if

f is said to be R;

\ {O)-pseudoconvex (or strongly pseudoconvex) on K if

Clearly strong pseudoconvexity implies weak pseudoconvexity, but not vice versa.

Theorem 9.2.3 If each component of f is pseudoconvex on K, then f is intlR;pseudoconvex on K. If in addition, at least one component of f is strictly pseudoconvex on K, then f is R: \ {O}-pseudoconvex on K. Proof: Let Vf (x)(y -X) gin, 0. Then there exists i, such that V fi(x)(y- X) 2 0.By the pseudoconvexity of fi, fi(y) 2 fi(x). Thus f (y) gi, f (X). Then f is intR:-pseudoconvex on K. If one component ( jsay) of f is strictly pseudoconvex, then by the strict pseudoconvexity of fj , fj (y) > fj (X) Thus f (y) &P, \{o) f (X). Then f is 1% \ {O)-pseudoconvex. B Theorem 9.2.4 If f is intlRr-pseudoconvex and

X

solves the following W V I :

then X solves the problem W O as defined in Definition 7.3.3, with K: as the feasible set. If f is R: \ {0) -pseudoconvex and X solves the following VVI Find then set.

X

X

E K,

Vf(x)(y - X) $R:\{O)

0,

V y E K,

solves the problem V 0 as defined in Definition 7.3.1 with K as the feasible

Proof: Follows from Definition 9.2.3. Let X E R:

\ {O).

B

Consider the scalar optimization problem (Problem P(X)) Min XT f (X) subject t o X E K ,

where f and K are as given in Definition 9.1.4. From Section 7.3, If the set f(K)+R: is convex, then a weakly minimal solution of WVO also solves P(X) (Theorem 7.3.2).

Lemma 9.2.5 If X is a globally minimum solution of P(X), then X is a solution of VI(X) with F ( x ) ~= Vf (X). If, in addition, that f is R:-convex, then the converse also holds, i.e., X is a solution of VI(X) implies that X is also a global minimum solution of P (X).

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

275

Proof: Assume that X is a global minimum solution of P(X). Since K is a convex set and f is differentiable, X is a solution of the variational inequality

It is clear that v ( X Tf ) (X) = XT~f (X). Thus X is a solution of VI(X) with F(X) = Vf (X). The second part of the result follows from Theorem 1.5.1 and the argument that f is R:-convex if and only if XTf is convex for all X E R:. I Remark 9.2.1 Note that the result of the first part in Lemma 9.2.5 is not true if X is only a local minimum solution. Theorem 9.2.6 (Necessary condition for WVVI in terms o f a scalar VI) Assume that F(x) is integrable, i.e., there exists f such that F ( x ) ~= V f (X). If f is intR:-pseudoconvex, f(K) is R:-bounded and X E K is a solution of WVVI, then there exists X E R : \ 10) such that X is a solution of VI(X). Proof: Following Theorem 9.2.4, X is a weakly minimal solution of WVO. BY Theorem 7.3.8, there exists e E intR:, a E RP such that X is a solution of the following optimization problem: Min subject to

tea (f (X)) X

E K,

where the Gerstewitz function teais as defined in Definition 7.3.5. From a corollary on page 52 of [C5], 0 E dO(tea0 f)(x) + N ( K , x ) , where N ( K 7X) is the normal cone defined by N ( K , X) = {X* E Rn I (z - x ) X*~ 5 0 Vz E K). Then X satisfies the following condition:

32

E

aO(cea0 f ) ( x ) such that zT (y - X) 2 0, Vy E

K,

where aOg(x)is the Clarke generalized subgradient [C5]. Note that

Thus

where I(x) is the set of index that achieves the maximum in (9.1.7). Thus solution of the variational inequality problem:

X

is a

276

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

where for some ti such that

Then X E R:

\ {O),

C:='=, ti = 1,ti 2 0, ti = 0, i @ Z(x). Let

and the proof is complete.

I

The above necessary and sufficient conditions for weak vector variational inequalities can be extended to vector variational inequalities as follows.

Theorem 9.2.7 (Sufficient condition for VVI in terms of a scalar VI) If there exists X E intlR? such that X E K: is a solution of VI(X), then X is a solution of VVI. Proof: If not, there exists y E K: such that

Since X E intlR:, Hence

X E

we have

x ~ F ( x (y ) ~- X) < O.

K: is not a solution of VI(X).

Theorem 9.2.8 (Necessary condition for VVI in terms of a scalar VI) Assume that F(x) is integrable, i.e., there exists f such that F ( x ) ~= Vf(x). If X E K is a solution of W I , and f is R : \ {O}-pseudoconvex, then there exists X E R: \ {O} such that X is a solution of VI(X). ProoE Let X E K be a solution of VVI. Since f is R:-pseudoconvex, it follows from part two of Theorem 9.2.4 that X solves VO. The rest of the proof is similar I to that of Theorem 9.2.6 and is omitted.

9.3 Duality in Extended Vector Variational

Inequalities

Like variational inequalities, duality plays a very important role in the analysis of vector variational inequalities. In this section, we study the duality of extended vector variational inequalities and establish their equivalences with VVI. Two examples are used to illustrate the role of VVI duality in multicriteria optimization and vector approximation problems. The main reference for this section is [Yl].

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

277

Like in the scalar case, the duality of W1 is more appropriately analysed in the extended case. For this section, the underlying vector ordering relationships are induced by a closed and convex cone C C RP.

Definition 9.3.1 (Extended vector variational inequality) Let F : Rn + Rnxpand f : Rn -+ (see Remark 9.3.1). The extended vector variational inequality (EVVI) is defined as: Find a point x0 E Rn such that

(Problem EVVI)

F(x')~(x - X') $c\10) f(xO)- f(x), b'x E Etn. (9.3.1)

Remark 9.3.1 Note that in order to be consistent with the definition of Fenchel Transform (Definition 7.2.1), f is assumed to be a function from Rn to RP.Thus, in general, VVI is not a special case of E W I . However, if we append an abstract ('m7'to W, written as = RPU { m ) (see [B4]), then the EVVI (9.3.1) includes the VVI (9.1.1) as a special case where f is just the following indicator function for the ground set K: f (X) =

0, if X E K; cm,if X $ K .

We make the following assumptions.

Assumption 9.3.1 (i) F is one-to-one (injective), (ii) f is continuous, and (iii) f*(U) # 0,VU E I R p X n . Under Assumption 9.3.1, we may define the adjoint function Ft : IRPXn-+ follows:

Rn as

If F is linear, then F't = F-'

Definition 9.3.2 (Dual extended vector variational inequality) The dual extended vector variational inequality (DEVVI) is defined by: (Problem DEVVI)

Find U' E dom(Ft) such that

( u - u ~ ) F ~ ( u ~ ) ~ ~ \ ~ ~V~U ~ ER *~(' "u ~ (9.3.2) )-~*(u

where the (set-valued) function

f*(U) = r n a ~ ~ \{Ux { ~ }- f (X) I is the type I Fenchel transform of f .

Theorem 9.3.1 Under Assumption 9.3.1, we have

X E

Rn}

278

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

(i) If x0 is a solution of EVVI, then U0 = -F(X')~ is a solution of the D E W . (ii) In addition to Assumption 9.3.1, we assume that the closed convex cone C is connected, i.e., C U (-C) = RP,and af(xO)# 0, where x0 = -F~(u'). 1f U0 solves DEVVI, then x0 solves EVVI. (iii) In both cases (i) and (ii), the following relation holds

Proof:(i) Let xo be a solution of EVVI, then

If UO = -F(x')~ is not a solution of D E W I , then there exists U E -Range(F) such that ( U - uO)F~(UO) f * ( u O )- f*(U). It follows from (9.3.4) that

This is a contradiction to the generalized Young's inequality of Theorem 7.2.1. Thus is a solution of DEVVI.

UO

(ii) Let U' be a solution of DEVVI. Let x0 = -F~(uO) and

(U + F(xO)~)(-XO) -F(X')~X' - UXO

$c\{o)

= -F(xO)~. Then

f*(-F(xO)) - f*(U),

+ f r ( U ) $C\{O)

f*(-F(x')).

Since af (xo) # 0, let U E af (xo). Then by Lemma 7.2.2,

uxO- f (xO)E f * (U). It follows from (9.3.5) and (9.3.6) that

UO

DUALITY IN VECTOR VARIATIONAL INEQUALITIES From the definition of f* and the assumption that C is connected, we have,

If x0 = - F t ( u O ) is not a solution of E W I , then there exists X E Rn such that

Then -F(X')~X'

-

f(xO)$! f*(-F'(xO)),

a contradiction. Thus x0 = -F+(UO) is a solution of EVVI. (iii) From (9.3.3), (9.3.6) and the fact that U' = -F(x')~,we have

Remark 9.3.2 Note that the interior of the closed convex cone C does not need t o be assumed nonempty. Note also that in order for Theorem 9.3.1 (ii) to hold, the restrictive assumption that C is connected is required. This is certainly not the case when the cone is assumed to be R:. Moreover, note that since the condition intC # 0 is not required, Theorem 9.3.1 includes the Mosco's result as a special case (see Theorem 6.1.1).

The weak vector variational inequality W W I and its dual are defined as follows: Definition 9.3.3 (Weak extended vector variational inequality WEVVI)

(Problem WWI)

Find x0 E IRn

such that

F(xO)~(X-XO) $int c f(xO)--(X),

V X E Etn.

Definition 9.3.4 (Dual weak extended vector variational inequality DWEVVI)

(Problem DWVVI)

Find U' E dom(Ft) such that c f;(u0) - f$(U), ( U - uO)F'(UO)

W E Rp79,3.8)

where the (set-valued) function :f (U) = maxintc{UX - f (X) I X E is the weak Fenchel transform of f , see Definition 7.2.1.

]Rn)

(9.3.9)

280

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

T h e o r e m 9.3.2 Under Assumption 9.3.1, we have (i) If x0 solves WEVVI, then U' = - F ( x ' ) ~ solves DWEVVI. (ii) In addition to Assumption 9.3.1, we assume that the closed convex cone C is connected, i.e., C U (-C) = lRP, and af(xO)# 0,where x0 = -Ft(UO). If UOsolves DWEVVI, then x0 solves WEVVI. (iii) In both cases (i) and (ii), the following relation holds

Proof: The proof follows essentially the same lines as in the proof of Theorem 9.3.1, with $C\io) replaced by Pint c and f * (U) replaced by f; (U). I In the following, two examples are given to illustrate the applications of a dual weak vector variational inequality. Example 9.3.1 Consider the vector approximation problem: (Vector Approximation Problem)

where K: C lR" is a closed and convex set, a', i = 1,- - - , p are fixed vectors in IR". We say that x0 E K: is a weakly minimal approximation to all the a2if

This is a weak extended vector variational inequality (WEVVI) with

Since Ft(UO)= -xO, U0 is a solution of the dual weak extended vector variational inequality. It should be easy t o verify that U' = -F(x')~ satisfies the D W E W I : ( U - U') (-X')

gint c :f

(U0) - f; (U), W E R P X n .

Example 9.3.2 Consider the multicriteria optimization problem VO: (Problem VO)

minint c h(x), subject to

X

E lRn

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

281

where h : Rn ---+ HtP is a differentiable vector-valued function. Let x0 be a weakly minimal solution of problem VO. Then by Theorem 9.1.1, x0 is a solution of the following weak vector variational inequality:

~ the dual weak extended vector From Theorem 9.3.1, U0 = - ~ h ( x O )satisfies variational inequality:

(U - U')(-X')

gint

c oW (U') - oW (U),

V U E Rpxn

(9.3.11)

where u,(U) = maxint c{Ux I X E IRn). This is the dual weak extended vector variational inequality of (9.3.10) if we let F = V h and -xO = Ft (U'). Note that the problem V 0 can be expressed as minint c+(x, 0), where

4 : ]Rn

X

subject to

X

E Etn

RnXP --+Etp is the perturbation function satisfying

r)

be the weak Fenchel transform of +(X, r ) . Then we may construct Let +;(X, the dual problem DV0 of V 0 as follows: (Problem DVO)

- +;l;(0, r ) , subject to E IRnxp.

max;,t c

It is easy to verify that the following weak duality relation of V 0 and DV0 holds. For any X E R n , r E IRnxp7

4 ( x , 0) 4 -+;(O,

F ) - int C.

(9.3.12)

The problem V 0 is said to be weakly stable if the set-valued mapping C : IRn

-+2"

has a weak subgradient at U = 0. From [SNTl], if V 0 is weakly stable, then there exists a solution r0E IRnxp of DV0 satisfying

Assuming that C is connected, then from the weak vector variational inequality (9.3.1), (9.3.15) - v h ( x O ) x 0.

Proof: Let y $ -int C and y E te - C . Then t 2 0. Otherwise t < 0 implies that te C -int C, thus y E te - C E -int C.

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

289

Let Jeo(y) 2 0. Then y 4 -int C. Otherwise there exists t < 0 such that y E te -C, a contradiction to Jeo(y) 2 0. D

Theorem 9.6.2 (Necessary and sufficient condition for solution of WVVI) Let e E int C. Then X E K: solves WVVI if and only if

Proof: Assume that

X E

K solves the problem WVVI and let e E int C, then,

F ( x ) ~ ( Y- X )

$in+,

C 0

Vy E E

* ~ ( x ) ~ ( X)y # -int C Vy E E * & o ( ~ ( x ) ~ ( y - x ) ) 2 0 VyJrK, -

b

min & O ( F ( X ) ~-(X~) ) YEK

> 0.

by definition byLemma9.6.1

By virtue of Theorem 9.6.2, the problem of solving WVVI essentially reduces to the following problem: (Problem P1) Find all X E E such that g(x) In the special case where C = R:, following equivalent form

> 0.

the Gerstewitz function may be expressed in the

Corollary 9.6.3 Let C = R : and let h(x) = min max {Fi(x)(y - X)), yEK l l i l p

where F, (X)be the ith row of the matrix F ( x ). Then if and only if h(x) 0.

>

X

E

E solves WVVI problem

Proof: The proof follows from Theorem 9.6.2 and (9.6.5) by letting a = 0 and e = (1,- - , E RP. D Based on the auxiliary function h or g, a scalar-valued gap function for the WVVI can be readily constructed. Note that this is in contrast with the set-valued gap function discussed in Section 9.4. Let y : Rn --+ IR be defined as:

DUALITY IN VECTOR VARIATIONAL INEQUALITIES

290

Theorem 9.6.4 y is a (scalar) gap function for WVVI in the sense that (i)y (X)

>0

Vx E

]Rn;

(ii)y(x) = 0 if and only if

X

solves W W I .

Proof: (i) holds trivially by construction. Corollary 9.6.3.

(ii) follows from Theorem 9.6.2 and I

If C = R : , then for a given X, the evaluation of the auxiliary function h requires the solution of a minimax problem, where the cost functions are clearly linear in the decision variable y. Furthermore, in many economics and network equilibrium models, the set K: is often a polyhedral set, in which case, the minimax problem reduces to a trivial linear programming problem:

which can be solved very quickly, even if F is highly nonlinear. In the case of VVI, the condition becomes significantly weaker. Theorem 9.6.5 (Necessary condition for solving VVI) Let e E int C. Then K: solves W 1 only if g(x) 2 0, where g(x) is defined by (9.6.4). If C = R : , then X E K: solves W 1 only if h(x) 2 0, where h(x) is defined by (9.6.6). X E

Proof: Note that a solution of VVI is also a solution of WVVI. The result follows W from Theorem 9.6.2 and Corollary 9.6.3. Theorem 9.6.6 (Sufficient condition for solving VVI) Let h(x) be defined by (9.6.6). If X E K: and h(x) > 0, then X E K solves VVI. Proof: If not, there is an y E K: such that

Then and thus h(x) = min max {Fi(x)(z - X)} 5 max {Fi(x)(y - X)} 5 0. zEK: l