Convex Analysis and Variational Problems

Convex Analysis and Variational Problems

SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimisation and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem *First time in print. i

Classics in Applied Mathematics (continued) Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimisation and Economic Theory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the NavierStokes Equations J. L. Hodges, Jr. and E. L. Lehmann, Basic Concepts of Probability and Statistics, Second Edition

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Convex Analysis and Variational Problems

Ivar Ekeland Universite Paris-Dauphine Paris, France

Roger Temam Universite Paris-Sud Orsay, France Indiana University Bloomington, Indiana

Society for Industrial and Applied Mathematics Philadelphia

Copyright ©1999 by the Society for Industrial and Applied Mathematics. This SIAM edition is an unabridged, corrected republication of the work first published in English by North-Holland and American Elsevier, Amsterdam and New York, 1976.

1098765432 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Ekeland, I. (Ivar), 1944[Analyse convexe et problemes variationnels. English] Convex analysis and variational problems / Ivar Ekeland, Roger Temam. p. cm. — (Classics in applied mathematics ; 28) English language ed. originally published: Amsterdam : North-Holland Pub. Co.; New York : American Elsevier Pub. Co. [distributor], 1976, in series: Studies in mathematics and its applications ; v.l. Includes bibliographical references and index. ISBN 0-89871-450-8 (pbk.) 1. Mathematical optimization. 2. Convex functions. 3. Calculus of variations. I. Temam, Roger. II. Title. III. Series QA402.5 .E3813 1999 519.3-dc21 is a registered trademark.

99-046956

CONTENTS Preface to the Classics Edition Preface

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PART ONE FUNDAMENTALS OF CONVEX ANALYSIS 3

Chapter I . Convex functions Chapter II. Minimization of convex functions and variational inequalities

34

Chapter III.

46

Duality i n convex optimization

PART TWO DUALITY AND CONVEX VARIATIONAL PROBLEMS Chapter IV. Applications of duality to the calculus of variations (I) 75 Chapter V. Applications of duality to the calculus of variations (II): 116 problems of the type minimal hypersurfaces 165 Chapter VI. Duality by the minimax theorem Chapter VII. Other applications of duality 186 PART THREE RELAXATION AND NON-CONVEX VARIATIONAL PROBLEMS Chapter VIII. Existence of solutions for variational problems . Chapter IX. Relaxation of non-convex variational problems (I) Chapter X. Relaxation of non-convex variational problems (II)

231

263 297

357 Appendix I. An a priori estimate in non-convex programming . Appendix II. Non-convex optimization problems depending on a parameter 375 Comments 385 Bibliography 391 Index 402

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PREFACE TO THE CLASSICS EDITION This edition of the book is the same as the initial one, except for a few corrections and the addition of a small number of references. Several parts of the book cover basic material which turns out to be useful in a number of applications and which is not expected to evolve; as far as we know this material has not appeared elsewhere in book form since it was published in this book. Here are some of the topics developed in this book, and their present standing in theoretical and applied research; references are provided in the "Additional References" at the end of the book: 1. Duality in the calculus of variation (convex variational problems in infinite dimension). Duality has important applications in mathematical economy, in continuum mechanics, in numerical analysis (mixed finite elements), and in control theory. New developments have occurred in convex analysis in finite dimension: we refer, for instance, to semi-definite programming. Duality for some (finite or infinite dimensional) nonconvex problems has been developed. Systematic use of duality in solid mechanics for plasticity related problems has been made. The (infinite dimensional) nonconvex problems of calculus of variation appearing in nonlinear elasticity have attracted much attention and effort (with little or no reference to duality). 2. Generalized solutions of minimal surface problems. Important developments have occurred in the parametric case (not considered in this book) with geometrical ideas totally different from those used here. Extensions of the methods of this book have been developed and studied for the time dependent (evolution nonparametric) minimal surface problem. 3. The minmax theorems stated in this book have also many useful applications— in particular in relation with duality for the same topics as in point 1 and most recently for control theory, namely the robust control of partial differential equations in finite time horizon.

Ivar Ekeland Roger Te mam

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PREFACE

In recent years, there has been a considerable expansion in the field of Convex Analysis, in conjunction with the development of various mathematical tools, certain of which have become standard. The initial motivation was provided by operations research: the success of linear programming, due to the duality theorem and the simplex method, has aroused the interest of managers and engineers in this type of problem, and similar results have therefore been sought for non-linear optimization. To this primary objective of Convex Analysis, others have been added, ranging from mathematical economics to mechanics, and encompassing more strictly mathematical problems such as the study of functional equations of monotone type. The result has been a deeper understanding of convex functions, together with the introduction of new concepts such as those of subdifferentiability and conjugate convex functions. The subdifferential of a convex function is a generalization of the notion of derivative, and has provided the theory of maximal monotone operators, so useful in the study of partial differential and integral equations, with its first examples. The concept of conjugate convex functions has emerged as an elegant and general formalization of duality in optimization. However, it does not seem to us that these methods have been used to their full advantage in the study of variational problems, that is, optimization problems in concrete functional spaces. The object of this book is to fill this gap in two main directions: by dualization of convex variational problems, by relaxation of non-convex variational problems. Duality allows us to associate a dual problem with a variational problem and to study the relationship between the two problems. This is useful in mathematical economics where the dual problem can be stated in terms of the price; in mechanics where the primal and the dual problems are two well-known forms of the conservation principles, characterizing the displacements and the constraints respectively; in numerical analysis where the dual problem may help us to solve the primal problem. In addition to these standard applications of duality, we have a new use for the calculus of variations in mind: the dual problem enables us to define the generalized solution of a variational problem which has no classical solution. xi

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The relaxation consists in associating a "convexified" problem with a non-convex variational problem. This approach to non-convex problems has been considerably developed for optimal control problems; the study is developed here in a more general framework, adapted to problems in the calculus of variations. The solutions of the "convexified" problem arise as generalized solutions of the initial problem. Thus we see that these two directions meet in the concept of generalized solutions: these are the cluster points of the minimizing sequences of the problem under consideration. The book is divided into three parts, dealing with a summary of convex analysis, duality for convex variational problems, and the relaxation of nonconvex variational problems respectively. We shall now describe the contents of the various chapters in more detail. Chapter I summarizes the essentials of the theory of convex functions. We have omitted those points which are not directly useful to us, such as the inf-convolution, so that we can concentrate on the fundamental concepts of conjugate convex functions and of subdifferentiability. Chapter II deals with the minimization of convex functions. Here we recall the principal results which guarantee the existence and uniqueness of the point where a convex function attains its minimum, characterizing it as the solution of a variational inequality. Chapter III develops the theory of duality in convex optimization following R. T. Rockafellar. Given a convex optimization problem, we embed it in a family of perturbed problems, and by using conjugate convex functions we associate a dual problem with it. This very flexible abstract theory can be adapted to a wide variety of situations. Chapter IV describes the application of duality to several problems in the calculus of variations, of mathematical physics, of mechanics and of filtering theory. In each case, we state the dual problem explicitly together with its relationship to the primal problem. Chapter V describes the application of duality to the classical problem of minimal hypersurfaces and to problems of related type. The dual problem still has a unique solution, and the primal-dual relationship enables us to associate a generalized solution of the primal problem with it. Hence we obtain the existence of a generalized solution to the problem of minimal hypersurfaces, for which it is well known that there is generally no classical solution. In addition to the systematic use of duality, we here have an unexpected application of e-subdifferentiability. Chapter VI describes a different theory of duality, based on the minimax theorems. This approach, which is older than Rockafellar's, adds nothing new and we develop it briefly for completeness.

PREFACE

Xlll

Chapter VII describes the application of duality to problems of numerical analysis, optimal control, mechanics and mathematical economics. All these applications of duality are still being developed at the present time and we do not pretend to be exhaustive. We have restricted ourselves to illustrating some typical methods with specific examples. Chapter VIII tackles non-convex variational problems, by studying those cases where the existence of a classical solution is assured. The existence theorem which we obtain is illustrated by examples taken from optimal control and the calculus of variations. Chapter IX considers variational problems devoid of a classical solution. We then define, by partial convexification, a relaxed problem, which can be shown to be near to the initial problem. In particular, the relaxed problem possesses classical solutions which are none other than the generalized solutions of the initial problem. These results are applied to a number of problems of optimal control and of the calculus of variations. Chapter X deals separately with the fundamental problem of the calculus of variations in dimension n > 1. Although it is not amenable to the methods of the preceding chapter, the results obtained are similar: obtaining the relaxed problem by partial convexification, characterizing the classical solutions of the relaxed problem as generalized solutions of the initial problem. Finally, we conclude with a study of variational equations. Clearly, if a problem in the calculus of variations has no classical solution, the corresponding Euler equation has no solution in general. However, we show that approximate solutions always exist. Chapters I to VII were the subject of a postgraduate course given by the second author at the Universite de Paris XI in 1970-71 and 1971-72. They contain some new results and others which have only recently been published (see R. Temam [l]-[4]). Chapters VIII to X continue and develop previous work of the first author (see I. Ekeland [1]) and also contain large borrowings from the work of H. Berliocchi and J. M. Lasry (see [1]). Obviously all the first part of this book owes much to J. J. Moreau and R. T. Rockafellar. We offer them our thanks. Our purpose was not to produce a systematic exposition of the topics considered here. We have only attempted to describe some methods linked with convex analysis, methods which have already been found to be productive and still seem to be promising. This should be the subject of future research. We dedicate this book to J. L. Lions who has profoundly influenced our mathematical thinking and to whom we owe much. We thank M. P. Lelong for welcoming our work into the series of which he is the general editor and also for his advice and suggestions.

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Our thanks also go to Mme Cartier and Mme Maynard who typed the greater part of the manuscript. Finally, we wish to thank Editions Dunod for their excellent typographic work Paris, November 1973. The English translation has been updated by incorporating recent work of the authors (Appendixes I and II) and their students. It has also benefited from the improvements suggested by the readers of the French edition.

PART ONE

Fundamentals of Convex Analysis

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CHAPTER I

Convex Functions

Introduction

This chapter assumes a basic knowledge of topological vector spaces. Moreover, we shall recall in Section 1 several fundamental aspects of this theory which will be constantly used in what follows. These reminders are in no way systematic and are centred on the notion of a convex set. We go on to consider convex functions (Sections 2-4) and their differentiability (Sections 5 and 6). All the vector spaces studied here are real. 1. CONVEX SETS AND THEIR SEPARATION 1.1. Convex sets Let V be a vector space over R. If u and v are two points of V, u and v are called the endpoints of the line-segment denoted by [u, v] where A set jtf