Convex Ana lysis General Vector Spaces C Zalinescu World Scientific Convex Analysis i n General Vector Spaces This page is intentionally left blank Convex Analysis 1 n General Vector Spaces C Zalinescu Faculty of Mathematics University "Al. I. Cuza" lasi, Romania B% World Scientific • New Jersey • London • Singapore •• Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Zalinescu, C, 1952- Convex analysis in general vector spaces / C. Zalinescu p. cm. Includes bibliographical references and index. ISBN 9812380671 (alk. paper) 1. Convex functions. 2. Convex sets. 3. Functional analysis. 4. Vector spaces. I. Title. QA331.5.Z34 2002 2002069000 515'.8-dc21 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by Uto-Print To the memory of my parents Casandra and Vasile Zalinescu This page is intentionally left blank Preface The text of this book has its origin in a course we delivered to students for Master Degree at the Faculty of Mathematics of the University "Al. I. Cuza" Ia§i, Romania. One can ask if another book on Convex Analysis is needed when there are many excellent books dedicated to this discipline like those written by R.T. Rockafellar (1970), J. Stoer and C. Witzgall (1970), J.-B. Hiriart- Urruty and C. Lemarechal (1993), J.M. Borwein and A. Lewis (2000) for finite dimensional spaces and by P.-J. Laurent (1972), I. Ekeland and R. Temam (1974), R.T. Rockafellar (1974), A.D. Ioffe and V.M. Tikhomirov (1974), V. Barbu and Th. Precupanu (1978, 1986), J.R. Giles (1982), R.R. Phelps (1989, 1993), D. Aze (1997) for infinite dimensional spaces. We think that such a book is necessary for taking into consideration new results concerning the validity of the formulas for conjugates and sub- differentials of convex functions constructed from other convex functions by operations which preserve convexity, results obtained in the last 10-15 years. Also, there are classes of convex functions like uniformly convex, uniformly smooth, well behaving, well conditioned functions that are not studied in other books. Characterizations of convex functions using other types of derivatives or subdifferentials than usual directional derivatives or Fenchel subdifferential are quite recent and deserve being included in a book. All these themes are treated in this book. We have chosen for studying convex functions the framework of locally convex spaces and the most general conditions met in the literature; even when restricted to normed vector spaces many results are stated in more general conditions than the corresponding ones in other books. To make vii Vlll Preface this possible, in the first chapter we introduce several interiority and closed- ness conditions and state two strong open mapping theorems. In the second chapter, besides the usual characterizations and properties of convex functions we study new classes of such functions: cs-convex, cs- closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions, respectively; note that the classes of li-convex and lcs-closed functions have very good stability properties. This will give the possibility to have a rich calculus for the conjugate and the subdifferential of convex functions under mild conditions. In obtaining these results we use the method of perturbation functions introduced by R.T. Rockafellar. The main tool is the fundamental duality formula which is stated under very general conditions by using open mapping theorems. The framework of the third chapter is that of infinite dimensional normed vector spaces. Besides some classical results in convex analysis we give characterizations of convex functions using abstract subdifferentials and study differentiability of convex functions. Also, we introduce and study well-conditioned convex functions, uniformly convex and uniformly smooth convex functions and their applications to the study of the geometry of Banach spaces. In connection with well-conditioned functions we study the sets of weak sharp minima, well-behaved convex functions and global error bounds for convex inequality systems. The chapter ends with the study of monotone operators by using convex functions. Every chapter ends with exercises and bibliographical notes; there are more than 80 exercises. The statements of the exercises are generally ex­ tracted from auxiliary results in recent articles, but some of them are known results that deserve being included in a textbook, but which do not fit very well our aims. The complete solutions of all exercises are given. The book ends with an index of terms and a list of symbols and notations. Even if all the results with the exception of those in the first section are given with their complete proofs, for a successful reading of the book a good knowledge of topology and topological vector spaces is recommended. Finally I would like to thank Prof. J.-P. Penot and Prof. A. Gopfert for reading the manuscript, for their remarks and encouragements. C. Zalinescu March 1, 2002 Ia§i, Romania Contents Preface vii Introduction xi Chapter 1 Preliminary Results on Functional Analysis 1 1.1 Preliminary notions and results 1 1.2 Closedness and interiority notions 9 1.3 Open mapping theorems 19 1.4 Variational principles 29 1.5 Exercises 34 1.6 Bibliographical notes 36 Chapter 2 Convex Analysis in Locally Convex Spaces 39 2.1 Convex functions 39 2.2 Semi-continuity of convex functions 60 2.3 Conjugate functions 75 2.4 The subdifferential of a convex function 79 2.5 The general problem of convex programming 99 2.6 Perturbed problems 106 2.7 The fundamental duality formula 113 2.8 Formulas for conjugates and e-subdifferentials, duality relations and optimality conditions 121 2.9 Convex optimization with constraints 136 2.10 A minimax theorem 143 2.11 Exercises 146 2.12 Bibliographical notes 155 ix x Contents Chapter 3 Some Results and Applications of Convex Analy­ sis in Normed Spaces 159 3.1 Further fundamental results in convex analysis 159 3.2 Convexity and monotonicity of subdifferentials 169 3.3 Some classes of functions of a real variable and differentiability of convex functions 188 3.4 Well conditioned functions 195 3.5 Uniformly convex and uniformly smooth convex functions . 203 3.6 Uniformly convex and uniformly smooth convex functions on bounded sets 221 3.7 Applications to the geometry of normed spaces 226 3.8 Applications to the best approximation problem 237 3.9 Characterizations of convexity in terms of smoothness 243 3.10 Weak sharp minima, well-behaved functions and global error bounds for convex inequalities 248 3.11 Monotone multifunctions 269 3.12 Exercises 288 3.13 Bibliographical notes 292 Exercises - Solutions 297 Bibliography 349 Index 359 Symbols and Notations 363 Introduction The primary aim of this book is to present the conjugate and subdifferential calculus using the method of perturbation functions in order to obtain the most general results in this field. The secondary aim is to give important applications of this calculus and of the properties of convex functions. Such applications are: the study of well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, best approximation prob­ lems, characterizations of convexity, the study of the sets of weak sharp minima, well-behaved functions and the existence of global error bounds for convex inequalities, as well as the study of monotone multifunctions by using convex functions. The method of perturbation functions is based on the "fundamental duality theorem" which says that under certain conditions one has inf $(i, 0) = max_ ( - $*(0,y*)). (FDF) For many problems in convex optimization one can associate a useful perturbation function. We give here four examples; see [Rockafellar (1974)] for other interesting ones. Example 1 (Convex programming; see Section 2.9) Let f,9i,---,9n '• X ->• E be proper convex functions with dom/ n C\7=i domgi ^ 0. The problem of minimizing f(x) over the set of those x S X satisfying gi{x) < 0 for alii = 1,..., n is equivalent to the minimization of $(x, 0) for x 6 X, where $:IxF4l, *{x,y):={ f{x) * *(x) <yiV 1 < i <n, -{ 1 +00 otherwise, Xll Introduction and Y := En; the element y* obtained from the right-hand side of (FDF) will furnish the Lagrange multipliers. Example 2 (Control problems) Let F : X xY -±Rbe a, proper convex function and A : X —> Y a linear operator. A control problem (in its abstract form) is to minimize F(x,y) for x € X and y = Ax + yo- The perturbation function to be considered is $ : X x Y —> M. defined by $(x,y) := F{x,Ax + y0+y). Example 3 (Semi-infinite programming) We are as in Example 1 but {1,... ,n} is replaced by a general nonempty set I\ In this case Y = W and $(x,y) := f(x) if gt(x) < yi for all i £ /, $(x,y) := co otherwise. Formula (FDF), or more precisely the Fenchel-Rockafellar duality for­ mula, can also be used for deriving results similar to that in the next ex­ ample.