Convex Ana lysis General Vector Spaces

388 Pages·2007·18.38 MB·English

C o n v ex
A na l y s is
G e n e r al
V e c t or
S p a c es
C Zalinescu
World Scientific
C o n v ex
Analysis
i n
G e n e r al
V e c t or
S p a c es
This page is intentionally left blank
C o n v ex
A n a l y s is
1 n
G e n e r al
V e c t or
S p a c es
C Zalinescu
Faculty of Mathematics
University "Al. I. Cuza" lasi, Romania
B% World Scientific
• NNeeww JJeerrsseeyy •• LLoonnddoonn •• SSiinnggaappoorree •• Hong Kong
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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.
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Printed in Singapore by Uto-Print
To the memory of my parents
Casandra and Vasile Zalinescu
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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