Convexity and Well-Posed Problems / Edition 1

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This book deals with the study of convex functions and of their behavior from the point of view of stability with respect to perturbations. Convex functions are considered from the modern point of view that underlines the geometrical aspect: thus a function is defined as convex whenever its graph is a convex set. A primary goal of this book is to study the problems of stability and well-posedness, in the convex case. Stability means that the basic parameters of a minimum problem do not vary much if we slightly change the initial data. On the other hand, well-posedness means that points with values close to the value of the problem must be close to actual solutions. In studying this, one is naturally led to consider perturbations of functions and of sets. This approach fits perfectly with the idea of regarding functions as sets. Thus the second part of the book starts with a short, yet rather complete, overview of the so-called hypertopologies, i.e. topologies in the closed subsets of a metric space. While there exist numerous classic texts on the issue of stability, there only exists one book on hypertopologies [Beer 1993]. The current book differs from Beer's in that it contains a much more condensed explication of hypertopologies and is intended to help those not familiar with hypertopologies learn how to use them in the context of optimization problems.

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Editorial Reviews

From the Publisher

From the reviews:

"In this book the author focuses on the study of convex functions and their properties under perturbations of data. In particular, he illustrates the ideas of stability and well-posedness and the connections between them. … This book is intended for graduate students and researchers especially in mathematics, physics and economics; to facilitate its use as a textbook, the author has included many exercises and examples of different levels of difficulty." (Davide La Torre, Mathematical Reviews, Issue 2006 h)

"This book studies convex functions in Banach spaces and the stable behavior under perturbations of the optimization problems associated to them. … An interesting feature of the book is the inclusion of some topics, like elements of game theory, hypertopologies and genericity of well-posedness, not usually found in textbooks devoted to convexity and optimization. … several useful examples, comments and remarks scattered throughout, and over 120 exercises of varying level difficulty. This book is suitable for graduate courses on convex optimization from a mathematical standpoint." (Tullio Zolezzi, Zentralblatt MATH, Vol. 1106 (8), 2007)

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Product Details

  • ISBN-13: 9780387287195
  • Publisher: Springer New York
  • Publication date: 11/16/2005
  • Series: CMS Books in Mathematics Series
  • Edition description: 2006
  • Edition number: 1
  • Pages: 305
  • Product dimensions: 6.20 (w) x 9.30 (h) x 0.80 (d)

Table of Contents

1 Convex sets and convex functions : the fundamentals 1
2 Continuity and [Gamma](X) 21
3 The derivatives and the subdifferential 31
4 Minima and quasi minima 55
5 The fenchel conjugate 79
6 Duality 99
7 Linear programming and game theory 117
8 Hypertopologies, hyperconvergences 139
9 Continuity of some operations between functions 169
10 Well-posed problems 185
11 Generic well-posedness 219
12 More exercises 249
A Functional analysis 257
B Topology 265
C More game theory 279
D Symbols, notations, definitions and important theorems 291
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