Convexity and Well-Posed Problems / Edition 1

Convexity and Well-Posed Problems / Edition 1

by Roberto Lucchetti, R. Lucchetti, Lucchetti R.
     
 

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

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Overview

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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Product Details

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

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Table of Contents

1Convex sets and convex functions : the fundamentals1
2Continuity and [Gamma](X)21
3The derivatives and the subdifferential31
4Minima and quasi minima55
5The fenchel conjugate79
6Duality99
7Linear programming and game theory117
8Hypertopologies, hyperconvergences139
9Continuity of some operations between functions169
10Well-posed problems185
11Generic well-posedness219
12More exercises249
AFunctional analysis257
BTopology265
CMore game theory279
DSymbols, notations, definitions and important theorems291

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