Convexity and Well-Posed Problems / Edition 1

Convexity and Well-Posed Problems / Edition 1

by Roberto Lucchetti, R. Lucchetti, Lucchetti R.
     
 

ISBN-10: 0387287191

ISBN-13: 9780387287195

Pub. Date: 11/16/2005

Publisher: Springer New York

This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can

Overview

This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.

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)

Table of Contents

Convex sets and convex functions: the fundamentals.- Continuity and ?(X).- The derivatives and the subdifferential.- Minima and quasi minima.- The Fenchel conjugate.- Duality.- Linear programming and game theory.- Hypertopologies, hyperconvergences.- Continuity of some operations between functions.- Well-posed problems.- Generic well-posedness.- More exercises.

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