Convexity and Well-Posed Problems (Paperback)

Roberto Lucchetti

ISBN 10: 1441921117 / ISBN 13: 9781441921116

Published by Springer-Verlag New York Inc., New York, NY, 2010

About this Item

Paperback. 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. 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. 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. 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. Shipping may be from multiple locations in the US or from the UK, depending on stock availability.

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Bibliographic Details

• Title: Convexity and Well-Posed Problems (Paperback)
• Publisher: Springer-Verlag New York Inc., New York, NY
• Publication Date: 2010
• Language: English
• ISBN 10: 1441921117
• ISBN 13: 9781441921116
• Binding: Paperback
• Condition: new

About this title

Synopsis

This book studies the problems of stability and well-posedness, in the convex case. Stability means the basic parameters of a minimum problem do not vary much if we slightly change the initial data, while well-posedness means 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 book contains a 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.

From the Back Cover

Intended for graduate students especially in mathematics, physics, and

economics, this book deals with the study of convex functions and of

their behavior from the point of view of stability with respect to

perturbations. The primary goal is the study of the problems of

stability and well-posedness, in the convex case. Stability means the

basic parameters of a minimum problem do not vary much if we slightly

change the initial data. 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

both functions and of sets.

The book includes a discussion of numerous topics, including:

* hypertopologies, ie, topologies on the closed subsets of a metric space;

* duality in linear programming problems, via cooperative game theory;

* the Hahn-Banach theorem, which is a fundamental tool for the study of convex functions;

* questions related to convergence of sets of nets;

* genericity and porosity results;

* algorithms for minimizing a convex function.

In order to facilitate use as a textbook, the author has included a

selection of examples and exercises, varying in degree of difficulty.

Robert Lucchetti is Professor of Mathematics at Politecnico di Milano. He has taught this material to graduate students at his own university, as well as the Catholic University of Brescia, and the University of Pavia.