Convexity and Well-Posed Problems
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Beschreibung
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.
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.
Contains a chapter on hypertopologies (only one other book on this topic) Author includes exercises, for use as a graduate text Over 45 figures are included Includes supplementary material: sn.pub/extras
Autor*in
Roberto Lucchetti
Themen in »Convexity and Well-Posed Problems«
Convexity calculus derivative differential equation functional analysis game theory linear optimization minimum optimization
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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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