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Convexity and Well-Posed Problems (CMS Books in Mathematics)

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Synopses & Reviews

Publisher Comments:

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.

Synopsis:

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.

Table of Contents

Preface.- Convex Sets and Convex Functions: the fundamentals.- Continuity and \Gamma (X).- The Derivatives and the Subdifferential.- Minima and Quasi Minima.- The Fenchel Conjugate.- Duality.- Linar Programming and Game Theory.- Hypertopologies, Hyperconvergences.- Continuity of Some Operations Between Functions.- Well-Posed Problems.- Generic Well-Posedness.- More Exercises.- Appendix A: Functional Analysis.- Appendix B: Topology.- Appendix C: More Game Theory.- Appendix D: Symbols, Notations, Definitions and Important Theorems.- References, Index.

Product Details

ISBN:
9780387287195
Author:
Lucchetti, Roberto
Publisher:
Springer
Author:
Lucchetti R.
Author:
Lucchetti, R.
Subject:
Calculus
Subject:
Linear Programming
Subject:
Mathematical Analysis
Subject:
Functional Analysis
Subject:
Convex functions
Subject:
Calculus of Variations and Optimal Control; Optimization
Subject:
Operations Research, Mathematical Programming
Subject:
Operations Research, Management Science
Subject:
Functional Analysis <P>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
Subject:
Mathematics-Computer
Copyright:
Edition Number:
1
Edition Description:
Book
Series:
CMS Books in Mathematics
Publication Date:
November 2005
Binding:
HARDCOVER
Language:
English
Illustrations:
Y
Pages:
319
Dimensions:
235 x 155 mm 616 gr

Related Subjects

Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Computer
Science and Mathematics » Mathematics » Functional Analysis
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry

Convexity and Well-Posed Problems (CMS Books in Mathematics) New Hardcover
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$99.25 In Stock
Product details 319 pages Springer - English 9780387287195 Reviews:
"Synopsis" by , 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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