Convexity and Well-Posed Problems

Roberto Lucchetti

ISBN 10: 0387287191 / ISBN 13: 9780387287195

Verlag: Springer New York Nov 2005, 2005

Beschreibung

Neuware - 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. TOC: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.

Bestandsnummer des Verkäufers 9780387287195

Bibliografische Details

• Titel: Convexity and Well-Posed Problems
• Verlag: Springer New York Nov 2005
• Erscheinungsjahr: 2005
• Sprache: Englisch
• ISBN-10: 0387287191
• ISBN-13: 9780387287195
• Einband: Buch
• Zustand: Neu
• Abmessungen: 244B x 160 H x 20 T Millimeter Breite: 244 mm, Höhe: 160 mm, Tiefe: 20 mm
• Gewicht: 559 Gramm

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Inhaltsangabe

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.

Von der hinteren Coverseite

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.