Convexity and Well-Posed Problems

Abstract

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. 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.