Convexity and Well-Posed Problems

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.