Convexity and Well-Posed Problems - Lucchetti, Roberto
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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…mehr

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Produktbeschreibung
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.
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Autorenporträt
Roberto Lucchetti, Politecnico di Milano, Italy
Rezensionen
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)