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Convexity and Well-Posed Problems

  • Textbook
  • © 2006

Overview

Authors:
  1. Roberto Lucchetti

    1. Dipto. Matematica, Politecnico di Milano, Milano, Italy

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  • Contains a chapter on hypertopologies (only one other book on this topic)
  • Author includes exercises, for use as a graduate text
  • Over 45 figures are included
  • Includes supplementary material: sn.pub/extras

Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Convex sets and convex functions: the fundamentals

    • Roberto Lucchetti
    Pages 1-19

  3. Continuity and Γ(X)

    • Roberto Lucchetti
    Pages 21-30

  4. The derivatives and the subdifferential

    • Roberto Lucchetti
    Pages 31-54

  5. Minima and quasi minima

    • Roberto Lucchetti
    Pages 55-77

  6. The Fenchel conjugate

    • Roberto Lucchetti
    Pages 79-97

  7. Duality

    • Roberto Lucchetti
    Pages 99-116

  8. Linear programming and game theory

    • Roberto Lucchetti
    Pages 117-137

  9. Hypertopologies, hyperconvergences

    • Roberto Lucchetti
    Pages 139-167

  10. Continuity of some operations between functions

    • Roberto Lucchetti
    Pages 169-183

  11. Well-posed problems

    • Roberto Lucchetti
    Pages 185-217

  12. Generic well-posedness

    • Roberto Lucchetti
    Pages 219-248

  13. More exercises

    • Roberto Lucchetti
    Pages 249-256

  14. Back Matter

    Pages 257-305
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Dipto. Matematica, Politecnico di Milano, Milano, Italy

    Roberto Lucchetti

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