Convexity and Well-Posed Problems

1. Front Matter

   Pages i-xiv

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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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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.

• Convexity
• calculus
• derivative
• differential equation
• functional analysis
• game theory
• linear optimization
• minimum
• optimization

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)

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

  Roberto Lucchetti

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