Fundamentals of Convex Analysis

Couverture
Springer Science & Business Media, 6 déc. 2012 - 259 pages
This book is an abridged version of our two-volume opus Convex Analysis and Minimization Algorithms [18], about which we have received very positive feedback from users, readers, lecturers ever since it was published - by Springer-Verlag in 1993. Its pedagogical qualities were particularly appreciated, in the combination with a rather advanced technical material. Now [18] hasa dual but clearly defined nature: - an introduction to the basic concepts in convex analysis, - a study of convex minimization problems (with an emphasis on numerical al- rithms), and insists on their mutual interpenetration. It is our feeling that the above basic introduction is much needed in the scientific community. This is the motivation for the present edition, our intention being to create a tool useful to teach convex anal ysis. We have thus extracted from [18] its "backbone" devoted to convex analysis, namely ChapsIII-VI and X. Apart from some local improvements, the present text is mostly a copy of the corresponding chapters. The main difference is that we have deleted material deemed too advanced for an introduction, or too closely attached to numerical algorithms. Further, we have included exercises, whose degree of difficulty is suggested by 0, I or 2 stars *. Finally, the index has been considerably enriched. Just as in [18], each chapter is presented as a "lesson", in the sense of our old masters, treating of a given subject in its entirety.
 

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Table des matières

Notation Elementary Results
2
Convex Sets Attached to a Convex Set
33
Projection onto Closed Convex Sets
46
Conical Approximations of Convex Sets
62
Exercises
70
Functional Operations Preserving Convexity
87
Local and Global Behaviour of a Convex Function
102
First and SecondOrder Differentiation 1 10
110
Sublinear Functions
123
The Support Function of a Nonempty Set
134
Correspondence Between Convex Sets and Sublinear Functions
143
Exercises
161
Subdifferentials of Finite Convex Functions
164
E Conjugacy in Convex Analysis
210
Bibliographical Comments
245
Droits d'auteur

Exercises
117

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Expressions et termes fréquents

Informations bibliographiques