Fundamentals of Convex Analysis
This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306). It presents an introduction to the basic concepts in convex analysis and a study of convex minimization problems (with an emphasis on numerical algorithms). The "backbone" of bot volumes was extracted, some material deleted which was deemed too advanced for an introduction, or too closely attached to numerical algorithms....
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This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched. The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.
PrefaceIntroduction: Notation, Elementary Results1AConvex Sets191Generalities192Convex Sets Attached to a Convex Set333Projection onto Closed Convex Sets464Separation and Applications515Conical Approximations of Convex Sets62BConvex Functions731Basic Definitions and Examples732Functional Operations Preserving Convexity873Local and Global Behaviour of a Convex Function1024First- and Second-Order Differentiation110CSublinearity and Support Functions1211Sublinear Functions1232The Support Function of a Nonempty Set1343Correspondence Between Convex Sets and Sublinear Functions143DSubdifferentials of Finite Convex Functions1631The Subdifferential: Definitions and Interpretations1642Local Properties of the Subdifferential1733First Examples1804Calculus Rules with Subdifferentials1835Further Examples1946The Subdifferential as a Multifunction199EConjugacy in Convex Analysis2091The Convex Conjugate of a Function2112Calculus Rules on the Conjugacy Operation2223Various Examples2334Differentiability of a Conjugate Function237Bibliographical Comments245The Founding Fathers of the Discipline249References251Index253
