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), 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. TOC:Introduction: Notation, Elementary Results.- Convex Sets: Generalities; Convex Sets Attached to a Convex Set; Projection onto Closed Convex Sets; Separation and Applications; Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples; Functional Operations Preserving Convexity; Local and Global Behaviour of a Convex Function; First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions; The Support Function of a Nonempty Set; Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations; Local Properties of the Subdifferential; First Examples; Calculus Rules with Subdifferentials; Further Examples; The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function; Calculus Rules on the Conjugacy Operation; Various Examples; Differentiability of a Conjugate Function.