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 thecorresponding 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.
Inhaltsverzeichnis
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.
