Convex Analysis and Minimization Algorithms I

Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those adapted to non-differentiable functions, provide an immediate application of convex analysis to various fields related to optimization and operations research. These two topics, making up the title of the book, reflect the two origins of authors, who belong respectively to the academic world and to that of applications. The approach of their book is very comprehensive without being encyclopaedic: the emphasis is on introducing readers in a gradual and digestible manner to the concepts of convex analysis, their interlinking and their implications, with algorithmic ideas worked in.
Theory is interspersed with application and vice versa; illustrative numerical results are given, and over 170 pictures illustrate and support geometric intuition. Throughout the book, ample comments help the reader further to master the concepts and methods, and to understand the motivations, the difficulties, and the relative significance of results.

In the second printing of these two volumes 305 and 306 the authors have made various local improvements and corrections, and updated the bibliography. The index has been considerably augmented and refined.

Zusammenfassung
From the reviews: "... The book is very well written, nicely illustrated, and clearly understandable even for senior undergraduate students of mathematics... Throughout the book, the authors carefully follow the recommendation by A. Einstein: 'Everything should be made as simple as possible, but not simpler.'"

Inhalt
Table of Contents Part I.- I. Convex Functions of One Real Variable.- II. Introduction to Optimization Algorithms.- III. Convex Sets.- IV. Convex Functions of Several Variables.- V. Sublinearity and Support Functions.- VI. Subdifferentials of Finite Convex Functions.- VII. Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory.- VIII. Descent Theory for Convex Minimization: The Case of Complete Information.- Appendix: Notations.- 1 Some Facts About Optimization.- 2 The Set of Extended Real Numbers.- 3 Linear and Bilinear Algebra.- 4 Differentiation in a Euclidean Space.- 5 Set-Valued Analysis.- 6 A Bird's Eye View of Measure Theory and Integration.- Bibliographical Comments.- References.