Convex Analysis and Minimization Algorithms I: Fundamentals (Grundlehren der mathematischen Wissenschaften Book 305)

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 the authors, who belong respectively to the academic world and to that of applications. Part I can be used as an introductory textbook (as a basis for courses, or for self-study); Part II continues this at a higher technical level and is addressed more to specialists, collecting results that so far have not appeared in books. Front Matter....Pages I-XVII Convex Functions of One Real Variable....Pages 1-46 Introduction to Optimization Algorithms....Pages 47-86 Convex Sets....Pages 87-141 Convex Functions of Several Variables....Pages 143-193 Sublinearity and Support Functions....Pages 195-236 Subdifferentials of Finite Convex Functions....Pages 237-289 Constrained Convex Minimization Problems: Minimality Conditions, Elements of Duality Theory....Pages 291-341 Descent Theory for Convex Minimization: The Case of Complete Information....Pages 343-384 Back Matter....Pages 385-420