Optimization is a rich and thriving mathematical discipline. Properties of minimizers and maximizers of functions rely intimately on a wealth of techniques from mathematical analysis, including tools from calculus and its generalizations, topological notions, and more geometric ideas. The powerful and elegant language of convex analysis unifies much of this theory. Hence our aim of writing a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. For students of optimization and analysis, there is great benefit to blurring the distinction between the two disciplines.
Good reference works on finite-dimensional convex analysis already exist.
Rockafellar's classic Convex Analysis has been indispensable and ubiquitous since the 1970's, and a more general sequel with Wets, Variational Analysis, appeared recently. Hiriart-Urruty and Lemarechal's Convex Analysis and Minimization Algorithms is a comprehensive but gentler introduction. Our goal is not to supplant these works, but on the contrary to promote them, and thereby to motivate future researchers. This book aims to make converts. We try to be succinct rather than systematic, avoiding becoming bogged down in technical details.