A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. Among its special features, the book: 1) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar 2) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality 3) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality 4) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming 5) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted on the internet. From the preface:This book focuses on the theory of convex sets and functions, and its connections with a number of topics that span a broad range from continuous to discrete optimization. These topics include Lagrange multiplier theory, Lagrangian and conjugate/Fenchel duality, minimax theory, and nondifferentiable optimization.The book evolved from a set of lecture notes for a graduate course at M.I.T. It is widely recognized that, aside from being an eminently useful subject in engineering, operations research, and economics, convexity is an excellent vehicle for assimilating some of the basic concepts of real analysis within an intuitive geometrical setting. Unfortunately, the subject's coverage in academic curricula is scant and incidental. We believe that at least part of the reason is the shortage of textbooks that are suitable for classroom instruction, particularly for nonmathematics majors. We have therefore tried to make convex analysis accessible to a broader audience by emphasizing its geometrical character, while maintaining mathematical rigor. We have included as many insightful illustrations as possible, and we have used geometric visualization as a principal tool for maintaining the students' interest in mathematical proofs.Our treatment of convexity theory is quite comprehensive, with all major aspects of the subject receiving substantial treatment. The mathematical prerequisites are a course in linear algebra and a course in real analysis in finite dimensional spaces (which is the exclusive setting of the book). A summary of this material, without proofs, is provided in Section 1.1. The coverage of the theory has been significantly extended in the exercises, which represent a major component of the book. Detailed solutions of all the exercises (nearly 200 pages) are internet-posted in the book's www pageSome of the exercises may be attempted by the reader without looking at the solutions, while others are challenging but may be solved by the advanced reader with the assistance of hints. Still other exercises represent substantial theoretical results, and in some cases include new and unpublished research. Readers and instructors should decide for themselves how to make best use of the internet-posted solutions.An important part of our approach has been to maintain a close link between the theoretical treatment of convexity and its application to optimization.