Convex Analysis and Optimization

A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization.

This major new book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several Athena Scientific books: Nonlinear Programming, Network Optimization, Introduction to Linear Optimization, and Network Flows and Monotropic Optimization.

Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including:

1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems.

2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization.

3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions.

Among its features, the book:

** develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar

** provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality

** includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality

** describes dual optimization, the associated computational methods, and applications in linear, quadratic, and integer programming

** contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted on the internet (see the Athena Scientific web site)

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