Convex analysis and optimization. (English) Zbl 1140.90001

Belmont, MA: Athena Scientific (ISBN 1-886529-45-0). 569 p. (2003).
Publisher’s description: 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 of our books: Nonlinear Programming (Athena Scientific, 1999), Network Optimization (Athena Scientific, 1998), Introduction to Linear Optimization (Athena Scientific, 1997), and Network Flows and Monotropic Optimization (Athena Scientific, 1998).
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: A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems; * 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; * 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.


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