Convex Analysis and Optimization

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Athena Scientific, Mar 1, 2003 - Mathematics - 560 pages

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

The 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: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997).

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:

a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar

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

c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality

d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming

e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html




 

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Contents

Basic Convexity Concepts
1
Convexity and Optimization
83
Polyhedral Convexity
165
Subgradients and Constrained Optimization
221
Lagrange Multipliers
269
Lagrangian Duality
345
Conjugate Duality
421
Dual Computational Methods
455
References
517
Index
529

Common terms and phrases

About the author (2003)

Dimitri P. Bertsekas is Fulton Professor of Computational Decision Making, at Arizona State University, and McAfee Professor of Engineering at the Massachusetts Institute of Technology. He is a winner of many awards and a member of the prestigious United States National Academy of Engineering

Angelia Nedic is a Professor and Electrical and Computer Engineering at Arizona State University

Asuman Ozdaglar is a professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology

Bibliographic information