Convex Optimization Theory
Dimitri P Bertsekas
158 x 240 mm
Year of Publishing
Territorial Rights
Universities Press

The book Convex Optimization Theory provides an insightful, concise and rigorous treatment of the basic theory of convex sets and functions in finite dimensions and the analytical/geometrical foundations of convex optimization and duality theory. The convexity theory is developed first in a simple accessible manner using easily visualized proofs. The focus then shifts to a transparent geometrical line of analysis to develop the fundamental duality between descriptions of convex sets and functions in terms of points and in terms of hyperplanes. Finally, convexity theory and abstract duality are applied to problems of constrained optimization, Fenchel and conic duality and game theory to develop the sharpest possible duality results within a highly visual geometric framework.

The Indian edition of the book alone carries a supplementary chapter containing the most popular convex optimization algorithms and some of the new optimization algorithms otherwise available at .

Key features:

  • Rigorous and comprehensive development of the theory of convex sets and functions in the classical tradition of Fenchel and Rockafellar
  • A geometric and highly visual treatment of convex optimization problems including duality, existence of solutions, and optimality conditions
  • An insightful and comprehensive presentation of minimax theory and zero sum games and its connection with duality
  • Contains many examples and illustrations in the text
  • Inclusion of many examples, illustrations, exercises with complete solutions and a supplementary chapter on the most popular convex optimization algorithms
  • Useable as a standalone text for a theoretically-oriented class on convex analysis and optimization, or as a theoretical supplement to either an applications/convex optimization models class or a nonlinear programming class
Dimitri P Bertsekas is McAfee Professor of Engineering at the Massachusetts Institute of Technology and a member of the prestigious United States National Academy of Engineering. He is the recipient of the 2001 A. R. Raggazini ACC education award and the 2009 INFORMS expository writing award.
1. Basic Concepts of Convex Analysis
1.1. Convex Sets and Functions
1.2. Convex and Affine Hulls
1.3. Relative Interior and Closure
1.4. Recession Cones
1.5. Hyperplanes
1.6. Conjugate Functions
1.7. Summary
2. Basic Concepts of Polyhedral Convexity
2.1. Extreme Points
2.2. Polar Cones
2.3. Polyhedral Sets and Functions
2.4. Polyhedral Aspects of Optimization
3. Basic Concepts of Convex Optimization
3.1. Constrained Optimization
3.2. Existence of Optimal Solutions
3.3. Partial Minimization of Convex Functions
3.4. Saddle Point and Minimax Theory
4. Geometric Duality Framework
4.1. Min Common/Max Crossing Duality
4.2. Some Special Cases
4.3. Strong Duality Theorem
4.4. Existence of Dual Optimal Solutions
4.5. Duality and Polyhedral Convexity
4.6. Summary
5. Duality and Optimization
5.1. Nonlinear Farkas’ Lemma
5.2. Linear Programming Duality
5.3. Convex Programming Duality
5.4. Subgradients and Optimality Conditions
5.5. Minimax Theory
5.6. Theorems of the Alternative
5.7. Nonconvex Problems
Appendix A: Mathematical Background
Notes and Sources
Supplementary Chapter 6 on Convex Optimization Algorithms
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