凸优化理论(清华版双语教学用书影印版)

    本书作者德梅萃·博赛克斯教授是优化理论的国际著名学者、美国国家工程院院士，现任美国麻省理工学院电气工程与计算机科学系教授，曾在斯坦福大学工程经济系和伊利诺伊大学电气工程系任教，在优化理论、控制工程、通信工程、计算机科学等领域有丰富的科研教学经验，成果丰硕。博赛克斯教授是一位多产作者，著有14本专著和教科书。本书是作者在优化理论与方法的系列专著和教科书中的一本，自成体系又相互对应。主要内容分为两部分：凸分析和凸问题的对偶优化理论。

    Dimitri Bertsekas studied Mechanical and Electrical Engineering at the National Technical University of Athens, Greece, and obtained his Ph.D. in system science from the Massachusetts Institute of Technology.He has held faculty positions with the Engineering-Economic Systems Department, Stanford University, and the Electrical Engineering Department of the University of Illinois, Urbana. Since 1979 he has been teaching at the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology (M.I.T.), where he is currently McAfee Professor of Engineering.
    His teaching and research spans several.fields, including deterministic optimization, dynamic programming and stochastic control, large-scale and distributed computation, and data communication networks. He has authored or coauthored numerous research papers and fourteen books, several of which are used as textbooks in MIT classes, including "Nonlinear Programming," "Dynamic Programming and Optimal Control," "Data Networks," "Introduction to Probability," as well as the present book. He often consults with priwte industry and has held editorial positions in several journals.
    Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for .his book "Neuro-Dynamic Programming" (co-authored with John Tsitsiklis), the 2000 Greek National Award for Operations Research, and the 2001 ACC John R. Ragazzini Education Award. In 2001,he was elected to the United States National Academy of Engineering.

1. Basic Concepts of Convex Analysis
  1.1. Convex Sets and Functions
  1.1.1. Convex Functions
  1.1.2. Closedness and Semicontinuity
  1.1.3. Operations with Convex Functions
  1.1.4. Characterizations of Differentiable Convex Functions
  1.2. Convex and Affine Hulls
  1.3. Relative Interior and Closure
  1.3.1. Calculus of Relative Interiors and Closures
  1.3.2. Continuity of Convex Functions
  1.3.3. Closures of Functions
  1.4. Recession Cones
  1.4.1. Directions of Recession of a Convex Function
  1.4.2. Nonemptiness of Intersections of Closed Sets
  1.4.3. Closedness Under Linear Transformations
  1.5. Hyperplanes
  1.5.1. Hyperplane Separation
  1.5.2. Proper Hyperplane Separation
  1.5.3. Nonvertical Hyperplane Separation
  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
  2.3.1. Polyhedral Cones and Farkas'' Lemma
  2.3.2. Structure of Polyhedral Sets
  2.3.3. Polyhedral 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 Comnon/Max Crossing Duality
  4.2. Some Special Cases
  4.2.1. Connection to Conjugate Convex Functions
  4.2.2. General Optimization Duality
  4.2.3. Optimization with Inequality Constraints
  4.2.4. Augmented Lagrangian Duality
  4.2.5. Minimax Problems
  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
  5.3.1. Strong Duality Theorem- Inequality Constraints
  5.3.2. Optimality Conditions
  5.3.3. Partially Polyhedral Constraints
  5.3.4. Duality and Existence of Optimal Primal Solutions
  5.3:5: Fenchel Duality
  5.3.6. Conic Duality
  5.4. Subgradients and Optimality Conditions
  5.4.1. Subgradients of Conjugate Functions
  5.4.2. Subdifferential Calculus
  5.4.3. Optimality Conditions
  5.4.4. Directional Derivatives
  5.5. Minimax Theory
  5.5.1. Minimax Duality Theorems
  5.5.2. Saddle Point Theorems
  5.6. Theorems of the Alternative
  5.7. Nonconvex Problems
  5.7.1. Duality Gap in Separable Problems
  5.7.2. Duality Gap in Minimax Problems
Appendix A: Mathematical Background
Notes and Sources
Supplementary Chapter 6 on Convex Optimization Algorithms