凸优化理论

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 Nonemptine~s 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 Common/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

《凸优化理论（影印版）》作者德梅萃·博赛克斯教授是优化理论的国际知名学者，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。《凸优化理论（影印版）》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。