《凸優化理論》作者德梅萃，博賽剋斯教授是優化理論的國際著名學者、美國國傢工程院院士，現任美國麻省理工學院電氣工程與計算機科學係教授，曾在斯坦福大學工程經濟係和伊利諾伊大學電氣工程係任教，在優化理論、控製工程、通信工程、計算機科學等領域有豐富的科研教學經驗，成果豐碩。博賽剋斯教授是一位多産作者，著有14本專著和教科書。《凸優化理論》是作者在優化理論與方法的係列專著和教科書中的一本，自成體係又相互對應。主要內容分為兩部分：凸分析和凸問題的對偶優化理論。

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 afiine 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 mon/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 algorithm