凸优化算法
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天津西青
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作者(美)博塞卡斯(Dimitri P.Bertsekas) 著
出版社清华大学出版社
ISBN9787302430704
出版时间2016-05
装帧平装
开本其他
定价89元
货号1201304493
上书时间2024-10-12
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作者简介
博塞斯(Dimitri P.Bertsekas)教授是优化理论的靠前有名学者、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。博塞斯教授是一位多产作者，著有14本专著和教科书。
目录
Contents
1. Convex Optimization Models: An Overview . . . . . . p. 1
1.1. LagrangeDuality .......... .......... p.2
1.1.1. Separable Problems – Decomposition . . . . . . . . . p. 7
1.1.2. Partitioning .................... p.9
1.2. Fenchel Duality and Conic Programming . . . . . . . . . . p. 10
1.2.1. LinearConicProblems . . . . . . . . . . . . . . . p.15
1.2.2. Second Order Cone Programming . . . . . . . . . . . p. 17
1.2.3. Semide.nite Programming . . . . . . . . . . . . . . p. 22
1.3. AdditiveCostProblems . . . . . . . . . . . . . . . . . p.25
1.4. LargeNumberofConstraints . . . . . . . . . . . . . . . p.34
1.5. ExactPenalty Functions . . . . . . . . . . . . . . . . p.39
1.6. Notes,Sources,andExercises . . . . . . . . . . . . . . p.47
2. Optimization Algorithms: An Overview . . . . . . . . p. 53
2.1. IterativeDescentAlgorithms . . . . . . . . . . . . . . . p.55
2.1.1. Di.erentiable Cost Function Descent – Unconstrained . . . . Problems ..................... p.58
2.1.2. Constrained Problems – Feasible Direction Methods . . . p. 71
2.1.3. Nondi.erentiable Problems – Subgradient Methods . . . p. 78
2.1.4. Alternative Descent Methods . . . . . . . . . . . . . p. 80
2.1.5. IncrementalAlgorithms . . . . . . . . . . . . . . . p.83
2.1.6. Distributed Asynchronous Iterative Algorithms . . . . p. 104
2.2. ApproximationMethods . . . . . . . . . . . . . . . p.106
2.2.1. Polyhedral Approximation . . . . . . . . . . . . . p. 107
2.2.2. Penalty, Augmented Lagrangian, and Interior . . . . . . . PointMethods .................. p.108
2.2.3. Proximal Algorithm, Bundle Methods, and . . . . . . . . . TikhonovRegularization . . . . . . . . . . . . . . p.110
2.2.4. Alternating Direction Method of Multipliers . . . . . p. 111
2.2.5. Smoothing of Nondi.erentiable Problems . . . . . . p. 113
2.3. Notes,Sources,andExercises . . . . . . . . . . . . . p.119
3. SubgradientMethods . . . . . . . . . . . . . . . p.135
3.1. Subgradients of Convex Real-Valued Functions . . . . . . p. 136
iv
Contents
3.1.1. Characterization of the Subdi.erential . . . . . . . . p. 146
3.2. Convergence Analysis of Subgradient Methods . . . . . . p. 148
3.3. .-SubgradientMethods ................ p.162
3.3.1. Connection with Incremental Subgradient Methods . . p. 166
3.4. Notes,Sources,andExercises . . . . . . . . . . . . . . p.167
4. Polyhedral Approximation Methods . . . . . . . . . p. 181
4.1. Outer Linearization – Cutting Plane Methods . . . . . . p. 182
4.2. Inner Linearization – Simpli Decomposition . . . . . . p. 188
4.3. Duality of Outer and Inner Linearization . . . . . . . . . p. 194
4.4. Generalized Polyhedral Approximation . . . . . . . . . p. 196
4.5. Generalized Simpli Decomposition . . . . . . . . . . p. 209
4.5.1. Di.erentiableCostCase . . . . . . . . . . . . . . p.213
4.5.2. Nondi.erentiable Cost and Side Constraints . . . . . p. 213
4.6. Polyhedral Approximation for Conic Programming . . . . p. 217
4.7. Notes,Sources,andExercises . . . . . . . . . . . . . . p.228
5. ProximalAlgorithms . . . . . . . . . . . . . . . p.233
5.1. Basic Theory of Proximal Algorithms . . . . . . . . . . p. 234
5.1.1. Convergence ................... p.235
5.1.2. RateofConvergence. . . . . . . . . . . . . . . . p.239
5.1.3. Gradient Interpretation . . . . . . . . . . . . . . p. 246
5.1.4. Fixed Point Interpretation, Overrelaxation, . . . . . . . . . andGeneralization ................ p.248
5.2. DualProximalAlgorithms . . . . . . . . . . . . . . . p.256
5.2.1. Augmented Lagrangian Methods . . . . . . . . . . p. 259
5.3. Proximal Algorithms with Linearization . . . . . . . . . p. 268
5.3.1. Proximal Cutting Plane Methods . . . . . . . . . . p. 270
5.3.2. BundleMethods ................. p.272
5.3.3. Proximal Inner Linearization Methods . . . . . . . . p. 276
5.4. Alternating Direction Methods of Multipliers . . . . . . . p. 280
5.4.1. Applications in Machine Learning . . . . . . . . . . p. 286
5.4.2. ADMM Applied to Separable Problems . . . . . . . p. 289
5.5. Notes,Sources,andExercises . . . . . . . . . . . . . . p.293
6. Additional Algorithmic Topics . . . . . . . . . . . p. 301
6.1. GradientProjectionMethods . . . . . . . . . . . . . . p.302
6.2. Gradient Projection with Extrapolation . . . . . . . . . p. 322
6.2.1. An Algorithm with Optimal Iteration Complexity . . . p. 323
6.2.2. Nondi.erentiable Cost – Smoothing . . . . . . . . . p. 326
6.3. ProximalGradientMethods . . . . . . . . . . . . . . p.330
6.4. Incremental Subgradient Proximal Methods . . . . . . . p. 340
6.4.1. Convergence for M
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