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作者：Stephen Boyd，Lieven Vandenberghe 页数：727 出版社：Cambridge University Press |
《Convex Optimization_Stephen Boyd》介绍
凸优化问题经常出现在许多不同的领域。这本书提供了一个全面的介绍这一主题，并详细说明了如何解决这些问题可以用数字与极大的效率。这本书从凸集和函数的基本元素开始，然后描述各种类型的凸优化问题。然后讨论了对偶和近似技术，以及统计估计技术。然后介绍了各种几何问题，并详细讨论了无约束和约束极小化问题以及内点法。这本书的重点是认识凸优化问题，然后找到最合适的技术来解决它们。它包含了许多工作的例子和家庭作业练习，将吸引学生，研究人员和从业人员等领域的工程，计算机科学，数学，统计，金融和经济。
Convex optimization problems arise frequently in many different fields. This book provides a comprehensive introduction to the subject, and shows in detail how such problems can be solved numerically with great efficiency. The book begins with the basic elements of convex sets and functions, and then describes various classes of convex optimization problems. Duality and approximation techniques are then covered, as are statistical estimation techniques. Various geometrical problems are then presented, and there is detailed discussion of unconstrained and constrained minimization problems, and interior-point methods. The focus of the book is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. It contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance and economics.
《Convex Optimization_Stephen Boyd》目录
Contents
Preface xi
1 Introduction 1
1.1 Mathematical optimization 1
1.2 Least-squares and linear programming4
1.3 Convex optimization 7
1.4 Nonlinear optimization9
1.5 Outline. 11
1.6 Notation14
Bibliography. 16
I Theory 19
2 Convex sets 21
2.1 Affine and convex sets. 21
2.2 Some important examples . 27
2.3 Operations that preserve convexity 35
2.4 Generalized inequalities43
2.5 Separating and supporting hyperplanes46
2.6 Dual cones and generalized inequalities51
Bibliography. 59
Exercises . 60
3 Convex functions 67
3.1 Basic properties and examples67
3.2 Operations that preserve convexity 79
3.3 The conjugate function90
3.4 Quasiconvex functions. 95
3.5 Log-concave and log-convex functions104
3.6 Convexity with respect to generalized inequalities 108
Bibliography. 112
Exercises . 113
viii Contents
4 Convex optimization problems 127
4.1 Optimization problems127
4.2 Convex optimization 136
4.3 Linear optimization problems. 146
4.4 Quadratic optimization problems . 152
4.5 Geometric programming160
4.6 Generalized inequality constraints . 167
4.7 Vector optimization 174
Bibliography. 188
Exercises . 189
5 Duality 215
5.1 The Lagrange dual function 215
5.2 The Lagrange dual problem 223
5.3 Geometric interpretation . 232
5.4 Saddle-point interpretation 237
5.5 Optimality conditions. 241
5.6 Perturbation and sensitivity analysis. 249
5.7 Examples253
5.8 Theorems of alternatives . 258
5.9 Generalized inequalities264
Bibliography. 272
Exercises . 273
II Applications 289
6 Approximation and fitting 291
6.1 Norm approximation 291
6.2 Least-norm problems. 302
6.3 Regularized approximation 305
6.4 Robust approximation. 318
6.5 Function fitting and interpolation . 324
Bibliography. 343
Exercises . 344
7 Statistical estimation 351
7.1 Parametric distribution estimation 351
7.2 Nonparametric distribution estimation359
7.3 Optimal detector design and hypothesis testing . 364
7.4 Chebyshev and Chernoff bounds . 374
7.5 Experiment design . 384
Bibliography. 392
Exercises . 393
Contents ix
8 Geometric problems 397
8.1 Projection on a set 397
8.2 Distance between sets. 402
8.3 Euclidean distance and angle problems405
8.4 Extremal volume ellipsoids 410
8.5 Centering . 416
8.6 Classification 422
8.7 Placement and location432
8.8 Floor planning. 438
Bibliography. 446
Exercises . 447
III Algorithms 455
9 Unconstrained minimization 457
9.1 Unconstrained minimization problems457
9.2 Descent methods . 463
9.3 Gradient descent method . 466
9.4 Steepest descent method . 475
9.5 Newton’s method . 484
9.6 Self-concordance496
9.7 Implementation508
Bibliography. 513
Exercises . 514
10 Equality constrained minimization 521
10.1 Equality constrained minimization problems. 521
10.2 Newton’s method with equality constraints 525
10.3 Infeasible start Newton method531
10.4 Implementation542
Bibliography. 556
Exercises . 557
11 Interior-point methods 561
11.1 Inequality constrained minimization problems561
11.2 Logarithmic barrier function and central path562
11.3 The barrier method 568
11.4 Feasibility and phase I methods579
11.5 Complexity analysis via self-concordance . 585
11.6 Problems with generalized inequalities596
11.7 Primal-dual interior-point methods 609
11.8 Implementation615
Bibliography. 621
Exercises . 623
x Contents
Appendices 631
A Mathematical background 633
A.1 Norms. 633
A.2 Analysis637
A.3 Functions . 639
A.4 Derivatives . 640
A.5 Linear algebra. 645
Bibliography. 652
B Problems involving two quadratic functions 653
B.1 Single constraint quadratic optimization . 653
B.2 The S-procedure655
B.3 The field of values of two symmetric matrices656
B.4 Proofs of the strong duality results 657
Bibliography. 659
C Numerical linear algebra background 661
C.1 Matrix structure and algorithm complexity 661
C.2 Solving linear equations with factored matrices664
C.3 LU, Cholesky, and LDLT factorization668
C.4 Block elimination and Schur complements 672
C.5 Solving underdetermined linear equations . 681
Bibliography. 684
References 685
Notation 697
Index 701
目录
序言席
1引言1
1.1数学优化1
1.2最小二乘法和线性规划4
1.3凸优化7
1.4非线性优化9
1.5概述。11
1.6注释14
参考文献。16
I理论19
2凸集21
2.1仿射集和凸集。21
2.2一些重要的例子。27
2.3保持凸性的操作35
2.4广义不等式43
2.5分离和支撑超平面46
2.6对偶锥和广义不等式51
参考文献。59
练习。60
3凸函数67
3.1基本特性和示例67
3.2保持凸性的操作79
3.3共轭函数90
3.4拟凸函数。95
3.5对数凹函数和对数凸函数104
3.6关于广义不等式的凸性108
参考文献。112
练习。113
八、目录
4凸优化问题127
4.1优化问题127
4.2凸优化136
4.3线性优化问题。146
4.4二次优化问题。152
4.5几何编程160
4.6广义不等式约束。167
4.7矢量优化174
参考文献。188
练习。189
5二元性215
5.1拉格朗日对偶函数215
5.2拉格朗日对偶问题223
5.3几何解释。232
5.4鞍点解释237
5.5最优性条件。241
5.6扰动和灵敏度分析。249
5.7示例253
5.8备选方案定理。258
5.9广义不等式264
参考文献。272
练习。273
二、应用289
6近似和拟合291
6.1范数近似291
6.2最小范数问题。302
6.3正则化近似305
6.4稳健近似。318
6.5函数拟合与插值。324
参考文献。343
练习。344
7统计估计351
7.1参数分布估计351
7.2非参数分布估计359
7.3最佳检测器设计和假设检验。364
7.4切比雪夫和切尔诺夫界限。374
7.5实验设计。384
参考文献。392
练习。393
目录九
8几何问题397
8.1集合上的投影397
8.2组间距离。402
8.3欧氏距离和角度问题405
8.4极值体积椭球410
8.5定心。416
8.6分类422
8.7放置和位置432
8.8楼层规划。438
参考文献。446
练习。447
III算法455
9无约束最小化457
9.1无约束极小化问题457
9.2下降方法。463
9.3梯度下降法。466
9.4最速下降法。475
9.5牛顿法。484
9.6自我和谐496
9.7实施508
参考文献。513
练习。514
10等式约束最小化521
10.1等式约束最小化问题。521
10.2等式约束牛顿法525
10.3不可行启动牛顿法531
10.4实施542
参考文献。556
练习。557
11内点法561
11.1不等式约束极小化问题561
11.2对数势垒函数和中心路径562
11.3屏障法568
11.4可行性和第一阶段方法579
11.5通过自协调进行复杂性分析。585
11.6广义不等式问题596
11.7原始-对偶内点法609
11.8实施615
参考文献。621
练习。623
x目录
附录631
数学背景633
A.1规范。633
A.2分析637
A.3功能。639
A.4衍生品。640
A.5线性代数。645
参考文献。652
涉及两个二次函数的B问题653
B.1单约束二次优化。653
B.2 S程序655
B.3两个对称矩阵的值域656
B.4强对偶结果的证明657
参考文献。659
C数值线性代数背景661
C.1矩阵结构和算法复杂度661
C.2用因子矩阵求解线性方程664
C.3 LU、Cholesky和LDLT因子分解668
C.4块消除和Schur补充672
C.5求解欠定线性方程组。681
参考文献。684
参考文献685
符号697
索引701