Dedication v
Biographies of the authors vii
Preface xv
Abbreviations xix
1. THE OPTIMIZATION PROBLEM 1
1.1 Introduction 1
1.2 The Basic Optimization Problem 4
1.3 General Structure of Optimization Algorithms 8
1.4 Constraints 10
1.5 The Feasible Region 17
1.6 Branches of Mathematical Programming 22
References 24
Problems 25
2. BASIC PRINCIPLES 27
2.1 Introduction 27
2.2 Gradient Information 27
2.3 The Taylor Series 28
2.4 Types of Extrema 31
2.5 Necessary and Sufficient Conditions for
Local Minima and Maxima 33
2.6 Classification of Stationary Points 40
2.7 Convex and Concave Functions 51
2.8 Optimization of Convex Functions 58
References 60
Problems 60
3. GENERAL PROPERTIES OF ALGORITHMS 65
3.1 Introduction 65
3.2 An Algorithm as a Point-to-Point Mapping 65
3.3 An Algorithm as a Point-to-Set Mapping 67
3.4 Closed Algorithms 68
3.5 Descent Functions 71
3.6 Global Convergence 72
3.7 Rates of Convergence 76
References 79
Problems 79
4. ONE-DIMENSIONAL OPTIMIZATION 81
4.1 Introduction 81
4.2 Dichotomous Search 82
4.3 Fibonacci Search 85
4.4 Golden-Section Search 92
4.5 Quadratic Interpolation Method 95
4.6 Cubic Interpolation 99
4.7 The Algorithm of Davies, Swann, and Campey 101
4.8 Inexact Line Searches 106
References 114
Problems 114
5. BASIC MULTIDIMENSIONAL GRADIENT METHODS 119
5.1 Introduction 119
5.2 Steepest-Descent Method 120
5.3 Newton Method 128
5.4 Gauss-Newton Method 138
References 140
Problems 140
6. CONJUGATE-DIRECTION METHODS 145
6.1 Introduction 145
6.2 Conjugate Directions 146
6.3 Basic Conjugate-Directions Method 149
6.4 Conjugate-Gradient Method 152
6.5 Minimization of Nonquadratic Functions 157
6.6 Fletcher-Reeves Method 158
6.7 Powell's Method 159
6.8 Partan Method 168
References 172
XI
Problems 172
7. QUASI-NEWTON METHODS 175
7.1 Introduction 175
7.2 The Basic Quasi-Newton Approach 176
7.3 Generation of Matrix Sk 177
7.4 Rank-One Method 181
7.5 Davidon-Fletcher-Powell Method 185
7.6 Broyden-Fletcher-Goldfarb-Shanno Method 191
7.7 Hoshino Method 192
7.8 The Broyden Family 192
7.9 The Huang Family 194
7.10 Practical Quasi-Newton Algorithm 195
References 199
Problems 200
8. MINIMAX METHODS 203
8.1 Introduction 203
8.2 Problem Formulation 203
8.3 Minimax Algorithms 205
8.4 Improved Minimax Algorithms 211
References 228
Problems 228
9. APPLICATIONS OF UNCONSTRAINED OPTIMIZATION 231
9.1 Introduction 231
9.2 Point-Pattern Matching 232
9.3 Inverse Kinematics for Robotic Manipulators 237
9.4 Design of Digital Filters 247
References 260
Problems 262
10. FUNDAMENTALS OF CONSTRAINED OPTIMIZATION 265
10.1 Introduction 265
10.2 Constraints 266
Xll
10.3 Classification of Constrained Optimization Problems 273
10.4 Simple Transformation Methods 277
10.5 Lagrange Multipliers 285
10.6 First-Order Necessary Conditions 294
10.7 Second-Order Conditions 302
10.8 Convexity 308
10.9 Duality 311
References 312
Problems 313
11. LINEAR PROGRAMMING PART I: THE SIMPLEX METHOD 321
11.1 Introduction 321
11.2 General Properties 322
11.3 Simplex Method 344
References 368
Problems 368
12. LINEAR PROGRAMMING PART II:
INTERIOR-POINT METHODS 373
12.1 Introduction 373
12.2 Primal-Dual Solutions and Central Path 374
12.3 Primal Affine-Scaling Method 379
12.4 Primal Newton Barrier Method 383
12.5 Primal-Dual Interior-Point Methods 388
References 402
Problems 402
13. QUADRATIC AND CONVEX PROGRAMMING 407
13.1 Introduction 407
13.2 Convex QP Problems with Equality Constraints 408
13.3 Active-Set Methods for Strictly Convex QP Problems 411
13.4 Interior-Point Methods for Convex QP Problems 417
13.5 Cutting-Plane Methods for CP Problems 428
13.6 Ellipsoid Methods 437
References 443
Xlll
Problems 444
14. SEMIDEFINITE AND SECOND-ORDER CONE
PROGRAMMING 449
14.1 Introduction 449
14.2 Primal and Dual SDP Problems 450
14.3 Basic Properties of SDP Problems 455
14.4 Primal-Dual Path-Following Method 458
14.5 Predictor-Corrector Method 465
14.6 Projective Method of Nemirovski and Gahinet 470
14.7 Second-Order Cone Programming 484
14.8 A Primal-Dual Method for SOCP Problems 491
References 496
Problems 497
15. GENERAL NONLINEAR OPTIMIZATION PROBLEMS 501
15.1 Introduction 501
15.2 Sequential Quadratic Programming Methods 501
15.3 Modified SQP Algorithms 509
15.4 Interior-Point Methods 518
References 528
Problems 529
16. APPLICATIONS OF CONSTRAINED OPTIMIZATION 533
16.1 Introduction 533
16.2 Design of Digital Filters 534
16.3 Model Predictive Control of Dynamic Systems 547
16.4 Optimal Force Distribution for Robotic Systems with Closed
Kinematic Loops 558
16.5 Multiuser Detection in Wireless Communication Channels 570
References 586
Problems 588
Appendices 591
A Basics of Linear Algebra 591
A. 1 Introduction 591
XIV
A.2 Linear Independence and Basis of a Span 592
A.3 Range, Null Space, and Rank 593
A.4 Sherman-Morrison Formula 595
A.5 Eigenvalues and Eigenvectors 596
A.6 Symmetric Matrices 598
A.7 Trace 602
A.8 Vector Norms and Matrix Norms 602
A.9 Singular-Value Decomposition 606
A. 10 Orthogonal Projections 609
A.l 1 Householder Transformations and Givens Rotations 610
A. 12 QR Decomposition 616
A. 13 Cholesky Decomposition 619
A. 14 Kronecker Product 621
A. 15 Vector Spaces of Symmetric Matrices 623
A. 16 Polygon, Polyhedron, Polytope, and Convex Hull 626
References 627
B Basics of Digital Filters 629
B.l Introduction 629
B.2 Characterization 629
B. 3 Time-Domain Response 631
B.4 Stability Property 632
B.5 Transfer Function 633
B.6 Time-Domain Response Using the Z Transform 635
B.7 Z-Domain Condition for Stability 635
B.8 Frequency, Amplitude, and Phase Responses 636
B.9 Design 639
Reference 644
Index 645