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Dedication v
Biographies of the authors vii
Preface xv
Abbreviations xix
The Optimization Problem 1
Introduction 1
The Basic Optimization Problem 4
General Structure of Optimization Algorithms 8
Constraints 10
The Feasible Region 17
Branches of Mathematical Programming 22
References 24
Problems 25
Basic Principles 27
Introduction 27
Gradient Information 27
The Taylor Series 28
Types of Extrema 31
Necessary and Sufficient Conditions for Local Minima and Maxima 33
Classification of Stationary Points 40
Convex and Concave Functions 51
Optimization of Convex Functions 58
References 60
Problems 60
General Properties of Algorithms 65
Introduction 65
An Algorithm as a Point-to-Point Mapping 65
An Algorithm as a Point-to-Set Mapping 67
Closed Algorithms 68
Descent Functions 71
Global Convergence 72
Rates of Convergence 76
References 79
Problems 79
One-Dimensional Optimization 81
Introduction 81
Dichotomous Search 82
Fibonacci Search 85
Golden-Section Search 92
Quadratic Interpolation Method 95
Cubic Interpolation 99
The Algorithm of Davies, Swann, and Campey 101
Inexact Line Searches 106
References 114
Problems 114
Basic Multidimensional Gradient Methods 119
Introduction 119
Steepest-Descent Method 120
Newton Method 128
Gauss-Newton Method 138
References 140
Problems 140
Conjugate-Direction Methods 145
Introduction 145
Conjugate Directions 146
Basic Conjugate-Directions Method 149
Conjugate-Gradient Method 152
Minimization of Nonquadratic Functions 157
Fletcher-Reeves Method 158
Powell's Method 159
Partan Method 168
References 172
Problems 172
Quasi-Newton Methods 175
Introduction 175
The Basic Quasi-Newton Approach 176
Generation of Matrix S[subscript k] 177
Rank-One Method 181
Davidon-Fletcher-Powell Method 185
Broyden-Fletcher-Goldfarb-Shanno Method 191
Hoshino Method 192
The Broyden Family 192
The Huang Family 194
Practical Quasi-Newton Algorithm 195
References 199
Problems 200
Minimax Methods 203
Introduction 203
Problem Formulation 203
Minimax Algorithms 205
Improved Minimax Algorithms 211
References 228
Problems 228
Applications of Unconstrained Optimization 231
Introduction 231
Point-Pattern Matching 232
Inverse Kinematics for Robotic Manipulators 237
Design of Digital Filters 247
References 260
Problems 262
Fundamentals of Constrained Optimization 265
Introduction 265
Constraints 266
Classification of Constrained Optimization Problems 273
Simple Transformation Methods 277
Lagrange Multipliers 285
First-Order Necessary Conditions 294
Second-Order Conditions 302
Convexity 308
Duality 311
References 312
Problems 313
Linear Programming Part I: The Simplex Method 321
Introduction 321
General Properties 322
Simplex Method 344
References 368
Problems 368
Linear Programming Part II: Interior-Point Methods 373
Introduction 373
Primal-Dual Solutions and Central Path 374
Primal Affine-Scaling Method 379
Primal Newton Barrier Method 383
Primal-Dual Interior-Point Methods 388
References 402
Problems 402
Quadratic and Convex Programming 407
Introduction 407
Convex QP Problems with Equality Constraints 408
Active-Set Methods for Strictly Convex QP Problems 411
Interior-Point Methods for Convex QP Problems 417
Cutting-Plane Methods for CP Problems 428
Ellipsoid Methods 437
References 443
Problems 444
Semidefinite and Second-Order Cone Programming 449
Introduction 449
Primal and Dual SDP Problems 450
Basic Properties of SDP Problems 455
Primal-Dual Path-Following Method 458
Predictor-Corrector Method 465
Projective Method of Nemirovski and Cabinet 470
Second-Order Cone Programming 484
A Primal-Dual Method for SOCP Problems 491
References 496
Problems 497
General Nonlinear Optimization Problems 501
Introduction 501
Sequential Quadratic Programming Methods 501
Modified SQP Algorithms 509
Interior-Point Methods 518
References 528
Problems 529
Applications of Constrained Optimization 533
Introduction 533
Design of Digital Filters 534
Model Predictive Control of Dynamic Systems 547
Optimal Force Distribution for Robotic Systems with Closed Kinematic Loops 558
Multiuser Detection in Wireless Communication Channels 570
References 586
Problems 588
Appendices 591
Basics of Linear Algebra 591
Introduction 591
Linear Independence and Basis of a Span 592
Range, Null Space, and Rank 593
Sherman-Morrison Formula 595
Eigenvalues and Eigenvectors 596
Symmetric Matrices 598
Trace 602
Vector Norms and Matrix Norms 602
Singular-Value Decomposition 606
Orthogonal Projections 609
Householder Transformations and Givens Rotations 610
QR Decomposition 616
Cholesky Decomposition 619
Kronecker Product 621
Vector Spaces of Symmetric Matrices 623
Polygon, Polyhedron, Polytope, and Convex Hull 626
References 627
Basics of Digital Filters 629
Introduction 629
Characterization 629
Time-Domain Response 631
Stability Property 632
Transfer Function 633
Time-Domain Response Using the Z Transform 635
Z-Domain Condition for Stability 635
Frequency, Amplitude, and Phase Responses 636
Design 639
Reference 644
Index 645
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