An Introduction to Optimization

Edwin K. P. Chong, PHD, is Professor of Electrical and Computer Engineering as well as Professor of Mathematics at Colorado State University. He is a Fellow of the IEEE and Senior Editor of IEEE Transactions on Automatic Control.

Stanislaw H. Zak, PHD, is Professor in the School of Electrical and Computer Engineering at Purdue University. He is former associate editor of Dynamics and Control and IEEE Transactions on Neural Networks

Preface xiii

PART I MATHEMATICAL REVIEW

1 Methods of Proof and Some Notation 3

1.1 Methods of Proof 3

1.2 Notation 5

Exercises 6

2 Vector Spaces and Matrices 7

2.1 Vector and Matrix 7

2.2 Rank of a Matrix 13

2.3 Linear Equations 17

2.4 Inner Products and Norms 19

Exercises 22

3 Transformations 25

3.1 Linear Transformations 25

3.2 Eigenvalues and Eigenvectors 26

3.3 Orthogonal Projections 29

3.4 Quadratic Forms 31

3.5 Matrix Norms 35

Exercises 40

4 Concepts from Geometry 45

4.1 Line Segments 45

4.2 Hyperplanes and Linear Varieties 46

4.3 Convex Sets 48

4.4 Neighborhoods 50

4.5 Polytopes and Polyhedra 52

Exercises 53

5 Elements of Calculus 55

5.1 Sequences and Limits 55

5.2 Differentiability 62

5.3 The Derivative Matrix 63

5.4 Differentiation Rules 67

5.5 Level Sets and Gradients 68

5.6 Taylor Series 72

Exercises 77

PART II UNCONSTRAINED OPTIMIZATION

6 Basics of Set-Constrained and Unconstrained Optimization 81

6.1 Introduction 81

6.2 Conditions for Local Minimizers 83

Exercises 93

7 One-Dimensional Search Methods 103

7.1 Introduction 103

7.2 Golden Section Search 104

7.3 Fibonacci Method 108

7.4 Bisection Method 116

7.5 Newton’s Method 116

7.6 Secant Method 120

7.7 Bracketing 123

7.8 Line Search in Multidimensional Optimization 124

Exercises 126

8 Gradient Methods 131

8.1 Introduction 131

8.2 The Method of Steepest Descent 133

8.3 Analysis of Gradient Methods 141

Exercises 153

9 Newton’s Method 161

9.1 Introduction 161

9.2 Analysis of Newton’s Method 164

9.3 Levenberg-Marquardt Modification 168

9.4 Newton’s Method for Nonlinear Least Squares 168

Exercises 171

10 Conjugate Direction Methods 175

10.1 Introduction 175

10.2 The Conjugate Direction Algorithm 177

10.3 The Conjugate Gradient Algorithm 182

10.4 The Conjugate Gradient Algorithm for Nonquadratic

Problems 186

Exercises 189

11 Quasi-Newton Methods 193

11.1 Introduction 193

11.2 Approximating the Inverse Hessian 194

11.3 The Rank One Correction Formula 197

11.4 The DFP Algorithm 202

11.5 The BFGS Algorithm 207

Exercises 211

12 Solving Linear Equations 217

12.1 Least-Squares Analysis 217

12.2 The Recursive Least-Squares Algorithm 227

12.3 Solution to a Linear Equation with Minimum Norm 231

12.4 Kaczmarz’s Algorithm 232

12.5 Solving Linear Equations in General 236

Exercises 244

13 Unconstrained Optimization and Neural Networks 253

13.1 Introduction 253

13.2 Single-Neuron Training 256

13.3 The Backpropagation Algorithm 258

Exercises 270

14 Global Search Algorithms 273

14.1 Introduction 273

14.2 The Nelder-Mead Simplex Algorithm 274

14.3 Simulated Annealing 278

14.4 Particle Swarm Optimization 282

14.5 Genetic Algorithms 285

Exercises 298

PART III LINEAR PROGRAMMING

15 Introduction to Linear Programming 305

15.1 Brief History of Linear Programming 305

15.2 Simple Examples of Linear Programs 307

15.3 Two-Dimensional Linear Programs 314

15.4 Convex Polyhedra and Linear Programming 316

15.5 Standard Form Linear Programs 318

15.6 Basic Solutions 324

15.7 Properties of Basic Solutions 327

15.8 Geometric View of Linear Programs 330

Exercises 335

16 Simplex Method 339

16.1 Solving Linear Equations Using Row Operations 339

16.2 The Canonical Augmented Matrix 346

16.3 Updating the Augmented Matrix 349

16.4 The Simplex Algorithm 350

16.5 Matrix Form of the Simplex Method 357

16.6 Two-Phase Simplex Method 361

16.7 Revised Simplex Method 364

Exercises 369

17 Duality 379

17.1 Dual Linear Programs 379

17.2 Properties of Dual Problems 387

Exercises 394

18 Nonsimplex Methods 403

18.1 Introduction 403

18.2 Khachiyan’s Method 405

18.3 Affine Scaling Method 408

18.4 Karmarkar’s Method 413

Exercises 426

19 Integer Linear Programming 429

19.1 Introduction 429

19.2 Unimodular Matrices 430

19.3 The Gomory Cutting-Plane Method 437

Exercises 447

PART IV NONLINEAR CONSTRAINED OPTIMIZATION

20 Problems with Equality Constraints 453

20.1 Introduction 453

20.2 Problem Formulation 455

20.3 Tangent and Normal Spaces 456

20.4 Lagrange Condition 463

20.5 Second-Order Conditions 472

20.6 Minimizing Quadratics Subject to Linear Constraints 476

Exercises 481

21 Problems with Inequality Constraints 487

21.1 Karush-Kuhn-Tucker Condition 487

21.2 Second-Order Conditions 496

Exercises 501

22 Convex Optimization Problems 509

22.1 Introduction 509

22.2 Convex Functions 512

22.3 Convex Optimization Problems 521

22.4 Semidefinite Programming 527

Exercises 540

23 Algorithms for Constrained Optimization 549

23.1 Introduction 549

23.2 Projections 549

23.3 Projected Gradient Methods with Linear Constraints 553

23.4 Lagrangian Algorithms 557

23.5 Penalty Methods 564

Exercises 571

24 Multiobjective Optimization 577

24.1 Introduction 577

24.2 Pareto Solutions 578

24.3 Computing the Pareto Front 581

24.4 From Multiobjective to Single-Objective Optimization 585

24.5 Uncertain Linear Programming Problems 588

Exercises 596

References 599

Index 609

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