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1 Introduction1

1.1 Mathematical optimization1

1.2 Least-squares and linear programming4

1.3 Convex optimization7

1.4 Nonlinear optimization9

1.5 Outline11

1.6 Notation14

Bibliography16

Ⅰ Theory19

2 Convex sets21

2.1 Affine and convex sets21

2.2 Some important examples27

2.3 Operations that preserve convexity35

2.4 Generalized inequalities43

2.5 Separating and supporting hyperplanes46

2.6 Dual cones and generalized inequalities51

Bibliography59

Exercises60

3 Convex functions67

3.1 Basic properties and examples67

3.2 Operations that preserve convexity79

3.3 The conjugate function90

3.4 Quasiconvex functions95

3.5 Log-concave and log-convex functions104

3.6 Convexity with respect to generalized inequalities108

Bibliography112

Exercises113

4 Convex optimization problems127

4.1 Optimization problems127

4.2 Convex optimization136

4.3 Linear optimization problems146

4.4 Quadratic optimization problems152

4.5 Geometric programming160

4.6 Generalized inequality constraints167

4.7 Vector optimization174

Bibliography188

Exercises189

5 Duality215

5.1 The Lagrange dual function215

5.2 The Lagrange dual problem223

5.3 Geometric interpretation232

5.4 Saddle-point interpretation237

5.5 Optimality conditions241

5.6 Perturbation and sensitivity analysis249

5.7 Examples253

5.8 Theorems of alternatives258

5.9 Generalized inequalities264

Bibliography272

Exercises273

Ⅱ Applications289

6 Approximation and fitting291

6.1 Norm approximation291

6.2 Least-norm problems302

6.3 Regularized approximation305

6.4 Robust approximation318

6.5 Function fitting and interpolation324

Bibliography343

Exercises344

7 Statistical estimation351

7.1 Parametric distribution estimation351

7.2 Nonparametric distribution estimation359

7.3 Optimal detector design and hypothesis testing364

7.4 Chebyshev and Chernoff bounds374

7.5 Experiment design384

Bibliography392

Exercises393

8 Geometric problems397

8.1 Projection on a set397

8.2 Distance between sets402

8.3 Euclidean distance and angle problems405

8.4 Extremal volume ellipsoids410

8.5 Centering416

8.6 Classification422

8.7 Placement and location432

8.8 Floor planning438

Bibliography446

Exercises447

Ⅲ Algorithms455

9 Unconstrained minimization457

9.1 Unconstrained minimization problems457

9.2 Descent methods463

9.3 Gradient descent method466

9.4 Steepest descent method475

9.5 Newton's method484

9.6 Self-concordance496

9.7 Implementation508

Bibliography513

Exercises514

10 Equality constrained minimization521

10.1 Equality constrained minimization problems521

10.2 Newton's method with equality constraints525

10.3 Infeasible start Newton method531

10.4 Implementation542

Bibliography556

Exercises557

11 Interior-point methods561

11.1 Inequality constrained minimization problems561

11.2 Logarithmic barrier function and central path562

11.3 The barrier method568

11.4 Feasibility and phase I methods579

11.5 Complexity analysis via self-concordance585

11.6 Problems with generalized inequalities596

11.7 Primal-dual interior-point methods609

11.8 Implementation615

Bibliography621

Exercises623

Appendices631

A Mathematical background633

A.1 Norms633

A.2 Analysis637

A.3 Functions639

A.4 Derivatives640

A.5 Linear algebra645

Bibliography652

B Problems involving two quadratic functions653

B.1 Single constraint quadratic optimization653

B.2 The S-procedure655

B.3 The field of values oftwo symmetric matrices656

B.4 Proofs of the strong duality results657

Bibliography659

C Numerical linear algebra background661

C.1 Matrix structure and algorithm complexity661

C.2 Solving linear equationswith factored matrices664

C.3 LU.Cholesky,and LDLT factorization668

C.4 Block elimination and Schur complements672

C.5 Solving underdetermined linear equations681

Bibliography684

References685

Notation697

Index701