凸优化

内容概要

作者：（美国）鲍迪（Stephen Boyd）

书籍目录

Preface

Introduction

1.1. Mathematical optimization

1.2 Least—squares and linear programming

1.3 Convex optimization

1.4 Nonlinear optimization

1.5 Outline

1.6 Notation

Bibliography

Theory

Convex sets

2.1 Affine and convex sets

2.2 Some important examples

2.3 Operations that preserve convexity

2.4 Generalized inequalities

2.5 Separating and supporting hyperplanes

2.6 Dual cones and generalized inequalities

Bibliography

Exercises

Convex functions

3.1 Basic properties and examples

3.2 Operations that preserve convexity

3.3 The conjugate function

3.4 Quasiconvex functions

3.5 Log—concave and log—convex functions

3.6 Convexity with respect to generalized inequalities

Bibliography

Exercises

Convex optimization problems

4.1 Optimization problems

4.2 Convex optimization

4.3 Linear optimization problems

4.4 Quadratic optimization problems

4.5 Geometric programming

4.6 Generalized inequality constraints

4.7 Vector optimization

Bibliography

Exercises

Duality

5.1 The Lagrange dual function

5.2 The Lagrange dual problem

5.3 Geometric interpretation

5.4 Saddle—point interpretation

5.5 Optimality conditions

5.0 Perturbation and sensitivity analysis

5.7 Examples

5.8 Theorems of alternatives

5，9 Generalized inequalities

Bibliography

Exercises

II Applications

6 Approximation and fitting

6.1 Norm approximation

0.2 Least—norm problems

6.3 Regularized approximation

6.4 Robust approximation

6.5 Function fitting and interpolation

Bibliography

Exercises

Statistical estimation

7.1 Parametric distribution estimation

7.2 Nonparametric distribution estimation

7.3 Optimal detector design and hypothesis testing

7.4 Chebyshev and Chernoff bounds

7.5 Experiment design

Bibliography

Exercises

8 Geometric problems

8.1 Projection on a set

8.2 Distance between sets

8.3 Euclidean distance and angle problems

8.4 Extremal volume ellipsoids

8.5 Centering

8.6 Classification

8.7 Placement and location

8.8 Floor planning

Bibliography

Exercises

III Algorithms

9 Unconstrained minimization

9.1 Unconstrained minimization problems

9.2 Descent methods

9.3 Gradient descent method

9.4 Steepest descent method

9.5 Newton's method

9.6 Self—concordance

9.7 Implementation

Bibliography

Exercises

10 Equality constrained minimization

10.1 Equality constrained minimization problems

10.2 Newton's method with equality constraints

10.3 Infeasible start Newton method

10.4 Implementation

Bibliography

Exercises

11 Interior—point methods

11.1 Inequality constrained minimization problems

11.2 Logarithmic barrier function and central path

11.3 The barrier method

11.4 Feasibility and phase I methods

11.5 Complexity analysis via self—concordance

11.6 Problems with generalized inequalities

11.7 Primal—dual interior—point methods

11.8 Implementation

Bibliography

Exercises

Appendices

A Mathematical background

A.1 Norms

A.2 Analysis

A.3 Functions

A.4 Derivatives

A.5 Linear algebra

Bibliography

B Problems involving two quadratic functions

B.1 Single constraint quadratic optimization

B.2 The S—procedure

B.3 The field of values of two symmetric matrices

B.4 Proofs of the strong duality results

Bibliography

C Numerical linear algebra background

C.1 Matrix structure and algorithm complexity

C.2 Solving linear equations with factored matrices

C.3 LU， Cholesky， and LDLT factorization

C.4 Block elimination and Schur complements

C.5 Solving underdetermined linear equations

Bibliography

References

Notation

Index

作者简介

《凸优化(英文)》由世界图书出版社出版。

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