凸优化

Preface
Introduction
1.1. Mathematical optimization
1.2 Least—squares and linear programming
1.3 Convex optimization
1.4 Nonlinear optimization
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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
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