Optimization Theory and Methods

Preface 1 Introduction 1.1 Introduction 1.2 Mathematics Foundations 1.2.1 Norm 1.2.2 Inverse and Generalized Inverse of a Matrix1.2.3 Properties of Eigenvalues 1.2.4 Rank-One Update 1.2.5 Function and Differential 1.3 Convex Sets and Convex Functions 1.3.1 Convex Sets 1.3.2 Convex Functions 1.3.3 Separation and Support of Convex Sets 1.4 Optimality Conditions for Unconstrained Case 1.5 Structure of Optimization Methods Exercises 2 Line Search 2.1 Introduction 2.2 Convergence Theory for Exact Line Search 2.3 Section Methods 2.3.1 The Golden Section Method 2.3.2 The Fibonacci Method 2.4 Interpolation Method 2.4.1 Quadratic Interpolation Methods 2.4.2 Cubic Interpolation Method 2.5 Inexact Line Search Techniques 2.5.1 Armijo and Goldstein Rule 2.5.2 Wolfe-Powell Rule 2.5.3 Goldstein Algorithm and Wolfe-Powell Algorithm 2.5.4 Backtracking Line Search 2.5.5 Convergence Theorems of Inexact Line Search Exercises 3 Newton’s Methods 3.1 The Steepest Descent Method 3.1.1 The Steepest Descent Method 3.1.2 Convergence of the Steepest Descent Method 3.1.3 Barzilai and Borwein Gradient Method 3.1.4 Appendix: Kantorovich Inequality 3.2 Newton’s Method 3.3 Modified Newton’s Method 3.4 Finite-Difference Newton’s Method 3.5 Negative Curvature Direction Method 3.5.1 Gill-Murray Stable Newton’s Method 3.5.2 Fiacco-McCormick Method 3.5.3 Fletcher-Freeman Method 3.5.4 Second-Order Step Rules 3.6 Inexact Newton’s Method Exercises 4 Conjugate Gradient Method 4.1 Conjugate Direction Methods 4.2 Conjugate Gradient Method 4.2.1 Conjugate Gradient Method 4.2.2 Beale’s Three-Term Conjugate Gradient Method4.2.3 Preconditioned Conjugate Gradient Method 4.3 Convergence of Conjugate Gradient Methods 4.3.1 Global Convergence of Conjugate Gradient Methods 4.3.2 Convergence Rate of Conjugate Gradient Methods Exercises 5 Quasi-Newton Methods 5.1 Quasi-Newton Methods 5.1.1 Quasi-Newton Equation 5.1.2 Symmetric Rank-One (SR1) Update 5.1.3 DFP Update 5.1.4 BFGS Update and PSB Update 5.1.5 The Least Change Secant Update 5.2 The Broyden Class 5.3 Global Convergence of Quasi-Newton Methods 5.3.1 Global Convergence under Exact Line Search 5.3.2 Global Convergence under Inexact Line Search 5.4 Local Convergence of Quasi-Newton Methods 5.4.1 Superlinear Convergence of General Quasi-Newton Methods 5.4.2 Linear Convergence of General Quasi-Newton Methods 5.4.3 Local Convergence of Broyden’s Rank-One Update 5.4.4 Local and Linear Convergence of DFP Method 5.4.5 Superlinear Convergence of BFGS Method 5.4.6 Superlinear Convergence of DFP Method 5.4.7 Local Convergence of Broyden’s Class Methods 5.5 Self-Scaling Variable Metric (SSVM) Methods 5.5.1 Motivation to SSVM Method 5.5.2 Self-Scaling Variable Metric (SSVM) Method 5.5.3 Choices of the Scaling Factor 5.6 Sparse Quasi-Newton Methods 5.7 Limited Memory BFGS Method Exercises 6 Trust-Region and Conic Model Methods 6.1 Trust-Region Methods 6.1.1 Trust-Region Methods 6.1.2 Convergence of Trust-Region Methods 6.1.3 Solving A Trust-Region Subproblem 6.2 Conic Model and Collinear Scaling Algorithm 6.2.1 Conic Model 6.2.2 Generalized Quasi-Newton Equation 6.2.3 Updates that Preserve Past Information 6.2.4 Collinear Scaling BFGS Algorithm 6.3 Tensor Methods 6.3.1 Tensor Method for Nonlinear Equations 6.3.2 Tensor Methods for Unconstrained Optimization Exercises