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Table of Contents
Intro
Preface
Contents
List of Algorithms
List of Applications
List of Figures
List of Tables
1: Introduction
1.1 Mathematical Modeling: Linguistic Models Versus Mathematical Models
1.2 Mathematical Modeling and Computational Sciences
1.3 The Modern Modeling Scheme for Optimization
1.4 Classification of Optimization Problems
1.5 Optimization Algorithms
1.6 Collections of Applications for Numerical Experiments
1.7 Comparison of Algorithms
1.8 The Structure of the Book
2: Fundamentals on Unconstrained Optimization. Stepsize Computation
2.1 The Problem
2.2 Fundamentals on the Convergence of the Line-Search Methods
2.3 The General Algorithm for Unconstrained Optimization
2.4 Convergence of the Algorithm with Exact Line-Search
2.5 Inexact Line-Search Methods
2.6 Convergence of the Algorithm with Inexact Line-Search
2.7 Three Fortran Implementations of the Inexact Line-Search
2.8 Numerical Studies: Stepsize Computation
3: Steepest Descent Methods
3.1 The Steepest Descent
Convergence of the Steepest Descent Method for Quadratic Functions
Inequality of Kantorovich
Numerical Study
Convergence of the Steepest Descent Method for General Functions
3.2 The Relaxed Steepest Descent
Numerical Study: SDB Versus RSDB
3.3 The Accelerated Steepest Descent
Numerical Study
3.4 Comments on the Acceleration Scheme
4: The Newton Method
4.1 The Newton Method for Solving Nonlinear Algebraic Systems
4.2 The Gauss-Newton Method
4.3 The Newton Method for Function Minimization
4.4 The Newton Method with Line-Search
4.5 Analysis of Complexity
4.6 The Modified Newton Method
4.7 The Newton Method with Finite-Differences
4.8 Errors in Functions, Gradients, and Hessians
4.9 Negative Curvature Direction Methods
4.10 The Composite Newton Method
5: Conjugate Gradient Methods
5.1 The Concept of Nonlinear Conjugate Gradient
5.2 The Linear Conjugate Gradient Method
The Linear Conjugate Gradient Algorithm
Convergence Rate of the Linear Conjugate Gradient Algorithm
Preconditioning
Incomplete Cholesky Factorization
Comparison of the Convergence Rate of the Linear Conjugate Gradient and of the Steepest Descent
5.3 General Convergence Results for Nonlinear Conjugate Gradient Methods
Preface
Contents
List of Algorithms
List of Applications
List of Figures
List of Tables
1: Introduction
1.1 Mathematical Modeling: Linguistic Models Versus Mathematical Models
1.2 Mathematical Modeling and Computational Sciences
1.3 The Modern Modeling Scheme for Optimization
1.4 Classification of Optimization Problems
1.5 Optimization Algorithms
1.6 Collections of Applications for Numerical Experiments
1.7 Comparison of Algorithms
1.8 The Structure of the Book
2: Fundamentals on Unconstrained Optimization. Stepsize Computation
2.1 The Problem
2.2 Fundamentals on the Convergence of the Line-Search Methods
2.3 The General Algorithm for Unconstrained Optimization
2.4 Convergence of the Algorithm with Exact Line-Search
2.5 Inexact Line-Search Methods
2.6 Convergence of the Algorithm with Inexact Line-Search
2.7 Three Fortran Implementations of the Inexact Line-Search
2.8 Numerical Studies: Stepsize Computation
3: Steepest Descent Methods
3.1 The Steepest Descent
Convergence of the Steepest Descent Method for Quadratic Functions
Inequality of Kantorovich
Numerical Study
Convergence of the Steepest Descent Method for General Functions
3.2 The Relaxed Steepest Descent
Numerical Study: SDB Versus RSDB
3.3 The Accelerated Steepest Descent
Numerical Study
3.4 Comments on the Acceleration Scheme
4: The Newton Method
4.1 The Newton Method for Solving Nonlinear Algebraic Systems
4.2 The Gauss-Newton Method
4.3 The Newton Method for Function Minimization
4.4 The Newton Method with Line-Search
4.5 Analysis of Complexity
4.6 The Modified Newton Method
4.7 The Newton Method with Finite-Differences
4.8 Errors in Functions, Gradients, and Hessians
4.9 Negative Curvature Direction Methods
4.10 The Composite Newton Method
5: Conjugate Gradient Methods
5.1 The Concept of Nonlinear Conjugate Gradient
5.2 The Linear Conjugate Gradient Method
The Linear Conjugate Gradient Algorithm
Convergence Rate of the Linear Conjugate Gradient Algorithm
Preconditioning
Incomplete Cholesky Factorization
Comparison of the Convergence Rate of the Linear Conjugate Gradient and of the Steepest Descent
5.3 General Convergence Results for Nonlinear Conjugate Gradient Methods