# An Introduction to Optimization, Third Edition

## Contents

Preface

### Part I. Mathematical Review

1 Methods of Proof and Some Notation
1.1 Methods of Proof
1.2 Notation
2 Vector Spaces and Matrices
2.1 Vector and Matrix
2.2 Rank of a Matrix
2.3 Linear Equations
2.4 Inner Products and Norms
3 Transformations
3.1 Linear Transformations
3.2 Eigenvalues and Eigenvectors
3.3 Orthogonal Projections
3.5 Matrix Norms
4 Concepts from Geometry
4.1 Line Segments
4.2 Hyperplanes and Linear Varieties
4.3 Convex Sets
4.4 Neighborhoods
4.5 Polytopes and Polyhedra
5 Elements of Calculus
5.1 Sequences and Limits
5.2 Differentiability
5.3 The Derivative Matrix
5.4 Differentiation Rules
5.5 Level Sets and Gradients
5.6 Taylor Series

### Part II. Unconstrained Optimization

6 Basics of Set-Constrained and Unconstrained Optimization
6.1 Introduction
6.2 Conditions for Local Minimizers
7 One-Dimensional Search Methods
7.1 Golden Section Search
7.2 Fibonacci Search
7.3 Newton's Method
7.4 Secant Method
7.5 Remarks on Line Search Methods
8.1 Introduction
8.2 The Method of Steepest Descent
8.3 Analysis of Gradient Methods
9 Newton's Method
9.1 Introduction
9.2 Analysis of Newton's Method
9.3 Levenberg-Marquardt Modification
9.4 Newton's Method for Nonlinear Least-Squares
10 Conjugate Direction Methods
10.1 Introduction
10.2 The Conjugate Direction Algorithm
10.3 The Conjugate Gradient Algorithm
10.4 The Conjugate Gradient Algorithm for Non-Quadratic Problems
11 Quasi-Newton Methods
11.1 Introduction
11.2 Approximating the Inverse Hessian
11.3 The Rank One Correction Formula
11.4 The DFP Algorithm
11.5 The BFGS Algorithm
12 Solving Linear Equations
12.1 Least-Squares Analysis
12.2 The Recursive Least-Squares Algorithm
12.3 Solution to Linear Equation with Minimum Norm
12.4 Kaczmarz's Algorithm
12.5 Solving Linear Equations in General
13 Unconstrained Optimization and Neural Networks
13.1 Introduction
13.2 Single-Neuron Training
13.3 The Backpropagation Algorithm
14 Global Search Algorithms
14.1 Introduction
14.2 The Nelder-Mead Simplex Algorithm
14.3 Simulated Annealing
14.4 Particle Swarm Optimization
14.5 Genetic Algorithms

### Part III. Linear Programming

15 Introduction to Linear Programming
15.1 Brief History of Linear Programming
15.2 Simple Examples of Linear Programs
15.3 Two-Dimensional Linear Programs
15.4 Convex Polyhedra and Linear Programming
15.5 Standard Form Linear Programs
15.6 Basic Solutions
15.7 Properties of Basic Solutions
15.8 Geometric View of Linear Programs
16 Simplex Method
16.1 Solving Linear Equations Using Row Operations
16.2 The Canonical Augmented Matrix
16.3 Updating the Augmented Matrix
16.4 The Simplex Algorithm
16.5 Matrix Form of the Simplex Method
16.6 Two-Phase Simplex Method
16.7 Revised Simplex Method
17 Duality
17.1 Dual Linear Programs
17.2 Properties of Dual Problems
18 Nonsimplex Methods
18.1 Introduction
18.2 Khachiyan's Method
18.3 Affine Scaling Method
18.4 Karmarkar's Method

### Part IV. Nonlinear Constrained Optimization

19 Problems with Equality Constraints
19.1 Introduction
19.2 Problem Formulation
19.3 Tangent and Normal Spaces
19.4 Lagrange Condition
19.5 Second-Order Conditions
19.6 Minimizing Quadratics Subject to Linear Constraints
20 Problems With Inequality Constraints
20.1 Karush-Kuhn-Tucker Conditions
20.2 Second-Order Conditions
21 Convex Optimization Problems
21.1 Introduction
21.2 Convex Functions
21.3 Convex Optimization Problems
21.4 }Semidefinite Programming
22 Algorithms for Constrained Optimization
22.1 Introduction
22.2 Projections
22.3 Projected Gradient Methods with Linear Constraints
22.4 Lagrangian Algorithms
22.5 Penalty Methods
23 Multiobjective Optimization
23.1 Introduction
23.2 Pareto Solutions
23.3 Computing the Pareto Front
23.4 From Multiobjective to Single-Objective Optimization
23.5 Uncertain Linear Programming Problems
References
Index

Professor Edwin K. P. Chong,