Nonlinear Programming: An Introduction

✍ Scribed by Peter Zörnig

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 360
• Category: Library

✦ Synopsis

This book is an introduction to nonlinear programming. It deals with the theoretical foundations and solution methods, beginning with the classical procedures and reaching up to “modern” methods like trust region methods or procedures for nonlinear and global optimization. A comprehensive bibliography including diverse web sites with information about nonlinear programming, in particular software, is presented. Without sacrificing the necessary mathematical rigor, excessive formalisms are avoided. Several examples, exercises with detailed solutions, and applications are provided, making the text adequate for individual studies.

The book is written for students from the fields of applied mathematics, engineering, economy, and computation.

• Didactic approach
• Many examples and solved exercises
• Comprehensive references - including web sites

✦ Table of Contents

Preface
Notations
1 Introduction
1.1 The model
1.2 Special cases and applications
1.2.1 Separable problem
1.2.2 Problem of quadratic optimization
1.2.3 Further examples of practical applications
1.3 Complications caused by nonlinearity
1.4 References for Chapter 1
Part I Theoretical foundations
2 Optimality conditions
2.1 Feasible directions
2.2 First and second-order optimality conditions
3 The convex optimization problem
3.1 Convex sets
3.2 Convex and concave functions
3.3 Differentiable convex functions
3.4 Subgradient and directional derivative
3.5 Minima of convex and concave functions
4 Karush–Kuhn–Tucker conditions and duality
4.1 Karush–Kuhn–Tucker conditions
4.2 Lagrange function and duality
4.3 The Wolfe dual problem
4.4 Second-order optimality criteria
4.5 References for Part I
Part II Solution methods
5 Iterative procedures and evaluation criteria
6 Unidimensional minimization
6.1 Delimitation of the search region
6.2 Newton’s method
6.3 Interpolation methods
6.4 On the use of the methods in practice
7 Unrestricted minimization
7.1 Analysis of quadratic functions
7.2 The gradient method
7.3 Multidimensional Newton’s method
7.4 Conjugate directions and quasi-Newton methods
7.5 Cyclic coordinate search techniques
7.6 Inexact line search
7.7 Trust region methods
8 Linearly constrained problems
8.1 Feasible direction methods
8.1.1 Rosen’s gradient projection method
8.1.2 Zoutendijk’s method
8.1.3 Advanced techniques: an outline
8.2 Linear equality constraints
9 Quadratic problems
9.1 An active-set method
9.2 Karush–Kuhn–Tucker conditions
9.3 Lemke’s method
10 The general problem
10.1 The penalty method
10.2 The barrier method
10.3 Sequential quadratic programming
11 Nondifferentiable and global optimization
11.1 Nondifferentiable optimization
11.1.1 Examples for nondifferentiable problems
11.1.2 Basic ideas of resolution
11.1.3 The concept of bundle methods
11.2 Global optimization
11.2.1 Specific cases of global optimization
11.2.2 Exact methods
11.2.3 Heuristic methods
11.3 References and software for Part II
Appendix: Solutions of exercises
References
Index