Nonlinear Programming: An Introduction

by Peter Zornig

Overview

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.

Product Details

ISBN-13:	9783110315271
Publisher:	De Gruyter
Publication date:	01/29/2014
Series:	De Gruyter Textbook Series
Pages:	360
Product dimensions:	6.69(w) x 9.45(h) x 0.03(d)
Age Range:	18 Years

About the Author

Peter Zörnig, University of Brasília, Brazil.

Table of Contents

Preface v

Notations ix

1 Introduction 1

1.1 The model 1

1.2 Special cases and applications 3

1.2.1 Separable problem 4

1.2.2 Problem of quadratic optimization 8

1.2.3 Further examples of practical applications 12

1.3 Complications caused by nonlinearity 19

1.4 References for Chapter 1 23

Part I Theoretical foundations

2 Optimality conditions 27

2.1 Feasible directions 27

2.2 First and second-order optimality conditions 33

3 The convex optimization problem 41

3.1 Convex sets 41

3.2 Convex and concave functions 48

3.3 Differentiable convex functions 51

3.4 Subgradient and directional derivative 53

3.5 Minima of convex and concave functions 62

4 Karush-Kuhn-Tucker conditions and duality 69

4.1 Karush-Kuhn-Tucker conditions 69

4.2 Lagrange function and duality 81

4.3 The Wolfe dual problem 94

4.4 Second-order optimality criteria 98

4.5 References for Part I 107

Part II Solution methods

5 Iterative procedures and evaluation criteria 111

6 undimensional minimization 115

6.1 Delimitation of the search region 116

6.2 Newton's method 120

6.3 Interpolation methods 126

6.4 On the use of the methods in practice 130

7 Unrestricted minimization 131

7.1 Analysis of quadratic functions 131

7.2 The gradient method 137

7.3 Multidimensional Newton's method 142

7.4 Conjugate directions and quasi-Newton methods 144

7.5 Cyclic coordinate search techniques 152

7.6 Inexact line search 155

7.7 Trust region methods 161

8 Linearly constrained problems 168

8.1 Feasible direction methods 168

8.1.1 Rosen's gradient projection method 168

8.1.2 Zoutendijk's method 176

8.1.3 Advanced techniques: an outline 180

8.2 Linear equality constraints 185

9 Quadratic problems 193

9.1 An active-set method 193

9.2 Karush-Kuhn-Tucker conditions 197

9.3 Lemke's method 199

10 The general problem 205

10.1 The penalty method 205

10.2 The barrier method 216

10.3 Sequential quadratic programming 221

11 Nondifferentiable and global optimization 226

11.1 Nondifferentiable optimization 226

11.1.1 Examples for nondifferentiable problems 226

11.1.2 Basic ideas of resolution 231

11.1.3 The concept of bundle methods 237

11.2 Global optimization 240

11.2.1 Specific cases of global optimization 241

11.2.2 Exact methods 242

11.2.3 Heuristic methods 249

11.3 References and software for Part II 254

Appendix: Solutions of exercises 258

References 343

Index 347

Editorial Reviews

„Es ist als Einstieg in das Thema auch für Studenten sehr gut geeignet, ebenso als Begleitmaterial zu einer eventuell auch mehrsemestrigen Vorlesung."
Prof. Dr. Matthias Grajewski, FH Aachen

From the Publisher