Zörnig Nonlinear Programming

An Introduction
1. Auflage 2014
ISBN: 978-3-11-031528-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 360 Seiten

Reihe: De Gruyter Textbook

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.

Autoren/Hrsg.

Zörnig, Peter

Fachgebiete

• Mathematik | Informatik EDV | Informatik Informatik Mathematik für Informatiker
• Mathematik | Informatik Mathematik Operations Research
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik
• Wirtschaftswissenschaften Betriebswirtschaft Unternehmensforschung

Inhaltsverzeichnis

1;Preface;5
2;Notations;9
3;1 Introduction;11
3.1;1.1 The model;11
3.2;1.2 Special cases and applications;13
3.2.1;1.2.1 Separable problem;14
3.2.2;1.2.2 Problem of quadratic optimization;18
3.2.3;1.2.3 Further examples of practical applications;22
3.3;1.3 Complications caused by nonlinearity;29
3.4;1.4 References for Chapter 1;33
4;Part I Theoretical foundations;35
4.1;2 Optimality conditions;37
4.1.1;2.1 Feasible directions;37
4.1.2;2.2 First and second-order optimality conditions;43
4.2;3 The convex optimization problem;51
4.2.1;3.1 Convex sets;51
4.2.2;3.2 Convex and concave functions;58
4.2.3;3.3 Differentiable convex functions;61
4.2.4;3.4 Subgradient and directional derivative;63
4.2.5;3.5 Minima of convex and concave functions;72
4.3;4 Karush–Kuhn–Tucker conditions and duality;79
4.3.1;4.1 Karush–Kuhn–Tucker conditions;79
4.3.2;4.2 Lagrange function and duality;91
4.3.3;4.3 The Wolfe dual problem;104
4.3.4;4.4 Second-order optimality criteria;108
4.3.5;4.5 References for Part I;117
5;Part II Solution methods;119
5.1;5 Iterative procedures and evaluation criteria;121
5.2;6 Unidimensional minimization;125
5.2.1;6.1 Delimitation of the search region;126
5.2.2;6.2 Newton’s method;130
5.2.3;6.3 Interpolation methods;136
5.2.4;6.4 On the use of the methods in practice;140
5.3;7 Unrestricted minimization;141
5.3.1;7.1 Analysis of quadratic functions;141
5.3.2;7.2 The gradient method;147
5.3.3;7.3 Multidimensional Newton’s method;152
5.3.4;7.4 Conjugate directions and quasi-Newton methods;154
5.3.5;7.5 Cyclic coordinate search techniques;162
5.3.6;7.6 Inexact line search;165
5.3.7;7.7 Trust region methods;171
5.4;8 Linearly constrained problems;178
5.4.1;8.1 Feasible direction methods;178
5.4.1.1;8.1.1 Rosen’s gradient projection method;178
5.4.1.2;8.1.2 Zoutendijk’s method;186
5.4.1.3;8.1.3 Advanced techniques: an outline;190
5.4.2;8.2 Linear equality constraints;195
5.5;9 Quadratic problems;203
5.5.1;9.1 An active-set method;203
5.5.2;9.2 Karush–Kuhn–Tucker conditions;207
5.5.3;9.3 Lemke’s method;209
5.6;10 The general problem;215
5.6.1;10.1 The penalty method;215
5.6.2;10.2 The barrier method;226
5.6.3;10.3 Sequential quadratic programming;231
5.7;11 Nondifferentiable and global optimization;236
5.7.1;11.1 Nondifferentiable optimization;236
5.7.1.1;11.1.1 Examples for nondifferentiable problems;236
5.7.1.2;11.1.2 Basic ideas of resolution;241
5.7.1.3;11.1.3 The concept of bundle methods;247
5.7.2;11.2 Global optimization;250
5.7.2.1;11.2.1 Specific cases of global optimization;251
5.7.2.2;11.2.2 Exact methods;252
5.7.2.3;11.2.3 Heuristic methods;259
5.7.3;11.3 References and software for Part II;264
6;Appendix: Solutions of exercises;268
7;References;353
8;Index;357

Über Autor(innen)

Peter Zörnig, University of Brasília, Brazil.