An Introduction to Nonlinear Optimization Theory

by Marius Durea, Radu Strugariu

Overview

The goal of this book is to present the main ideas and techniques in the field of continuous smooth and nonsmooth optimization. Starting with the case of differentiable data and the classical results on constrained optimization problems, and continuing with the topic of nonsmooth objects involved in optimization theory, the book concentrates on both theoretical and practical aspects of this field. This book prepares those who are engaged in research by giving repeated insights into ideas that are subsequently dealt with and illustrated in detail.

Product Details

ISBN-13:	9783110427356
Publisher:	De Gruyter Poland
Publication date:	03/30/2015
Sold by:	Barnes & Noble
Format:	eBook
Pages:	328
File size:	12 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Marius Durea, “Alexandru Ioan Cuza” University of Iasi; Radu Strugariu, “Gheorghe Asachi” Technical University of Iasi, Iasi, Romania

Table of Contents

Preface vii

1 Preliminaries 1

1.1 R^p Space 1

1.2 Limits of functions and Continuity 5

1.3 Differentiability 10

1.4 The Riemann Integral 15

2 Nonlinear Analysis Fundamentals 18

2.1 Convex Sets and Cones 18

2.2 Convex Functions 30

2.2.1 General Results 30

2.2.2 Convex Functions of One Variable 38

2.2.3 Inequalities 42

2.3 Banach Fixed Point Principle 52

2.3.1 Contractions and Fixed Points 53

2.3.2 The Case of One Variable Functions 62

2.4 Graves Theorem 71

2.5 Semicontinuous Functions 73

3 The Study of Smooth Optimization Problems 78

3.1 General Optimality Conditions 78

3.2 Functional Restrictions 91

3.2.1 Fritz John Optimality Conditions 92

3.2.2 Karush-Kuhn-Tucker Conditions 95

3.2.3 Qualification Conditions 102

3.3 Second-order Conditions 109

3.4 Motivations for Scientific Computations 113

4 Convex Nonsmooth Optimization 117

4.1 Further Properties and Separation of Convex Sets 117

4.2 The Subdifferential of a Convex Function 120

4.3 Optimality Conditions 125

5 Lipschitz Nonsmooth Optimization 131

5.1 Clarke Generalized Calculus 131

5.1.1 Clarke Subdifferential 131

5.1.2 Clarke Tangent and Normal Cones 148

5.1.3 Optimality Conditions in Lipschitz Optimization 156

5.2 Mordukhovich Generalized Calculus 159

5.2.1 Fréchet and Mordukhovich Normal Cones 160

5.2.2 Fréchet and Mordukhovich Subdifferentials 167

5.2.3 The Extremal Principle 177

5.2.4 Calculus Rules 181

5.2.5 Optimality Conditions 193

6 Basic Algorithms 196

6.1 Algorithms for Nonlinear Equations 197

6.1.1 Picard's Algorithm 197

6.1.2 Newton's Method 203

6.2 Algorithms for Optimization Problems 206

6.2.1 The Case of Unconstrained Problems 206

6.2.2 The Case of Constraint Problems 213

6.3 Scientific Calculus Implementations 223

7 Exercises and Problems, and their Solutions 240

7.1 Analysis of Real Functions of One Variable 240

7.2 Nonlinear Analysis 252

7.3 Smooth Optimization 263

7.4 Nonsmooth Optimization 291

Bibliography 313

List of Notations 315

Index 317