Vector Optimization with Infimum and Supremum

by Andreas Löhne

Overview

The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is a subset of the power set, in fact, the space of self-infimal sets. Based on this idea we establish solution concepts, existence and duality results and algorithms for the linear case. The main advantage of this approach is the high degree of analogy to corresponding results of Scalar Optimization. The concepts and results are used to explain and to improve practically relevant algorithms for linear vector optimization problems.

Product Details

ISBN-13:	9783642268410
Publisher:	Springer Berlin Heidelberg
Publication date:	07/14/2013
Series:	Vector Optimization
Edition description:	2011
Pages:	206
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

Introduction 1

Part I General and Convex Problems

1 A complete lattice for vector optimization 7

1.1 Partially ordered sets and complete lattices 8

1.2 Conlinear spaces 11

1.3 Topological vector spaces 13

1.4 Infimal and supremal sets 19

1.5 Hyperspaces of upper closed sets and self-infimal sets 25

1.6 Subspaces of convex elements 29

1.7 Scalarization methods 34

1.8 A topology on the space of self-infimal sets 37

1.9 Notes on the literature 41

2 Solution concepts 43

2.1 A solution concept for lattice-valued problems 45

2.2 A solution concept for vector optimization 51

2.3 Semicontinuity concepts 55

2.4 A vectorial Weierstrass theorem 61

2.5 Mild solutions 62

2.6 Maximization problems and saddle points 67

2.7 Notes on the literature 73

3 Duality 75

3.1 A general duality concept applied to vector optimization 76

3.2 Conjugate duality 78

3.2.1 Conjugate duality of type I 79

3.2.2 Duality result of type II and dual attainment 83

3.2.3 The finite dimensional and the polyhedral case 91

3.3 Lagrange duality 92

3.3.1 The scalar case 93

3.3.2 Lagrange duality of type I 95

3.3.3 Lagrange duality of type II 98

3.4 Existence of saddle points 102

3.5 Connections to classic results 103

3.6 Notes on the literature 106

Part II Linear Problems

4 Solution concepts and duality 111

4.1 Scalarization 112

4.1.1 Basic methods 112

4.1.2 Solutions of scalarized problems 114

4.2 Solution concept for the primal problem 116

4.3 Set-valued duality 122

4.4 Lattice theoretical interpretation of duality 130

4.5 Geometric duality 137

4.6 Homogeneous problems 151

4.7 Identifying faces of minimal vectors 156

4.8 Notes on the literature 158

5 Algorithms 161

5.1 Benson's algorithm 162

5.2 A dual variant of Benson's algorithm 169

5.3 Solving bounded problems 177

5.4 Solving the homogeneous problem 178

5.5 Computing an interior point of the lower image 190

5.6 Degeneracy 192

5.7 Notes on the literature 194

References 197

Index 205