Vector Optimization

By: Andreas LÃ¶hne

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Hardcover | 27 May 2011

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218 Pages

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24.77 x 16.51 x 1.27

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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.

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Introduction	p. 1
General and Convex Problems
A complete lattice for vector optimization	p. 7
Partially ordered sets and complete lattices	p. 8
Conlinear spaces	p. 11
Topological vector spaces	p. 13
Infimal and supremal sets	p. 19
Hyperspaces of upper closed sets and self-infimal sets	p. 25
Subspaces of convex elements	p. 29
Scalarization methods	p. 34
A topology on the space of self-infimal sets	p. 37
Notes on the literature	p. 41
Solution concepts	p. 43
A solution concept for lattice-valued problems	p. 45
A solution concept for vector optimization	p. 51
Semicontinuity concepts	p. 55
A vectorial Weierstrass theorem	p. 61
Mild solutions	p. 62
Maximization problems and saddle points	p. 67
Notes on the literature	p. 73
Duality	p. 75
A general duality concept applied to vector optimization	p. 76
Conjugate duality	p. 78
Conjugate duality of type I	p. 79
Duality result of type II and dual attainment	p. 83
The finite dimensional and the polyhedral case	p. 91
Lagrange duality	p. 92
The scalar case	p. 93
Lagrange duality of type I	p. 95
Lagrange duality of type II	p. 98
Existence of saddle points	p. 102
Connections to classic results	p. 103
Notes on the literature	p. 106
Linear Problems
Solution concepts and duality	p. 111
Scalarization	p. 112
Basic methods	p. 112
Solutions of scalarized problems	p. 114
Solution concept for the primal problem	p. 116
Set-valued duality	p. 122
Lattice theoretical interpretation of duality	p. 130
Geometric duality	p. 137
Homogeneous problems	p. 151
Identifying faces of minimal vectors	p. 156
Notes on the literature	p. 158
Algorithms	p. 161
Benson's algorithm	p. 162
A dual variant of Benson's algorithm	p. 169
Solving bounded problems	p. 177
Solving the homogeneous problem	p. 178
Computing an interior point of the lower image	p. 190
Degeneracy	p. 192
Notes on the literature	p. 194
References	p. 197
Index	p. 205
Table of Contents provided by Ingram. All Rights Reserved.

Vector Optimization

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Hardcover

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