Vector Optimization with Infimum and Supremum

by Andreas Löhne

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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Year of publication:	2011
Authors:	Löhne, Andreas
Publisher:	Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg
Subject:	Mathematische Optimierung \| Mathematical programming \| Theorie \| Theory \| Mehrkriterielle Optimierung
Description of contents:	Table of Contents [swbplus.bsz-bw.de] ; Description [swbplus.bsz-bw.de] ; Description [zbmath.org]

Extent:	Online-Ressource (X, 206p. 30 illus, digital)
Series:	Vector Optimization SpringerLink / Bücher
Type of publication:	Book / Working Paper
Language:	English
Notes:	Includes bibliographical references and index Vector Optimization with Infimum and Supremum; Preface; Contents; Introduction; Part I General and Convex Problems; Chapter 1 A complete lattice for vector optimization; 1.1 Partially ordered sets and complete lattices; 1.2 Conlinear spaces; 1.3 Topological vector spaces; 1.4 Infimal and supremal sets; 1.5 Hyperspaces of upper closed sets and self-infimal sets; 1.6 Subspaces of convex elements; 1.7 Scalarization methods; 1.8 A topology on the space of self-infimal sets; 1.9 Notes on the literature; Chapter 2 Solution concepts; 2.1 A solution concept for lattice-valued problems 2.2 A solution concept for vector optimization2.3 Semicontinuity concepts; 2.4 A vectorial Weierstrass theorem; 2.5 Mild solutions; 2.6 Maximization problems and saddle points; 2.7 Notes on the literature; Chapter 3 Duality; 3.1 A general duality concept applied to vector optimization; 3.2 Conjugate duality; 3.2.1 Conjugate duality of type I; 3.2.2 Duality result of type II and dual attainment; 3.2.3 The finite dimensional and the polyhedral case; 3.3 Lagrange duality; 3.3.1 The scalar case; 3.3.2 Lagrange duality of type I; 3.3.3 Lagrange duality of type II; 3.4 Existence of saddle points 3.5 Connections to classic results3.6 Notes on the literature; Part II Linear Problems; Chapter 4 Solution concepts and duality; 4.1 Scalarization; 4.1.1 Basic methods; 4.1.2 Solutions of scalarized problems; 4.2 Solution concept for the primal problem; 4.3 Set-valued duality; 4.4 Lattice theoretical interpretation of duality; 4.5 Geometric duality; 4.6 Homogeneous problems; 4.7 Identifying faces of minimal vectors; 4.8 Notes on the literature; Chapter 5 Algorithms; 5.1 Benson's algorithm; 5.2 A dual variant of Benson's algorithm; 5.3 Solving bounded problems 5.4 Solving the homogeneous problem5.5 Computing an interior point of the lower image; 5.6 Degeneracy; 5.7 Notes on the literature; References; Index
ISBN:	978-3-642-18351-5 ; 978-3-642-18350-8
Other identifiers:	10.1007/978-3-642-18351-5 [DOI]
Classification:	Angewandte Mathematik ; Theoretische Informatik
Source:	ECONIS - Online Catalogue of the ZBW

Persistent link: https://ebvufind01.dmz1.zbw.eu/10014015337