A new approach to duality in vector optimization

In this article, we develop a new approach to duality theory for convex vector optimization problems. We modify a given (set-valued) vector optimization problem such that the image space becomes a complete lattice (a sublattice of the power set of the original image space), where the corresponding i...

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Bibliographic Details
Published inOptimization Vol. 56; no. 1-2; pp. 221 - 239
Main Authors Löhne, Andreas, Tammer, Christiane
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Group 01.02.2007
Taylor & Francis LLC
Subjects
Complete lattice
Duality
Fenchel duality
Image space
Mathematics Subject Classifications 2000: 90C29
Optimization
Self-infimal sets
Set inclusion problems
Studies
Upper closed sets
Vector optimization
Vector space
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ISSN0233-1934
1029-4945
DOI10.1080/02331930600819720

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Summary:In this article, we develop a new approach to duality theory for convex vector optimization problems. We modify a given (set-valued) vector optimization problem such that the image space becomes a complete lattice (a sublattice of the power set of the original image space), where the corresponding infimum and supremum are sets that are related to the set of (minimal and maximal) weakly efficient points. In doing so we can carry over the structures of the duality theory in scalar convex programming. Exemplarily this is demonstrated for the case of Fenchel duality. We also show the relationship to set-valued optimization based on the ordering 'set inclusion'. Finally, some consequences for duality in linear vector optimization are discussed.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331930600819720