A new approach to duality in vector optimization

Abstract

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.