Optimal Elements in Vector Optimization with a Variable Ordering Structure

Abstract

Optimality concepts for vector optimization problems with a variable ordering structure are examined. These considerations are motivated by an application in medical image registration, where the preferences vary depending on the element in the image space.The idea of variable ordering structures was first introduced by Yu (J. Opt. Theory Appl. 14:319–377, [1974]) in terms of domination structures. Variable ordering structures mean that there is a set-valued map with cone values that associates to each element an ordering. A candidate element is called a nondominated element iff it is not dominated by other reference elements w.r.t. their corresponding ordering. In addition to nondominated elements, another notion of optimal elements, called minimal elements, has also been discussed. For that notion, only the ordering of the candidate element itself is considered. This paper shows that these two different optimality concepts are connected by duality properties. Characterizations and existence results of the above two solution concepts are also given.