Necessary conditions for approximate solutions of vector and set optimization problems with variable domination structure
Marius Durea, Christian Günther, Radu Strugariu, Christiane Tammer
TL;DR
The paper addresses necessary conditions for approximate solutions to vector and set optimization problems under variable domination structures (vds). It develops an Ekeland-type variational principle tailored to vds and couples it with Mordukhovich generalized differentiation to derive necessary conditions for approximately nondominated points in both single-valued and set-valued objective settings. A novel Gerstewitz-type scalarization functional is introduced to obtain scalarized EVP results, and a directional openness framework for sums of set-valued maps yields coderivative-based optimality conditions, with symmetric treatment for $K$- and $Q$-driven domination. The results advance the theory of approximate optimality under variable domination structures and suggest directions for numerical algorithms and broader extensions.
Abstract
We consider vector and set optimization problems with respect to variable domination structures given by set-valued mappings acting between the preimage space and the image space of the objective mapping, as well as by set-valued mappings with the same input and output space, that coincides with the image space of the objective mapping. The aim of this paper is to derive necessary conditions for approximately nondominated points of problems with a single-valued objective function, employing an extension of Ekeland's Variational Principle for problems with respect to variable domination structures in terms of generalized differentiation in the sense of Mordukhovich. For set-valued objective mappings, we derive necessary conditions for approximately nondominated points of problems with variable domination structure taking into account the incompatibility between openness and optimality and a directional openness result for the sum of set-valued maps. We describe the necessary conditions for approximately nondominated points of set optimization problems with variable domination structure in terms of the limiting (Mordukhovich) generalized differentiation objects.