Variational principles in set optimization with domination structures and application to changing jobs

Abstract : This paper is devoted to new versions of Ekeland’s variational principle in set optimization with domination structure, where set optimization is an extension of vector optimization from vector-valued functions to set-valued maps using Kuroiwa’s set-less relations to compare one entire image set with another whole image set, and where domination structure is an extension of ordering cone in vector optimization; it assigns each element of the image space to its own domination set. We use Gerstewitz’s nonlinear scalarization function to convert a set-valued map into an extended real-valued function and the idea of the proof of Dancs-Hegedüs-Medvegyev’s fixed-point theorem. Our setting is applicable to dynamic processes of changing jobs in which the cost function does not satisfy the symmetry axiom of metrics and the class of set-valued maps acting from a quasimetric space into a real linear space. The obtained result is new even in simpler settings.