A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization

In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map $F$ respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for $F$. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar- single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of $F$.


Optimization(2018) Vol. 67, No.7, 1031-1050