Higher-order variational set of the benson proper perturbation map in set-valued optimization

Abstract

In the paper, we study sensitivity analysis in set-valued optimization, a research direction has been attracting much attention of many mathematicians in the world recently. The main derivative used in the paper is higher-order variational set (introduced by Khanh and Tuan in 2008) which is considered as a generalization of the contingent derivative (known as the first and the most popular derivative in set-valued optimization). Firstly, we establish relationships between higher-order variational sets of a given set-valued map and those of its profile (extended by a ordering cone). Then, we give results on higher-order variational set of the Benson proper perturbation map for a kind of set-valued optimization problem, the perturbation map is defined in the objective space. Finally, we apply the obtained results to sensitivity analysis for optimal-value map of a parametrized constrained set-valued optimization problem whose the objective map and constrained maps depends on some parameter. More precisely, some results on sensitivity analysis for parametrized constrained set-valued optimization problem are obtained. The content of the paper gives us more applications of higher-order variational set in set-valued optimization.