Abstract
The aim of this paper is to study stability of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems with respect to the perturbations of feasible sets and objective mappings. We introduce a new metric between two set-valued mappings by utilizing a Hausdorff-type distance proposed by Han [A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems. Appl Anal. 2022;101:1243–1260]. The new metric between two set-valued mappings allows us to discuss set optimization problems with respect to the perturbation of objective mappings. Then, we establish semicontinuity and Lipschitz continuity of l-minimal approximate solution mapping and weak l-minimal approximate solution mapping to parametric set optimization problems by using the scalarization method and a density result. Finally, our main results are applied to stability of the approximate solution sets for vector optimization problems.
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