On Levitin–Polyak well-posedness and stability in set optimization

In this paper, Levitin–Polyak (in short LP) well-posedness in the set and scalar sense are defined for a set optimization problem and a relationship between them is found. Necessary and sufficiency criteria for the LP well-posedness in the set sense are established. Some characterizations in terms of Hausdorff upper semicontinuity and closedness of approximate solution maps for the LP well-posedness have been obtained. Further, a sequence of solution sets of scalar problems is shown to converge in the Painlevé–Kuratowski sense to the minimal solution sets of the set optimization problem. Finally, the perturbations of the ordering cone and the feasible set of the set optimization problem are considered and the convergence of its weak minimal and minimal solution sets in terms of Painlevé–Kuratowski convergence is discussed.