Stability of Optimal Points with Respect to Improvement Sets

The aim of this paper is to study the stability of optimal point sets based on the improvement set E by using the scalarization method and the density results. Under the convergence of a sequence of sets in the sense of Wijsman, we derive the convergence of the sets of E-optimal points, weak E-optimal points, E-quasi-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points in the sense of Wijsman. Moreover, we obtain the semicontinuity of E-optimal point mapping, weak E-optimal point mapping, E-quasi-optimal point mapping, E-Benson proper optimal point mapping, E-super optimal point mapping and E-strictly optimal point mapping. Finally, we make a new attempt to establish Lipschitz continuity of these E-optimal point mappings under some suitable conditions.