Stability and Scalarization for a Unified Vector Optimization Problem

Abstract

This paper aims at investigating the Painlevé-Kuratowski convergence of solution sets of a sequence of perturbed vector problems, obtained by perturbing the feasible set and the objective function of a unified vector optimization problem, in real normed linear spaces. We establish convergence results, both in the image and given spaces, under the assumptions of domination and strict domination properties. Moreover, scalarization techniques are employed to establish the Painlevé-Kuratowski convergence in terms of the solution sets of a sequence of scalarized problems.

Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency

“…To treat this problem, many papers in the literature have imposed additional conditions under which the efficient solution set coincides with the weakly efficient solution one, whose closedness can be easier to obtain. For more details, we would like to refer the reader to [10,11,15,25,36]. Taking this observation into account, we are interested in establishing the closedness of Eff(K , F) in which Eff(K , F) and weakly efficient solution set are distinct.…”

Scalarization and convergence in unified set optimization

“…In order to study stability via scalarization the associated scalarized problem is perturbed instead of the original problem and then the convergence of solution sets of perturbed scalarized problems to the solution set of the given problem is established. In vector case, a lot of work has been done in this direction (see [20,30] and references therein). Moreover, in case of set optimization, the study of stability on the ground of scalarization techniques is not much advanced.…”

On the Generic Uniqueness of Pareto-Efficient Solutions of Vector Optimization Problems

“…Tykhonov well-posedness plays a very important role in solving optimization problems by the iterative method (see, e.g., [16]). In vector optimization theory, there are a lot of approaches which give rise to various well-posedness notions (see, e.g., [17][18][19][20][21][22][23][24][25]). e Pareto-efficient solutions of vector optimization problems are based on partial order, so they are not generally unique and even not generically unique (see Example 1).…”