External and internal stability in set optimization

Abstract

The main aim of this paper is to establish stability in set optimization in terms of convergence of a sequence of solution sets of perturbed set optimization problems to the solution set of the original set optimization problem both in the image space and the given space. The perturbed problems are obtained by perturbing the feasible set without changing the objective map. Formulations of external stability and internal stability are considered in the image space. External stability, which pertains to complete convergence of a subsequence of weak minimal solution sets, both in the sense of Hausdorff and $\text{Kuratowski}-\text{Painlev}\acute{\text{e}}$ convergence of sets, is established under certain compactness and continuity assumptions. This leads to the upper $\text{Kuratowski}-\text{Painlev}\acute{\text{e}}$ convergence of the sequence of solution sets in the given space. Internal stability is established for minimal solution sets under certain continuity, compactness and domination assumptions which leads to the lower $\text{Kuratowski- Painlev}\acute{\text{e}}$ convergence of the sequence of solution sets in the given space. External stability for minimal solution sets and internal stability for weak minimal solution sets are deduced under the strict quasiconvexity assumption. In particular, the results are also deduced for a vector optimization problem.

Keywords:

• $\text{Kuratowski- Painlev}\acute{\text{e}}$ convergence
• Hausdorff convergence
• strict quasiconvexity
• stability

AMS CLASSIFICATIONS:

• 49J53
• 90C31
• 90C29

Acknowledgements

The authors are thankful to referees for their valuable comments and suggestions which helped to improve the presentation of the paper. Also, the authors are thankful to the associate editor for providing an alternate proof of Lemma 2.3.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Karuna http://orcid.org/0000-0002-3498-6029

Additional information

Funding

The research of C. S. Lalitha was supported by MATRIC scheme of Department of Science and Technology, India.