Density and connectedness of optimal points with respect to improvement sets

Abstract

Based on the improvement set E, this paper aims at investigating density and connectedness 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. By virtue of the separation theorem for convex sets, we obtain the scalarizaiton results for the sets of E-optimal points, weak E-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. Then we make a new attempt to establish some density theorems for the sets of these optimal points. Finally, we investigate connectedness and arcwise connectedness of the sets of various notions of E-optimality by using the scalarization method and the density theorems.

Keywords:

• Vector optimization
• density theorem
• connectedness
• improvement set

AMS Subject Classifications:

• 90C26
• 90C29

Acknowledgments

The author is grateful to the editor and the two anonymous reviewers for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China (11801257).