Proper minimal points of nonconvex sets in Banach spaces in terms of the limiting normal cone

Abstract

In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results.

Keywords:

• Proper minimal point
• limiting (Mordukhovich) normal cone
• nonsmooth analysis
• ordered Banach space
• Asplund space

Acknowledgements

The authors would like to express their gratitude to the anonymous referee and the associate editor for their helpful comments on the first version of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was in part supported by a grant from School of Mathematics, Institute for Research in Fundamental Sciences (IPM) [grant number 94260124].