Strict efficiency conditions for nonsmooth optimization with inclusion constraint under Hölder directional metric subregularity

ABSTRACT

In this paper, we investigate the higher-order optimality conditions for strict efficient solutions to a nonsmooth optimization problem subject to inclusion constraint. A concept of higher-order contingent derivative type for set-valued maps, some main calculus rules of which are obtained, is proposed and employed with the Robinson–Ursescu open mapping theorem to get a Karush–Kuhn–Tucker condition under assumptions of Hölder direction metric subregularity. Sufficient conditions for these solutions are established without convexity assumptions and possibly without the existence of derivatives. As an application, we extend some optimality conditions for a nonsmooth optimization problem subject to generalized inequality constraint. Another application is presented for necessary and sufficient conditions for robust local strict efficient solutions in uncertain vector optimization. Some examples are provided to illustrate our theorems as well. Our results are new and improve the existing ones in the literature substantially.

KEYWORDS:

• Karush–Kuhn–Tucker condition
• contingent-type derivative
• Studniarski's derivative
• Hölder directional metric subregularity
• strict efficient solution

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49J52
• 90C29
• 90C46
• 9C48

Acknowledgments

The author is very grateful to the Editors and the Anonymous Referee for their helpful comments and suggestions that have contributed to improve the paper. A part of this work was done during research stays of the authors at Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for its hospitality and support. This work was supported by the National Foundation for Science and Technology Development (NAFOSTED) grand 101.01-2021.13.

Disclosure statement

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