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

• Published in: Optimization Vol. 72; no. 3; pp. 763 - 788
• Main Author: Minh Tung, Nguyen
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.03.2023; Taylor & Francis LLC
• Subjects: 9C48; Calculus; contingent-type derivative; Convexity; Hölder directional metric subregularity; Karush-Kuhn-Tucker condition; Kuhn-Tucker method; Optimization; strict efficient solution; Studniarski's derivative; Theorems
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2021.1984470

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.