Báo cáo : Karush-Kuhn-Tucker conditions with higher-order complementarity slackness under Holder metric regularity for nonsmooth equilibrium problems
Người trình bày: TS. Nguyễn Minh Tùng (ĐH Khoa học Tự nhiên TPHCM)
Thời gian: Chủ nhật - 24.02.2019 - 09h30
Địa điểm: phòng F.300 - ĐH Khoa học Tự nhiên, 227 Nguyễn Văn Cừ, Quận 5, TpHCM
Abstract: In [1], higher-order Karush-Kuhn-Tucker multiplier rules with higher-order complementarity slackness for the objective were established in terms of contingent derivatives for set-valued optimization problems, under linear metric regularity. When, relaxing the metric regularity assumptions to Holder regularity, the Studniarski derivative needs to replace the contingent derivatives, but then the obtained multiplier rules do not include higher-order complementarity slackness. Though only one novelty of the two, higher-order complementarity slackness or Holder regularity, can be received for each obtained multiplier rule, and the involved derivatives need be different for one type or the other one of rules, the results in [1] are significant developments in optimality conditions.
The main aim of this paper is to complete the above lacking results. Namely, we propose a new kind of derivatives called (higher-order) quasi-contingent derivatives, prove its calculus rules, and design new sets of m-order critical directions in terms of the quasi-contingent derivative and involving information of optimality conditions of orders 1, …, m-1. Then, applying these new objects, we establish higher-order Karush-Kuhn-Tucker multiplier rules with higher-order complementarity slackness for both the objective and the constraints, under Holder regularity conditions. We discuss both cases of the solid and nonsolid ordering cone of the constraint space and other related facts such as several important kinds of constraint qualification conditions and higher-order envelope-like phenomena. These results are significantly new even in second-order optimality conditions for the classical smooth mathematical programming, but to encompass recent developments in the literature, we choose a presentation for equilibrium problems (more general than optimization problems) and for open-cone solutions and proper solutions (more general than weak solutions).
Reference
1. Phan Quoc Khanh, Nguyen Minh Tung, Higher-order Karush-Kuhn-Tucker conditions in nonsmooth optimization, SIAM J. Optimization, Vol. 28, no. 1, 820-848, 2018.