Higher-Order Karush--Kuhn--Tucker Conditions in Nonsmooth Optimization

Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

“…In most known necessary conditions, (F, G) and its derivatives are used, for example in [11,13,26,27,46]. Inspired by the idea in [28,30], the Aubin property is employed to obtain a sharper second-order necessary conditions involving separately derivatives of F and G. From there, constraint qualifications of the Kurcyusz-Robinson-Zowe type, not qualification condition in terms of (F, G), can be invoked to get Karush-Kuhn-Tucker multiplier rules for problem (P).…”

“…Our constraint qualifications in Theorems 4.2 and 4.3 are form of Kurcyusz-Robinson-Zowe condition, see more details in [12,28,30,46]. With the help of Robinson-Ursescu open mapping theorem, a sufficient condition for the qualification (4.4) is studied.…”

High order tangent vectors to sets with applications to constrained optimization problems

“…Notions of first, second, and higher order tangent vectors to sets are crucial in variational analysis. A far from complete but very numerous list of (mainly English-language) works devoted to this subject can be found in the review [1], the monograph [2], and also in relatively recent papers [3][4][5]. We supplement this list with works [6][7][8][9][10][11][12][13][14][15][16][17][18] related to this area of research, written mainly in Russian and not included in the surveys of the publications mentioned above.…”

“…Somewhat later, under the name high-order tangent sets, they were considered in the monograph [20]. In subsequent years various high-order variational (contingent, adjacent and others) sets and based on them definitions of high-order derivatives for (set-valued) mappings with applications to high-order optimality conditions were the subject of many papers (see [3,[33][34][35][36][37][38][39][40][41][42] and references therein). Following [14], in the present paper we refer to high order variational sets introduced by Hoffmann and Kornstaedt in [32] as high order proper tangent sets.…”