Slater Condition for Tangent Derivatives

Xi Yin Zheng ()
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Xi Yin Zheng: Department of Mathematics, Yunnan University, Kunming 650091, People’s Republic of China

Mathematics of Operations Research, 2022, vol. 47, issue 4, 3282-3303

Abstract: Noting that the existing Slater condition, as a fundamental constraint qualification in optimization, is only applicable in the convex setting, we introduce and study the Slater condition for the Bouligand and Clarke tangent derivatives of a general vector-valued function F with respect to a closed convex cone K . Without any assumption, it is proved that the Slater condition for the Clarke (respectively, Bouligand) tangent derivative with respect to K is always stable when the objective function F undergoes small Lipschitz (calm) perturbations. Based on this, we prove that if the Clarke (Bouligand) tangent derivative of F satisfies the Slater condition (with respect to K ) then the conic inequality determined by F has a stable metric subregularity when F undergoes small Lipschitz (calm regular) perturbations. In the composite-convexity case, the converse implication is also proved to be true. Moreover, under the Slater condition for the tangent derivative of F , it is proved that the normal cone to the sublevel set of F can be formulated by the subdifferential of F , which improves the corresponding results in either the smooth or convex case. As applications, without any qualification assumption, we improve and generalize formulas for the normal cone to a convex sublevel set by Cabot and Thibault [(2014), Sequential formulae for the normal cone to sublevel sets. Transactions of the American Mathematical Society 366(12):6591–6628]. With the help of these formulas, some new Karush–Kuhn–Tucker optimality conditions are established.

Keywords: Primary: 49J52; 49K40; 90C31; tangent derivative; Slater condition; normal cone; metric subregularity; KKT condition (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:4:p:3282-3303

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