CALMNESS AND THE ABADIE CQ FOR MULTIFUNCTIONS AND LINEAR REGULARITY FOR A COLLECTION OF CLOSED SETS

摘要

contrast to many existing results on calmness (metric subregularity) of multifunctions established by using dual notions like the normal cones and coderivatives, this paper is devoted to provide primal issues for calmness. In terms of the Bouligand tangent cone and Clarke tangent cone, we introduce and study the Abadie constraint qualification (ACQ) and the strong ACQ for a nonconvex multifunction. With the help of the strong ACQ, we establish several primal characterizations for calmness and strong calmness of multifunctions with the Shapiro property (an extension of both convexity and smoothness). As applications, we consider linear regularity for a finite or infinite collection of closed (not necessarily convex) sets and establish some primal sufficient and necessary conditions for linear regularity, which extend and improve some existing results to either the nonconvex or the infinite index set case.