Metric Subregularity of Multifunctions and Applications to Characterizations of Asplund Spaces

Abstract: In this paper, we investigate metric subregularity of multifunctions between Asplund spaces. Using Mordukhovich normal cones and coderivatives, we introduce the limiting Basic Constraint Qualification (BCQ) associated with a given multifunction. This BCQ provides necessary dual conditions for the metric subregularity of multifunctions in the Asplund space setting. Furthermore, we establish characterizations of Asplund spaces in terms of the limiting BCQ condition implied by metric subregularity. By employing Frechet normal cones and coderivatives, we derive necessary dual conditions for metric subregularity expressed as fuzzy inclusions, and we also obtain characterizations of Asplund spaces via these fuzzy inclusions. As an application, we examine metric subregularity of the conic inequality defined by a vector-valued function and a closed (not necessarily convex) cone with a nontrivial recession cone. By using Mordukhovich and Frechet subdifferentials relative to the given cone, we establish necessary dual conditions for the metric subregularity of such inequalities in Asplund spaces. The results based on Mordukhovich subdifferentials characterize Asplund spaces, while those based on Frechet subdifferentials yield necessary or sufficient conditions for Asplund spaces. These conditions recover, as special cases, the known error-bound results for inequalities defined by extended-real-valued functions on Asplund spaces. Overall, this work highlights that the validity of necessary conditions formulated via normal cones and subdifferentials for error bounds of convex or nonconvex inequalities depends crucially on the Asplund property of the underlying space.