Stability of Implicit Multifunctions via Point-Based Criteria and Applications

Abstract

We first establish new point-based sufficient conditions for an implicit multifunction to achieve the metric subregularity. These conditions are expressed in terms of limit set critical for metric subregularity of the corresponding parametric multifunction formulated the implicit multifunction. We then show that the sufficient conditions obtained turn out to be also necessary for the metric subregularity of the implicit multifunction in the case, where the corresponding parametric multifunction is (locally) convex and closed. In this way, we give criteria ensuring the calmness for the implicit multifunction. As applications, we derive point-based sufficient and necessary conditions for a multifunction (resp., its inverse multifunction) to have the metric subregularity (resp., the calmness) and for the efficient solution map of a parametric vector optimization problem to admit the metric subregularity as well as the calmness.

Primal necessary characterizations of transversality properties

“…, n) with ϕ(max 1≤i≤n x i ) < δ , and any number ρ satisfying ϕ (max 1≤i≤n x i ) < ρ < δ , condition (1) holds. The latter condition obviously implies (9).…”

“…[14,15]. The general approach of the current paper can be applied to the analysis of implicit multifunctions, e.g., like those studied in [8,9].…”