A Subdifferential Condition for Calmness of Multifunctions

Abstract

A condition ensuring calmness of a class of multifunctions between finite-dimensional spaces is derived in terms of subdifferential concepts developed by Mordukhovich. The considered class comprises general constraint set mappings as they occur in optimization or mappings associated with a certain type of variational system. The condition ensuring calmness is obtained as an appropriate reduction of Mordukhovich's well-known characterization of the stronger Aubin property. (Roughly spoken, one may pass to the boundaries of normal cones or subdifferentials when aiming at calmness.) It allows one to derive dual constraint qualifications in nonlinear optimization that are weaker than conventional ones (e.g., Mangasarian–Fromovitz) but still sufficient for the existence of Lagrange multipliers.