Although the property of strong metric sub regularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong metric regularity. In this talk I will try to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. In particular, it obeys the inverse function theorem paradigm also for nonsmooth perturbations.
A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fr\’echet coderivative will be presented, and it will be shown by a counterexample that subjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton’s method for solving variational inequalities will be considered including inexact and semismooth methods. A characterization of the strong metric subregularity of the KKT mapping will be demonstrated, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.
Asen Dontchev, Mathematical Reviews and University of Michigan