Strong Subregularity of Variational Inequalities: General Results and Case Studies

Abstract

Metric sub-regularity of mappings associated with variational inequalities (VIs) is a property that proved to be fundamentally important in the analysis of approximation methods for VIs, such as finite-dimensional approximations, gradient projection methods, Newton-type methods, etc. Such VIs arise, in particular, when considering the system of optimality conditions for control-constrained optimal control problems. The variables in the VIs in this case consist basically of the triples (state function, co-state function, control function). The choice of an appropriate (metric) space for the control function depends on the problem structure: L2, L∞, L1 or other (not induced by norms) metric spaces are relevant for specific classes of problems. The talk will present new sufficient conditions for strong metric sub-regularity of VIs in (nonreflexive) Banach spaces, and, as application, several new result about sub-regularity of the Pontryagin optimality system for several classes of ODE or PDE optimal control problems.