Metric Subregularity for a Piecewise Linear Mapping between two Banach Spaces

Observe that the graph of a piecewise linear mapping to an infinite dimensional Banach space is not the union of finitely many convex polyhedra. This and Robinson's theorem on local metric subregularity for a polyhedral mapping motivate us to consider the metric subregualrity for a piecewise linear mapping between two general Banach spaces. We prove that a piecewise linear mapping G between two Banach spaces is boundedly metrically subregular at any point in its graph gph(G) if and only if G is metrically subregular at some point in gph(G) if and only if G has the L-closed range property, which complements Robinson's theorem. As an complement of Mordukhovich's criterion on the metric regularity in the finite dimension case, we also provide a dual characterization for a piecewise linear mapping to be metrically regular.