Lipschitzian multifunctions and a Lipschitzian inverse mapping theorem

We introduce a new class of multifunctions whose graphs under certain ‘kernel inverting’ matrices, are locally equal to the graphs of Lipschitzian (single-valued) mappings. We characterize the existence of Lipschitzian localizations of these multifunctions in terms of a natural condition on a generalized Jacobian mapping. One corollary to our main result is a Lipschitzian inverse mapping theorem for the broad class of ‘max hypomonotone’ multifunctions. We apply our theoretical results to the sensitivity analysis of solution mappings associated with parameterized optimization problems. In particular, we obtain new characterizations of the Lipschitzian stability of stationary points and Karush–Kuhn–Tucker pairs associated with parametrized nonlinear programs.