We introduce a new class of multifunctions whose graphs under certain "kern
el inverting" matrices, are locally equal to the graphs of Lipschitzian (si
ngle-valued) mappings. We characterize the existence of Lipschitzian locali
zations of these multifunctions in terms of a natural condition on a genera
lized Jacobian mapping. One corollary to our main result is a Lipschitzian
inverse map ping theorem for the broad class of "max hypomonotone" multifun
ctions. We apply our theoretical results to the sensitivity analysis of sol
ution mappings associated with parameterized optimization problems. In part
icular, we obtain new characterizations of the Lipschitzian stability of st
ationary points and Karush-Kuhn-Tucker pairs associated with parameterized
nonlinear programs.