In this paper, we show that for a polyhedral multifunction F : Rn → Rm with convex range, the inverse function F–1 is locally lower Lipschitzian at every point of the range of F (equivalently Lipschitzian on the range of F) if and only if the function F is open. As a consequence, we show that for a piecewise affine function f : Rn → Rn, f is surjective and f–1 is Lipschitzian if and only if f is coherently oriented. An application, via Robinson’s normal map formulation, leads to the following result in the context of affine variational inequalities: the solution mapping (as a function of the data vector) is nonempty-valued and Lipschitzian on the entire space if and only if the solution mapping is single-valued. This extends a recent result of Murthy, Parthasarathy and Sabatini, proved in the setting of linear complementarity problems.
