On the Lipschitzian properties of polyhedral multifunctions

In this paper, we show that for a polyhedral multifunction F : R n ! R m 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 aane function f : R n ! R n , 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 aane 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.