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On the Lipschitzian properties of polyhedral multifunctions. (English) Zbl 0854.49010
Summary: 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 affine 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 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 results of Murthy, Parthasarathy and Sabatini, proved in the setting of linear complementarity problems.
49J40Variational methods including variational inequalities
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)