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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 $F:{R}^{n}\to {R}^{m}$ 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 $F\right)$ if and only if the function $F$ is open. As a consequence, we show that for a piecewise affine function $f:{R}^{n}\to {R}^{n}$, $f$ 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.
##### MSC:
 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)