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Zbl 0854.49010
Gowda, M. Seetharama ; Sznajder, Roman
On the Lipschitzian properties of polyhedral multifunctions. (English)
[J] Math. Program. 74, No.3 (A), 267-278 (1996). ISSN 0025-5610; ISSN 1436-4646/e

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 $F)$ 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 2000:: *49J40 Variational methods including variational inequalities
90C33 Complementarity problems

Keywords: polyhedral multifunction; affine variational inequalities

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