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On the Lipschitzian properties of polyhedral multifunctions

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Abstract

In this paper, we show that for a polyhedral multifunctionF:R n →R m with convex range, the inverse functionF −1 is locally lower Lipschitzian at every point of the range ofF (equivalently Lipschitzian on the range ofF) if and only if the functionF is open. As a consequence, we show that for a piecewise affine functionf:R n →R n,f is surjective andf −1 is Lipschitzian if and only iff 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland Baltimore County, 21228, Baltimore, MD, USA

    M. Seetharama Gowda & Roman Sznajder

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  1. M. Seetharama Gowda
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  2. Roman Sznajder
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Correspondence to M. Seetharama Gowda.

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Research supported by the National Science Foundation Grant CCR-9307685.

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Seetharama Gowda, M., Sznajder, R. On the Lipschitzian properties of polyhedral multifunctions. Mathematical Programming 74, 267–278 (1996). https://doi.org/10.1007/BF02592199

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  • DOI: https://doi.org/10.1007/BF02592199

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