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On the uniqueness of solutions to linear complementarity problems

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Abstract

This paper characterizes the classU of all realn×n matricesM for which the linear complementarity problem (q, M) has a unique solution for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. It is shown that restricting the uniqueness property to the interior ofK(M) is necessary because whenU, the problem (q, M) has infinitely many solutions ifq belongs to the boundary of intK(M). It is shown thatM must have nonnegative principal minors whenU andK(M) is convex. Finally, it is shown that whenM has nonnegative principal minors, only one of which is 0, andK(M)≠R n, thenU andK(M) is a closed half-space.

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Author notes

  1. Richard E. Stone

    Present address: Graduate School of Business, Harvard University, 02163, Boston, MA, USA

Authors and Affiliations

  1. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA

    Richard W. Cottle & Richard E. Stone

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  1. Richard W. Cottle
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  2. Richard E. Stone
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Research supported by National Science Foundation Grant MCS76-81259.

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Cottle, R.W., Stone, R.E. On the uniqueness of solutions to linear complementarity problems. Mathematical Programming 27, 191–213 (1983). https://doi.org/10.1007/BF02591945

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  • DOI: https://doi.org/10.1007/BF02591945

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