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Linear complementarity as absolute value equation solution

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Abstract

We consider the linear complementarity problem (LCP): \(Mz+q\ge 0, z\ge 0, z^{\prime }(Mz+q)=0\) as an absolute value equation (AVE): \((M+I)z+q=|(M-I)z+q|\), where \(M\) is an \(n\times n\) square matrix and \(I\) is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generated random solvable instances of the LCP with \(n=10, 50, 100, 500\) and 1,000. The algorithm solved \(100\,\%\) of the test problems to an accuracy of \(10^{-8}\) by solving 2 or less linear programs per LCP problem.

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Acknowledgments

The research described here, based on Data Mining Institute Report 13-02, March 2013, was supported by the Microsoft Corporation and ExxonMobil.

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Authors and Affiliations

  1. Computer Sciences Department, University of Wisconsin, Madison, WI, 53706, USA

    Olvi L. Mangasarian

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Olvi L. Mangasarian

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  1. Olvi L. Mangasarian
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Correspondence to Olvi L. Mangasarian.

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Mangasarian, O.L. Linear complementarity as absolute value equation solution. Optim Lett 8, 1529–1534 (2014). https://doi.org/10.1007/s11590-013-0656-z

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