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A hybrid algorithm for solving the absolute value equation

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Abstract

We propose a hybrid algorithm for solving the NP-hard absolute value equation (AVE): \(Ax-|x|=b\), where \(A\) is an \(n\times n\) square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving iteratively a linear system of equations followed by a linear program. The algorithm was tested on 100 consecutively generated random solvable instances of the AVE with \(n=\) 50, 100, 200, 500 and 1000. The algorithm solved \(100\,\%\) of the test problems to an accuracy of \(10^{-8}\) by solving an average of 2.77 systems of linear equations and linear programs per AVE.

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Acknowledgments

The research described here, based on Data Mining Institute Report 14-02, April 2014, 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. A hybrid algorithm for solving the absolute value equation. Optim Lett 9, 1469–1474 (2015). https://doi.org/10.1007/s11590-015-0893-4

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