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Absolute Value Equation Solution Via Linear Programming

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Abstract

By utilizing a dual complementarity property, we propose a new linear programming method for solving the NP-hard absolute value equation (AVE): Ax−|x|=b, where A is an n×n square matrix. The algorithm makes no assumptions on the AVE other than solvability and consists of solving a few linear programs, typically less than four. The algorithm was tested on 500 consecutively generated random solvable instances of the AVE with n=10, 50, 100, 500 and 1000. The algorithm solved 100 % of the test problems to an accuracy of 10−8 by solving an average of 3.3 linear programs per AVE problem.

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Notes

  1. I am indebted to a reviewer for pointing out this simple transformation.

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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.

Additional information

Research described here is available as Data Mining Institute Report 13-01, February 2013: ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/13-01.pdf.

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Mangasarian, O.L. Absolute Value Equation Solution Via Linear Programming. J Optim Theory Appl 161, 870–876 (2014). https://doi.org/10.1007/s10957-013-0461-y

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