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Absolute Value Equation Solution Via Linear Programming

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  • Olvi L. Mangasarian

    (University of Wisconsin
    University of California at San Diego)

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.

Suggested Citation

  • Olvi L. Mangasarian, 2014. "Absolute Value Equation Solution Via Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 870-876, June.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:3:d:10.1007_s10957-013-0461-y
    DOI: 10.1007/s10957-013-0461-y
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    References listed on IDEAS

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    1. Oleg Prokopyev, 2009. "On equivalent reformulations for absolute value equations," Computational Optimization and Applications, Springer, vol. 44(3), pages 363-372, December.
    2. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
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    Cited by:

    1. Milan Hladík, 2018. "Bounds for the solutions of absolute value equations," Computational Optimization and Applications, Springer, vol. 69(1), pages 243-266, January.
    2. J. Y. Bello Cruz & O. P. Ferreira & L. F. Prudente, 2016. "On the global convergence of the inexact semi-smooth Newton method for absolute value equation," Computational Optimization and Applications, Springer, vol. 65(1), pages 93-108, September.
    3. Yu Han & Haiyan Xu & Ginger Y. Ke, 2020. "Construction and application of hyper-inverse conflict models based on the sequential stability," EURO Journal on Decision Processes, Springer;EURO - The Association of European Operational Research Societies, vol. 8(3), pages 237-259, November.
    4. Li Fangping & Yang Yuguo & Wu Yue, 2016. "The Optimization of Reservoir Based on the Combination of ABC Classification Method and Linear Programming Method," International Journal of Business and Management, Canadian Center of Science and Education, vol. 11(11), pages 156-156, October.
    5. Miao, Xin-He & Yang, Jiantao & Hu, Shenglong, 2015. "A generalized Newton method for absolute value equations associated with circular cones," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 155-168.
    6. L. Abdallah & M. Haddou & T. Migot, 2019. "A sub-additive DC approach to the complementarity problem," Computational Optimization and Applications, Springer, vol. 73(2), pages 509-534, June.

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