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A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

Author

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  • Porfirio Suñagua

    (San Andres University of La Paz)

  • Aurelio R. L. Oliveira

    (University of Campinas)

Abstract

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra’s predictor-corrector method for large scale linear programming problems needs to find a basis through a sophisticated process based on the application of a rectangular LU factorization. This class of splitting preconditioners works better near a solution of the linear programming problem when the matrices are highly ill-conditioned. In this study, we develop and implement a new approach to find a basis for the splitting preconditioner, based on standard rectangular LU factorization with partial permutation of the scaled transpose linear programming constraint matrix. In most cases, this basis is better conditioned than the existing one. In addition, we include a penalty parameter in Mehrotra’s predictor-corrector method in order to reduce ill-conditioning of the normal equations matrix. Computational experiments show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the increased efficiency and robustness of the new approach become evident by the performance profile.

Suggested Citation

  • Porfirio Suñagua & Aurelio R. L. Oliveira, 2017. "A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods," Computational Optimization and Applications, Springer, vol. 67(1), pages 111-127, May.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:1:d:10.1007_s10589-016-9887-0
    DOI: 10.1007/s10589-016-9887-0
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    References listed on IDEAS

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    1. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    2. Harry M. Markowitz, 1957. "The Elimination form of the Inverse and its Application to Linear Programming," Management Science, INFORMS, vol. 3(3), pages 255-269, April.
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    Cited by:

    1. Luciana Casacio & Aurelio R. L. Oliveira & Christiano Lyra, 2018. "Using groups in the splitting preconditioner computation for interior point methods," 4OR, Springer, vol. 16(4), pages 401-410, December.
    2. Manolo Rodriguez Heredia & Aurelio Ribeiro Leite Oliveira, 2020. "A new proposal to improve the early iterations in the interior point method," Annals of Operations Research, Springer, vol. 287(1), pages 185-208, April.

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