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

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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.

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Acknowledgements

Thanks to CNPq, FAPESP and UMSA for their financial support.

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Authors and Affiliations

  1. Department of Mathematics, San Andres University of La Paz, 1995 Villazon Avenue, La Paz, Bolivia

    Porfirio Suñagua

  2. Department of Applied Mathematics, University of Campinas, Campinas, SP, 13083-859, Brazil

    Aurelio R. L. Oliveira

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  1. Porfirio Suñagua
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  2. Aurelio R. L. Oliveira
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Correspondence to Aurelio R. L. Oliveira.

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Suñagua, P., Oliveira, A.R.L. A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods. Comput Optim Appl 67, 111–127 (2017). https://doi.org/10.1007/s10589-016-9887-0

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