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Sparsity preserving preconditioners for linear systems in interior-point methods

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  • Milan Dražić
  • Rade Lazović
  • Vera Kovačević-Vujčić

Abstract

Systems of normal equations arising in interior-point methods for linear programming in the case of a degenerate optimal face have highly ill-conditioned coefficient matrices. In 2004, Monteiro et al. (SIAM J Optim 15:96–100, 2004 ) proposed a preconditioner which guarantees uniform well-conditionedness. However, the proposed preconditioner may lead to considerable loss of sparsity. Our approach is directed towards a generalization of the proposed preconditioner which makes a balance between sparsity and well-conditionedness. Experimental results on Netlib instances show the effects of the new approach. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Milan Dražić & Rade Lazović & Vera Kovačević-Vujčić, 2015. "Sparsity preserving preconditioners for linear systems in interior-point methods," Computational Optimization and Applications, Springer, vol. 61(3), pages 557-570, July.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:3:p:557-570
    DOI: 10.1007/s10589-015-9735-7
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    References listed on IDEAS

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    1. 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.
    2. María Gonzalez-Lima & Aurelio Oliveira & Danilo Oliveira, 2013. "A robust and efficient proposal for solving linear systems arising in interior-point methods for linear programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 573-597, December.
    3. Ilan Adler & Narendra Karmarkar & Mauricio G. C. Resende & Geraldo Veiga, 1989. "Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm," INFORMS Journal on Computing, INFORMS, vol. 1(2), pages 84-106, May.
    4. Bocanegra, Silvana & Castro, Jordi & Oliveira, Aurelio R.L., 2013. "Improving an interior-point approach for large block-angular problems by hybrid preconditioners," European Journal of Operational Research, Elsevier, vol. 231(2), pages 263-273.
    5. Maria Gonzalez-Lima & Hua Wei & Henry Wolkowicz, 2009. "A stable primal–dual approach for linear programming under nondegeneracy assumptions," Computational Optimization and Applications, Springer, vol. 44(2), pages 213-247, November.
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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.

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