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



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.


Linear programming Splitting preconditioner Rectangular LU factorization Transpose basis 



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


  1. 1.
    Amestoy, P.R., Davis, T.A., Duff, I.S.: Algorithm 837: Amd, an approximate minimum degree ordering algorithm. ACM Trans. Math. Softw. (TOMS) 30(3), 381–388 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28(2), 149–171 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bocanegra, S., Campos, F., Oliveira, A.R.: Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods. Comput. Optim. Appl. 36(2–3), 149–164 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Campos, F.F., Rollet, J.S.: Controlled Cholesky Factorization for Preconditioning the Conjugate Gradient Method. Oxford University Computing Laboratory, Numerical Analysis Group (1995)Google Scholar
  5. 5.
    Carolan, W.J., Hill, J.E., Kennington, J.L., Niemi, S., Wichmann, S.J.: An empirical evaluation of the korbx \(\textregistered \) algorithms for military airlift applications. Oper. Res. 38(2), 240–248 (1990)CrossRefGoogle Scholar
  6. 6.
    Chai, J., Toh, K.: Preconditioning and iterative solution of symmetric indefinite linear system arising from interior point methods for linear programming. Comput. Optim. Appl. 36, 221–247 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Czyzyk, J., Mehrotra, S., Wagner, M., Wright, S.J.: PCx an interior point code for linear programming. Optim. Methods Softw. 11–2(1–4), 397–430 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Davis, T.A.: Algorithm 832: Umfpack v4. 3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. (TOMS) 30(2), 196–199 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Davis, T.A., Duff, I.S.: An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM J. Matrix Anal. Appl. 18(1), 140–158 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dolan, E.D., More, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Drazic, M.D., Lazovic, R.P., Kovacevic-Vujcic, V.V.: Sparsity preserving preconditioners for linear systems in interior-point methods. Comput. Optim. Appl. 61(3), 557–570 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Duff, I.S.: Ma57–a code for the solution of sparse symmetric definite and indefinite systems. ACM Trans. Math. Softw. (TOMS) 30(2), 118–144 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Duff, I.S., Pralet, S.: Towards stable mixed pivoting strategies for the sequential and parallel solution of sparse symmetric indefinite systems. SIAM J. Matrix Anal. Appl. 29(3), 1007–1024 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ghidini, C.T.L.S., Oliveira, A.R.L., Silva, J., Velazco, M.I.: Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods. Linear Algebra Appl. 218, 1267–1284 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ghidini, C.T.L.S., Oliveira, A.R.L., Sorensen, D.C.: Computing a hybrid preconditioner approach to solve the linear systems arising from interior point methods for linear programming using the gradient conjugate method. Ann. Manag. Sci. 3, 45–66 (2014)CrossRefGoogle Scholar
  16. 16.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. JHU Press, Baltimore (2013)MATHGoogle Scholar
  17. 17.
    Gondzio, J.: Interior point methods 25 years later. Eur. J. Oper. Res. 218(3), 587–601 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kershaw, D.S.: The incomplete cholesky-conjugate gradient method for the iterative solution of systems of linear equations. J. Comput. Phys. 26(1), 43–65 (1978)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kuzmin, A., Luisier, M., Schenk, O.: Fast methods for computing selected elements of the greens function in massively parallel nanoelectronic device simulations. In: Wolf, F., Mohr, B., Mey, D. (eds.) Euro-Par 2013 Parallel Processing. Lecture Notes in Computer Science, vol. 8097, pp. 533–544. Springer, Berlin (2013). doi: 10.1007/978-3-642-40047-6_54 CrossRefGoogle Scholar
  20. 20.
    Luenberger, D.G.: Linear and Nonlinear Programming. Springer, New York (2003)MATHGoogle Scholar
  21. 21.
    Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J. Optim. 2(3), 435–449 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Markowitz, H.M.: The elimination form of the inverse and its application to linear programming. Manag. Sci. 3(3), 255–269 (1957)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2, 575–601 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Oliveira, A.R., Sorensen, D.C.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl. 394, 1–24 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Schenk, O., Bollhöfer, M., Römer, R.A.: On large-scale diagonalization techniques for the Anderson model of localization. SIAM Rev. 50(1), 91–112 (2008). doi: 10.1137/070707002 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schenk, O., Wächter, A., Hagemann, M.: Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization. Comput. Optim. Appl. 36(2–3), 321–341 (2007). doi: 10.1007/s10589-006-9003-y MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Velazco, M., Oliveira, A.R., Campos, F.: A note on hybrid preconditioners for large-scale normal equations arising from interior-point methods. Optim. Methods Softw. 25(2), 321–332 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wright, S.J.: Primal-Dual Interior-Point Methods, vol. 54. SIAM (1997)Google Scholar
  29. 29.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discret. Methods 2(1), 77–79 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsSan Andres University of La PazLa PazBolivia
  2. 2.Department of Applied MathematicsUniversity of CampinasCampinasBrazil

Personalised recommendations