Numerical Analysis and Applications

, Volume 8, Issue 4, pp 285–292 | Cite as

A simplex method algorithm using a double basis



A simplex method algorithm not requiring explicit LU decomposition updating in the iterations is considered. The solutions obtained with fixed LU factors are corrected using small auxiliary matrices. The results of some numerical experiments are presented.


LU decomposition decomposition updating sparse matrices simplex method linear programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Murtagh, B., Sovremennoe lineinoe programmirovanie: teoriya i praktika (Advanced Linear Programming: Computation and Practice), Moscow: Mir, 1984.Google Scholar
  2. 2.
    Bartels, R.H. and Golub, G.H., The Simplex Method of Linear Programming Using the LU Decomposition, Comm. ACM, 1969, vol. 12, pp. 266–268.MATHCrossRefGoogle Scholar
  3. 3.
    Forrest, J.J.H. and Tomlin, J.A., Updating Triangular Factors of the Basis toMaintain Sparsity in the Product Form Simplex Method, Math. Program., 1972, vol. 2, pp. 263–278.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Reid, J.K., A Sparsity-Exploiting Variant of the Bartels–Golub Decomposition for Linear Programming Bases, Math. Program., 1982, vol. 24, pp. 55–69.MATHCrossRefGoogle Scholar
  5. 5.
    Forrest, J.J.H. and Tomlin, J.A., Vector Processing in Simplex and Interior Methods for Linear Programming, IBM Res. Report, 1988, no. RJ 6390 (62372).Google Scholar
  6. 6.
    Bisschop, J. and Meeraus, F., Matrix Augmentation and Partitioning in the Updating of the Basis Inverse, Math. Program., 1977, vol. 13, pp. 241–254.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Proctor, P.E., Implementation of the Double-Basis Simplex Method for the General Linear Programming Problem, SIAM J. Alg. Discr.Meth., 1985, vol. 6, no. 4, pp. 567–575.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Eldersveld, S.K. and Saunders, M.A., A Block-LU Update for Large-Scale Linear Programming, SIAM J. Matrix An. Appl., 1992, vol. 13, no. 1, pp. 191–201.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Markowitz, H.M., The Elimination Form of the Inverse and Its Application to Linear Programming, Manag. Sci., 1957, vol. 3, pp. 255–269.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Saunders, M.A., Large-Scale Numerical Optimization, 2014; http://webstanfordedu/class/msande318/notes/notes05-updatespdf.Google Scholar
  11. 11.
    Olschowka, M. and Neumaier, A., A New Pivoting Strategy for Gaussian Elimination, Lin. Alg. Appl., 1996, vol. 240, nos. 1–3, pp. 131–151.MathSciNetCrossRefGoogle Scholar
  12. 12.
    http://wwwopensourceorg/licenses/cpl1.0php.Google Scholar
  13. 13.
    Pissanetzki, S., Tekhnologiya razrezhennykh matrits (SparseMatrix Technology), Moscow: Mir, 1988.Google Scholar
  14. 14.
    Jonker, R. and Volgenant, A., A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems, Computing, 1987, vol. 38, pp. 325–340.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jonker, R. and Vogenant, A., Linear Assignment Problem; http://wwwassignmentproblemscom/LAPJVhtm.Google Scholar
  16. 16.
    Karypis, G. and Kumar, V., Metis: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices, vers. 4.0, University of Minnesota, Department of Computer Science/Army HPC Research Center Minneapolis, MN 55455, 1998; http://glarosdtcumnedu/gkhome/views/metis.Google Scholar
  17. 17.
    Gay, D.M., Electronic Mail Distribution of Linear Programming Test Problems, Math. Programm. Soc. COAL Newsletter, 1985, vol. 13, pp. 10–12.Google Scholar
  18. 18.
    Zabinyako, G.I. and Kotelnikov, E.A., Linear Optimization Programs, NCC Bull., Ser. Numer. An., Novosibirsk, 2002, iss. 11, pp. 103–112.MATHGoogle Scholar
  19. 19.
    Davis, T.A., University of Florida Sparse Matrix Collection; http://wwwciseufledu/~davis/sparse.Google Scholar
  20. 20.
    Duff, I.S. and Koster, J., The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices, SIAM J. Matrix An. Appl.. 1999, vol. 20, no. 4, pp. 889–901.MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Zabinyako, G.I., Re-Construction of Inverse Matrices, Sib. Zh. Industr. Mat., 2009, vol. 12, no. 3, pp. 41–51.MATHMathSciNetGoogle Scholar
  22. 22.
    Schenk, O. and Gartner, K., Solving Unsymmetric Sparse Systems of Linear Equation with PARDISO, Future Gener. Computer Systems, 2004, vol. 20, pp. 475–487.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsSiberian Branch, Russian Academy of SciencesNovosibirskRussia

Personalised recommendations