Basis Inverse and Update Methods

  • Nikolaos Ploskas
  • Nikolaos Samaras
Part of the Springer Optimization and Its Applications book series (SOIA, volume 127)


The computation of the basis inverse is the most time-consuming step in simplex-type algorithms. The basis inverse does not have to be computed from scratch at each iteration, but updating schemes can be applied to accelerate this calculation. This chapter presents two basis inverse and two basis update methods used in simplex-type algorithms: (i) Gauss-Jordan elimination basis inverse method, (ii) LU Decomposition basis inverse method, (iii) Product Form of the Inverse basis update method, and (iv) Modification of the Product Form of the Inverse basis update method. Each technique is presented with: (i) its mathematical formulation, (ii) a thorough numerical example, and (iii) its implementation in MATLAB. Finally, a computational study is performed. The aim of the computational study is to compare the execution time of the basis inverse and update methods and highlight the significance of the choice of the basis update method on simplex-type algorithms and the reduction that it can offer to the solution time.

Supplementary material (2 kb)
chapter 7 (Zip 2 kb)


  1. 1.
    Badr, E. S., Moussa, M., Papparrizos, K., Samaras, N., & Sifaleras, A. (2006). Some computational results on MPI parallel implementations of dense simplex method. In Proceedings of World Academy of Science, Engineering and Technology, 23, (CISE 2006), Cairo, Egypt, 39–32.Google Scholar
  2. 2.
    Bartels, R. H., & Golub, G. H. (1969). The simplex method of linear programming using LU decomposition. Communications of the ACM, 12, 266–268.CrossRefGoogle Scholar
  3. 3.
    Benhamadou, M. (2002). On the simplex algorithm revised form. Advances in Engineering Software, 33(11–12), 769–777.CrossRefGoogle Scholar
  4. 4.
    Brayton, R. K., Gustavson, F. G., & Willoughby, R. A. (1970) Some results on sparse matrices. Mathematics of Computation, 24, 937–954.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chvatal, V. (1983). Linear programming. New York, USA: W.H. Freeman and Company.zbMATHGoogle Scholar
  6. 6.
    Dantzig, G. B., & Orchard–Hays, W. (1954). The product form of the inverse in the simplex method. Mathematical Tables and Other Aids to Computation, 8, 64–67.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Forrest, J. J. H., & Tomlin, J. A. (1972). Updated triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical Programming, 2, 263–278.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goldfarb, D. (1977). On the Bartels-Golub decomposition for linear programming bases. Mathematical Programming, 13, 272–279.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed). JHU Press.Google Scholar
  10. 10.
    Lim, S., Kim, G., & Park, S. (2003). A comparative study between various LU update methods in the simplex method. Journal of the Military Operations Research Society of Korea, 29(1), 28–42.Google Scholar
  11. 11.
    Markowitz, H. (1957). The elimination form of the inverse and its applications to linear programming. Management Science, 3, 255–269.MathSciNetCrossRefGoogle Scholar
  12. 12.
    McCoy, P. F., & Tomlin, J. A. (1974). Some experiments on the accuracy of three methods of updating the inverse in the simplex method. Technical Report, Stanford University.Google Scholar
  13. 13.
    Nazareth, J. L. (1987). Computer solution of linear programs. Oxford, UK: Oxford University Press.zbMATHGoogle Scholar
  14. 14.
    Ploskas, N. (2014). Hybrid optimization algorithms: implementation on GPU. Ph.D. thesis, Department of Applied Informatics, University of Macedonia.Google Scholar
  15. 15.
    Ploskas, N., & Samaras, N. (2013). Basis update on simplex type algorithms. In Book of Abstracts of the EWG-DSS Thessaloniki 2013, p. 11, Thessaloniki, Greece.Google Scholar
  16. 16.
    Ploskas, N., & Samaras, N. (2013). A computational comparison of basis updating schemes for the simplex algorithm on a CPU-GPU system. American Journal of Operations Research, 3, 497–505.CrossRefGoogle Scholar
  17. 17.
    Ploskas, N., Samaras, N., & Margaritis, K. (2013). A parallel implementation of the revised simplex algorithm using OpenMP: Some Preliminary Results. In A. Migdalas et al. (Eds.), Optimization Theory, Decision Making, and Operations Research Applications, Series Title: Springer Proceedings in Mathematics & Statistics 31 (pp. 163–175). New York: Springer.Google Scholar
  18. 18.
    Ploskas, N., Samaras, N., & Papathanasiou, J. (2012). LU decomposition in the revised simplex algorithm. In Proceedings of the 23th National Conference, Hellenic Operational Research Society (pp. 77–81), Athens, Greece.Google Scholar
  19. 19.
    Ploskas, N., Samaras, N., & Papathanasiou, J. (2013). A web-based decision support system using basis update on simplex type algorithms. In J. Hernandez et al. (Eds.), Decision Support Systems II Recent Developments Applied to DSS Network Environments, Lecture Notes in Business Information Processing (LNBIP 164) (pp. 102–114). New York: Springer.CrossRefGoogle Scholar
  20. 20.
    Ploskas, N., Samaras, N., & Sifaleras, A. (2009). A parallel implementation of an exterior point algorithm for linear programming problems. In Proceedings of the 9th Balcan Conference on Operational Research (BALCOR 2009), 2–6 September, Constanta, Romania.Google Scholar
  21. 21.
    Reid, J. (1982). A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. Mathematical Programming, 24, 55–69.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Saunders, M. (1976). A fast and stable implementation of the simplex method using Bartels-Golub updating. In J. Bunch, & S. T. Rachev (Eds.), Sparse Matrix Computation (pp. 213–226). New York: Academic Press.CrossRefGoogle Scholar
  23. 23.
    Suhl, L. M., & Suhl, U. H. (1993). A fast LU update for linear programming. Annals of Operations Research, 43(1), 33–47.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Nikolaos Ploskas
    • 1
  • Nikolaos Samaras
    • 1
  1. 1.Department of Applied InformaticsUniversity of MacedoniaThessalonikiGreece

Personalised recommendations