Annals of Operations Research

, Volume 5, Issue 2, pp 501–515 | Cite as

Modifying the forrest-tomlin and saunders updates for linear programming problems with variable upper bounds

  • M. J. Todd
Mathematical Programming
  • 39 Downloads

Abstract

The author previously described a modification of the simplex method to handle variable upper bounds implicitly. This method can easily be used when the representation of the basis inverse (e.g. a triangular decomposition of the basis) is maintained as a dense matrix in core, but appears to cause difficulties for large problems where secondary storage and packed matrices may be employed. Here we show how the Forrest-Tomlin and Saunders updating schemes, which are designed for such large problems, can be adapted.

Keywords and phrases

Linear programming variable upper bounds rank-one updates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.H. Bartels, A stabilization of the simplex method, Numerische Mathematik 16(1971)414.Google Scholar
  2. [2]
    R.H. Bartels and G.H. Golub, The simplex method for linear programming using LU decomposition, Communications of the ACM 12(1969)266.Google Scholar
  3. [3]
    M. Bastian, Updated triangular factors of the working basis in Winkler's decomposition approach, Math. Progr. 17(1979)391.Google Scholar
  4. [4]
    M. Bastian, Implicit representation of generalized variable upper bounds using the elimination form of the inverse on secondary storage, Math. Progr. 30(1984)357.Google Scholar
  5. [5]
    J.J.H. Forrest and J.A. Tomlin, Updating triangular factors of the basis to maintain sparsity in the product form of the simplex method, Math. Progr. 2(1972)263.Google Scholar
  6. [6]
    F. Glover, Compact LP bases for a class of IP problems, Math. Progr. 12(1977)102.Google Scholar
  7. [7]
    E. Hellerman and D. Rarick, Reinversion with the preassigned pivot procedure, Math. Progr. 6(1971)195.Google Scholar
  8. [8]
    E. Hellerman and D. Rarick, The partitioned preassigned pivot procedure (P4), in:Sparse Matrices and their Applications, ed. D.J. Rose and R.A. Willoughby (Plenum Press, New York, 1972) p. 67.Google Scholar
  9. [9]
    J.K. Reid, A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases, Math. Progr. 24(1982)55.Google Scholar
  10. [10]
    M.A. Saunders, A fast stable implementation of the simplex method using Bartels—Golub updating, in:Sparse Matrix Computations, ed. J.R. Bunch and D.J. Rose (Academic Press, New York, 1976) p. 213.Google Scholar
  11. [11]
    L. Schrage, Implicit representation of variable upper bounds in linear programming, Math. Progr. Study 4(1975)118.Google Scholar
  12. [12]
    L. Schrage, Implicit representation of generalized variable upper bounds in linear programming, Math. Progr. 14(1978)11.Google Scholar
  13. [13]
    M.J. Todd, An implementation of the simplex method for linear programming problems with variable upper bounds, Math. Progr. 23(1982)34.Google Scholar

Copyright information

© J.C. Baltzer A.G., Scientific Publishing Company 1986

Authors and Affiliations

  • M. J. Todd
    • 1
  1. 1.School of Operations Research and Industrial Engineering, College of EngineeringCornell UniversityIthacaUSA

Personalised recommendations