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Updated triangular factors of the basis to maintain sparsity in the product form simplex method

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Abstract

In recent years triangular factorization of the basis has greatly enhanced the efficiency of linear programming inversion routines, leading to greater speed, accuracy and a sparser representation. This paper describes a new product form method for updating the triangular factors at each iteration of the simplex method which has proved extremely effective in reducing the rate of growth of the transformation (eta) files, thus reducing the amount of work per iteration and the frequency of re-inversion. We indicate some of the programming measures required to implement the method and give computational experience on real problems of up to 3500 rows.

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References

  1. R.H. Bartels, “A numerical investigation of the simplex method,” Computer Science Department, Stanford University, Technical Report No. CS-104, July 31, 1968.

  2. R.H. Bartels and G.H. Golub, “The simplex method of linear programming using LU decomposition,”Communications ACM 12 (1969) 266–268, 275–278.

    Google Scholar 

  3. E.M.L. Beale, “Sparseness in linear programming,” in:Large sparse sets of linear equations, Ed. J.K. Reid (Academic Press, London, 1970) pp. 1–15.

    Google Scholar 

  4. J.M. Bennett and D.R. Green, “Updating the inverse or the triangular factors of a modified matrix,” Basser Computing Department, University of Sydney, Technical Report No. 42, April, 1966.

  5. R.K. Brayton, F.G. Gustavson and R.A. Willoughby, “Some results on sparse matrices,” RC-2332, IBM Research Centre, Yorktown Heights, N.Y., February 14, 1969.

    Google Scholar 

  6. J. de Buchet, “How to take into account the low density of matrices to design a mathematical programming package — Relevant effects on optimization and inversion algorithms,” in:Large sparse sets of linear equations, Ed. J.K. Reid (Academic Press, London, 1970) pp. 211–217.

    Google Scholar 

  7. G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, 1963).

    Google Scholar 

  8. G.B. Dantzig, “Compact basis triangularization for the simplex method,” in:Recent Advances in mathematical programming, eds. R.L. Graves and P. Wolfe (McGraw-Hill, New York, 1963) pp. 125–132.

    Google Scholar 

  9. G.B. Dantzig, R.P. Harvey, R.D. McKnight and S.S. Smith, “Sparse matrix techniques in two mathematical programming codes,” in:Sparse matrix proceedings, Ed. R. Willoughby (RA-1, IBM Research Centre, Yorktown Heights, N.Y., 1969) pp. 85–99.

    Google Scholar 

  10. E. Hellerman and D. Rarick, “Reinversion with the preassigned pivot procedure,” Paper presented at the 7th Mathematical Programming Symposium, The Hague, September 1970.

  11. H.M. Markowitz, “The elimination form of inverse and its application to linear programming,” Management Science 3 (1957) 255–269.

    Google Scholar 

  12. W. Orchard-Hays,Advanced linear programming computing techniques (McGraw-Hill, New York, 1968).

    Google Scholar 

  13. D.M. Smith, “Data logistics for matrix inversion,” in:Sparse matrix proceedings, ed. R. Willoughby (RA-1, IBM Research Centre, Yorktown Heights, N.Y., 1969) pp. 127–138.

    Google Scholar 

  14. R.P. Tewarson, “On the product form of inverses of sparse matrices,”SIAM Reviews 8 (1966) 336–342.

    Google Scholar 

  15. J.A. Tomlin, “Maintaining a sparse inverse in the simplex method,” Operations Research Department, Stanford University, Technical Report 70-15, November 1970.

  16. J.H. Wilkinson,Rounding errors in algebraic processes (Prentice-Hall, Englewood Cliffs, N.J., 1963).

    Google Scholar 

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Forrest, J.J.H., Tomlin, J.A. Updated triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical Programming 2, 263–278 (1972). https://doi.org/10.1007/BF01584548

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