Advertisement

Sparse in-core linear programming

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 506)

Abstract

Linear programming in core using a variant of the Bartels-Golub decomposition of the basis matrix will be considered. This variant is particularly well-adapted to sparsity preservation, being capable of revising the factorisation without any fill-in whenever this is possible by permutations alone. In addition strategies for colum pivoting in the simplex method itself will be discussed and in particular it will be shown that the “steepest edge” algorithm is practical. This algorithm has long been known to give good results in respect of number of iterations, but has been thought to be impractical.

Test results on genuine problems with hundreds of rows and thousands of columns will be reported. These tests include comparisons with other methods.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bartels, R.H. (1971). A stabilization of the simplex method. Num. Math., 16, 414–434.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Beale, E.M.L. (1971). Sparseness in linear programming. In “Large sparse sets of linear equations”. Ed. J.K. Reid, Academic Press.Google Scholar
  3. Dantzig G.B. (1963). Linear programming and extensions. Princeton University Press.Google Scholar
  4. Duff, I.S. and Reid, J.K. (1974). A comparison of sparsity orderings for obtaining a pivotal sequence in Gaussian elimination. J. Inst. Maths. Applics., 14, 281–291.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Forrest, J.J.H. and Tomlin, J.A. (1972). Updating triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical programming, 2, 263–278.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Gill, P.E. and Murray, W. (1973). A numerically stable form of the simplex algorithm. Linear Alg. Appl., 7, 99–138.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Goldfarb, D. (1975). On the Bartels-Golub decomposition for linear programming bases. To appear.Google Scholar
  8. Goldfarb, D. and Reid, J.K. (1975a). A practical steepest edge simplex algorithm. To appear.Google Scholar
  9. Goldfarb, D. and Reid, J.K. (1975b). Fortran subroutines for sparse in-core linear programming. A.E.R.E. Report to appear.Google Scholar
  10. Gill, P.E. (1974). Recent developments in numerically stable methods for linear programming. Bull. Inst. Maths. Applics. 10, 180–186.Google Scholar
  11. Gustavson, F.G. (1972). Some basic techniques for solving sparse systems of linear equations. In “Sparse matrices and their applications”. Ed. D.J. Rose and R.A. Willoughby, Plenum Press.Google Scholar
  12. Harris, P.M.J. (1973). Pivot selection methods of the Devex LP code. Mathematical programming 5, 1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Householder, A.S. (1964). The theory of matrices in numerical analysis. Blaisdell.Google Scholar
  14. Kuhn, H.W. and Quant, R.E. (1963). An experimental study of the simplex method. Proc. of Symposia in Applied Maths, Vol. XV, Ed. Metropolis et al. A.M.S.Google Scholar
  15. Markowitz, H.M. (1957). The elimination form of the inverse and its applications to linear programming. Management Sci., 3, 255–269.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Reid, J.K. (1973). Sparse linear programming using the Bartels-Golub decomposition. Verbal presentation at VIII International Symposium on Mathematical Programming, Stanford University.Google Scholar
  17. Reid, J.K. (1975). A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. To appear.Google Scholar
  18. Saunders, M.A. (1972). Large-scale linear programming using the Cholesky factorization. Report STAN-CS-72-252, Stanford University.Google Scholar
  19. Tomlin, J.A. (1972). Pivoting for size and sparsity in linear programming inversion routines. J. Inst. Maths. Applics., 10, 289–295.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

There are no affiliations available

Personalised recommendations