Mathematical Programming

, Volume 45, Issue 1–3, pp 475–501 | Cite as

New crash procedures for large systems of linear constraints

  • Nicholas I. M. Gould
  • John K. Reid


Many algorithms for solving linearly constrained optimization problems maintain sets of basic variables. The calculation of the initial basis is of great importance as it determines to a large extent the amount of computation that will then be required to solve the problem. In this paper, we suggest a number of simple methods for obtaining an initial basis and perform tests to indicate how they perform on a variety of real-life problems.

Key words

Linear constraints simple bounds initial basis feasible point 


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  1. D.M. Carstens, “Crashing techniques,” in: W. Orchard-Hays,Advanced Linear-Programming Computing Techniques (McGraw-Hill, New York, 1968) pp. 131–139.Google Scholar
  2. V. Chvátal,Linear Programming (Freeman, New York and San Francisco, 1983).Google Scholar
  3. A.R. Curtis and J.K. Reid, “On the automatic scaling of matrices for Gaussian elimination,”Journal of the Institute of Mathematics and its Applications 10 (1972) 118–124.Google Scholar
  4. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NY, 1963).Google Scholar
  5. J.J. Dongarra and E. Grosse, “Distribution of mathematical software via electronic mail,”Communications of the ACM 30 (1987) 403–407.Google Scholar
  6. I.S. Duff, A.M. Erisman and J.K. Reid,Direct Methods for Sparse Matrices (Oxford University Press, London, 1986).Google Scholar
  7. A.M. Erisman, R.G. Grimes, J.G. Lewis and W.G. Poole Jr., “A structurally stable modification of Hellerman-Rarick's P4 algorithm for reordering unsymmetric sparse matrices,”SIAM Journal on Numerical Analysis 22 (1985) 369–385.Google Scholar
  8. D.M. Gay, “Electronic mail distribution of linear programming test problems,”Mathematical Programming Society COAL Newsletter (December 1985).Google Scholar
  9. N.E. Gibbs, “A hybrid profile reduction algorithm,”ACM Transactions on Mathematical Software 2 (1976) 378–387.Google Scholar
  10. N.E. Gibbs, W.G. Poole Jr. and P.K. Stockmeyer, “An algorithm for reducing the bandwidth and profile of a sparse matrix,”SIAM Journal on Numerical Analysis 13 (1976) 236–250.Google Scholar
  11. D. Goldfarb and J.K. Reid, “A practicable steepest-edge simplex algorithm,”Mathematical Programming 12 (1977) 361–371.Google Scholar
  12. Harwell Subroutine Library, “A catalogue of subroutine,” Report R9185, HMSO (London, 1988).Google Scholar
  13. J.G. Lewis, “Implementation of the Gibbs-Poole-Stockmeyer and Gibbs-King algorithms,”ACM Transactions on Mathematical Software 8 (1982) 180–194.Google Scholar
  14. B.A. Murtagh,Advanced Linear Programming (McGraw-Hill, New York, 1981).Google Scholar
  15. B.A. Murtagh and M.A. Saunders, “MINOS 5.1 User's guide,” Report SOL 83-20R, Department of Operations Research, Stanford University (Stanford, CA, 1987).Google Scholar
  16. W. Orchard-Hays,Advanced Linear-Programming Computing Techniques (McGraw-Hill, New York, 1968).Google Scholar
  17. C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982).Google Scholar
  18. J.K. Reid, “A sparsity exploiting variant of the Bartels-Golub decomposition for linear programming bases,”Mathematical Programming 24 (1982) 55–69.Google Scholar
  19. S.W. Sloan, “An algorithm for profile and wavefront reduction of sparse matrices,”International Journal for Numerical Methods in Engineering 23 (1986) 239–251.Google Scholar

Copyright information

© North-Holland 1989

Authors and Affiliations

  • Nicholas I. M. Gould
    • 1
  • John K. Reid
    • 1
  1. 1.Computer Science and Systems DivisionHarwell LaboratoryUK

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