Annals of Operations Research

, Volume 5, Issue 1–4, pp 413–424 | Cite as

Advanced basis construction in linear programming

  • H. J. Greenberg
II. Mathematical Programming


This paper considers basis construction in a linear program when the number of activities with basic status is not equal to the number of rows in the particular scenario. This arises when starting with an ‘advanced basis’, obtained from a solution to another scenario. The goal here is to provide a triangular-seeking algorithm that takes advantage of structural properties during the construction of a basis agenda. For completeness, some facts, which are known but have not been published, are given about choosing an advanced basis and about spikes.

Keywords and phrases

Linear programming basis construction matrix rearrangement 


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  1. [1]
    K. Darby-Dowman and G. Mitra, An investigation of algorithms used in the restructuring of linear programming basis matrices prior to inversion, 10th Int. Symp. on Mathematical Programming, Montreal, Canada (1979).Google Scholar
  2. [2]
    I.S. Duff, On permutations to block triangular form, J. Inst. Maths. Applics. 19(1977)339.CrossRefGoogle Scholar
  3. [3]
    J.J.H. Forrest and J.A. Tomlin, Updating triangular factors of the basis to maintain sparsity in the product form simplex method, Math. Progr. 2(1972)263.CrossRefGoogle Scholar
  4. [4]
    H.J. Greenberg and D.C. Rarick, Determining GUB sets via an invert agenda algorithm, Math. Progr. 7(1977)240.CrossRefGoogle Scholar
  5. [5]
    E. Hellerman and D.C. Rarick, Reinversion with the pre-assigned pivot procedure, Math. Progr. 1(1971)195.CrossRefGoogle Scholar
  6. [6]
    E. Hellerman and D.C. Rarick, The partitioned pre-assigned pivot procedure (P4), in:Sparse Matrices and Their Applications, ed. D.J. Rose and R.A. Willoughby (Plenum Press, New York, 1972) p. 65.Google Scholar
  7. [7]
    J.E. Hopcroft and R.M. Karp, An n5/2 algorithm for maximum matchings in bipartite graphs, Siam J. Comput. 2(1973)225.CrossRefGoogle Scholar
  8. [8]
    J.E. Kalen, Aspects of large-scale in-core linear programming,Proc. 1971 Ann. Conf. (ACM, New York, 1971) p. 304.CrossRefGoogle Scholar
  9. [9]
    R.D. McBride, A spike collective dynamic factorization algorithm for the simplex method, Mgt. Sci. 24(1978)1031.CrossRefGoogle Scholar
  10. [10]
    R.D. McBride, A bump triangular dynamic factorization algorithm for the simplex method, USC Working Paper No. 19, Los Angeles, California (1977).Google Scholar
  11. [11]
    W. Orchard-Hays,Advanced Computing Techniques in Linear Programming (McGraw-Hill, New York, 1968).Google Scholar

Copyright information

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

Authors and Affiliations

  • H. J. Greenberg
    • 1
  1. 1.Mathematics DepartmentUniversity of Colorado at DenverDenverUSA

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