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New purification algorithms for linear programming

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  • K. O. Kortanek
  • Zhu Jishan

Abstract

Two new algorithms are presented for solving linear programs which employ the opposite‐sign property defined for a set of vectors in m space. The first algorithm begins with a strictly positive feasible solution and purifies it to a basic feasible solution having objective function value no less under maximization. If this solution is not optimal, then it is drawn back into the interior with the same objective function value, and a restart begins. The second algorithm can begin with any arbitrary feasible point. If necessary this point is purified to a basic feasible solution by dual‐feasibility–seeking directions. Should dual feasibility be attained, then a duality value interval is available for estimating the unknown objective function value. If at this juncture the working basis is not primal feasible, then further purification steps are taken tending to increase the current objective function value, while simultaneously seeking another dual feasible solution. Both algorithms terminate with an optimal basic solution in a finite number of steps.

Suggested Citation

  • K. O. Kortanek & Zhu Jishan, 1988. "New purification algorithms for linear programming," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(4), pages 571-583, August.
    • Handle: RePEc:wly:navres:v:35:y:1988:i:4:p:571-583
    DOI: 10.1002/1520-6750(198808)35:43.0.CO;2-L
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    References listed on IDEAS

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    1. Robert G. Bland, 1977. "New Finite Pivoting Rules for the Simplex Method," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 103-107, May.
    2. Kortanek, K. O. & Shi, M., 1987. "Convergence results and numerical experiments on a linear programming hybrid algorithm," European Journal of Operational Research, Elsevier, vol. 32(1), pages 47-61, October.
    3. Harvey J. Greenberg, 1986. "An analysis of degeneracy," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(4), pages 635-655, November.
    4. BLAND, Robert G., 1977. "New finite pivoting rules for the simplex method," LIDAM Reprints CORE 315, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Hanif D. Sherali, 1987. "Algorithmic insights and a convergence analysis for a Karmarkar‐type of algorithm for linear programming problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(3), pages 399-416, June.
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    Cited by:

    1. Erling D. Andersen, 1999. "On Exploiting Problem Structure in a Basis Identification Procedure for Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 11(1), pages 95-103, February.

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