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A variation on Karmarkar’s algorithm for solving linear programming problems

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Abstract

The algorithm described here is a variation on Karmarkar’s algorithm for linear programming. It has several advantages over Karmarkar’s original algorithm. In the first place, it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function. The minimum value of the objective function does not have to be known in advance. Secondly, in the absence of degeneracy, the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero. This makes it possible to identify an optimal basis before the algorithm converges.

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References

  1. T.M. Cavalier and A.L. Soyster, “Some computational experience and a modification of the Karmarkar algorithm,” The Pennsylvania State University, ISME Working Paper 85-105, 1985.

  2. P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, “On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method,” Manuscript, Stanford University, 1985.

  3. N. Karmarkar, “A new polynomial-time algorithm for linear programming,” Proceedings of the 16th Annual ACM Symposium on Theory of Computing, 1984, pp. 302–311.

  4. R.J. Vanderbei, M.S. Meketon and B.A. Freedman, “A modification of Karmarkar’s linear programming algorithm,” Manuscript, AT & T Bell Laboratories, Holmdel, New Jersey, June 1985.

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Barnes, E.R. A variation on Karmarkar’s algorithm for solving linear programming problems. Mathematical Programming 36, 174–182 (1986). https://doi.org/10.1007/BF02592024

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  • DOI: https://doi.org/10.1007/BF02592024

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