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A double-pivot simplex algorithm and its upper bounds of the iteration numbers

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Abstract

In this paper, a double-pivot simplex method is proposed. Two upper bounds of iteration numbers are derived. Applying one of the bounds to some special linear programming (LP) problems, such as LP with a totally unimodular matrix and Markov decision problem with a fixed discount rate, indicates that the double-pivot simplex method solves these problems in a strongly polynomial time. Applying the other bound to a variant of Klee–Minty cube shows that this bound is actually attainable. Numerical test on three variants of Klee–Minty cubes is performed for the problems with sizes as big as 200 constraints and 400 variables. The test result shows that the proposed algorithm performs extremely good for all three variants. Dantzig’s simplex method cannot handle the Klee–Minty cube problems with 200 constraints because it needs about \(2^{200} \approx 10^{60}\) iterations. Numerical test is also performed for randomly generated problems for both the proposed and Dantzig’s simplex methods. This test shows that the proposed method is promising for large-size problems.

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Notes

  1. For \(m=n/2\), the matrix \(\mathbf{M} \) has \(\frac{n^2}{4}\) nonzeros, but for most Netlib problems, there are only \({\mathcal {O}}(n)\) nonzeros. Therefore, it is not a surprise that solving the randomly generated dense LP problems is time-consuming.

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Acknowledgements

After this author finished the first version of this paper, he realized that Vitor and Easton [26] have recently proposed a similar idea to update two pivots at a time. This author would like to thank Dr. F. Vitor for sharing his paper which is very useful in comparing the proposed work to his excellent work. The author would also like to thank the anonymous reviewer for his constructive and detailed comments which lead to significant improvements of the paper.

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  1. US NRC, Office of Research, 11555 Rockville Pike, Rockville, 20850, USA

    Yaguang Yang

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Yang, Y. A double-pivot simplex algorithm and its upper bounds of the iteration numbers. Res Math Sci 7, 34 (2020). https://doi.org/10.1007/s40687-020-00235-2

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