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Hyper-Sparsity in the Revised Simplex Method and How to Exploit it

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Abstract

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems the result of one or more of these operations is usually sparse, a property we call hyper-sparsity. Analysis of the commonly-used techniques for implementing each step of the revised simplex method shows them to be inefficient when hyper-sparsity is present. Techniques to exploit hyper-sparsity are developed and their performance is compared with the standard techniques. For the subset of our test problems that exhibits hyper-sparsity, the average speedup in solution time is 5.2 when these techniques are used. For this problem set our implementation of the revised simplex method which exploits hyper-sparsity is shown to be competitive with the leading commercial solver and significantly faster than the leading public-domain solver.

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  1. School of Mathematics, University of Edinburgh, JCMB, King's Buildings, EDINBURGH, EH9 3JZ, UK

    J. A. J. Hall & K. I. M. McKinnon

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  1. J. A. J. Hall
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  2. K. I. M. McKinnon
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Hall, J.A.J., McKinnon, K.I.M. Hyper-Sparsity in the Revised Simplex Method and How to Exploit it. Comput Optim Applic 32, 259–283 (2005). https://doi.org/10.1007/s10589-005-4802-0

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  • DOI: https://doi.org/10.1007/s10589-005-4802-0

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