When solving large sparse linear programming (LP) problems, a typical implementation of the revised simplex method with a sparse factorisation of the basis matrix avoids the prohibitive cost of the standard simplex method or the revised simplex method with the explicit inverse of the basis matrix. In significant classes of LP problems, in particular those with a near or pure network structure, there is little or no fill-in in the factors of the basis matrix. For such very sparse problems, this paper illustrates how the computational cost of a commercial-quality implementation of the revised simplex method is dominated by operations involving zero values. Techniques by which this hyper-sparsity may be exploited are described. It is demonstrated that these techniques result in a computational performance which is vastly superior to that of at least one major commercial implementation of the revised simplex method. The extent to which the performance approaches that of a dedicated network LP solver is also examined.
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