## On the Factorization of Simplex Basis Matrices

Please always quote using this URN: urn:nbn:de:0297-zib-11392
• In the simplex algorithm, solving linear systems with the basis matrix and its transpose accounts for a large part of the total computation time. We investigate various methods from modern numerical linear algebra to improve the computation speed of the basis updates arising in LPs. The experiments are executed on a large real-world test set. The most widely used solution technique is sparse LU factorization, paired with an updating scheme that allows to use the factors over several iterations. Clearly, small number of ﬁll-in elements in the LU factors is critical for the overall performance. Using a wide range of LPs we show numerically that after a simple permutation the non-triangular part of the basis matrix is so small, that the whole matrix can be factorized with (relative) ﬁll-in close to the optimum. This permutation has been exploited by simplex practitioners for many years. But to our knowledge no systematic numerical study has been published that demonstrates the eﬀective reduction to a surprisingly small non-triangular problem, even for large scale LPs. For the factorization of the non-triangular part most existing simplex codes use some variant of dynamic Markowitz pivoting, which originated in the late 1950s. We also show numerically that, in terms of ﬁll-in and in the simplex context, dynamic Markowitz is quite consistently superior to other, more recently developed techniques.

Author: Robert Luce, Jurjen Duintjer Tebbens, Jörg Liesen, Robert Nabben, Martin Grötschel, Thorsten Koch, Olaf Schenk ZIB-Report 65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F05 Direct methods for linear systems and matrix inversion 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C05 Linear programming TU Berlin, Zuse Institute Berlin, University of Basel 2009/07/29 ZIB-Report (09-24) 1438-0064

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