Title: A review of the LU update in the simplex algorithm

Authors: Joseph M. Elble; Nikolaos V. Sahinidis

Addresses: Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 104 S. Mathews Ave., Urbana, IL 61801, USA ' Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA

Abstract: In the simplex method for linear optimisation, two linear systems involving a square non-singular basis matrix B and its transpose are solved at each iteration. A column of B is then replaced, and the solution of these two linear systems is required again. For the past four decades, researchers have made considerable advancements in updating the sparse LU factors of the basis matrix in this algorithm. This paper reviews the state-of-the-art in LU update procedures, while analysing each procedure to gain a full understanding of the progress made in this area.

Keywords: simplex algorithm; LU update; linear optimisation; matrix factorisation; computational stability; LU factorisation; LU decompostion.

DOI: 10.1504/IJMOR.2012.048901

International Journal of Mathematics in Operational Research, 2012 Vol.4 No.4, pp.366 - 399

Published online: 23 Dec 2014 *

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