A review of the LU update in the simplex algorithm

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.