A review of LU factorisation in the simplex algorithm

This paper reviews the computational aspects of the LU factorisation of large-scale linear optimisation bases. In the simplex algorithm, the solution of two linear systems, involving a square basis matrix and its transpose, accounts for a large portion of the computation time. The most widely used solution technique is LU factorisation and accompanying update routines. The utilisation of a column replacement update procedure means that a given factorisation may be kept over a large number of simplex iterations. Therefore, a significant emphasis is placed on the sparsity of the LU factors. This emphasis means that a suitable factorisation routine must be chosen with care. A review of LU factorisation is offered, including stability monitoring and analysis, sparsity analysis, error analysis, algorithmic aspects, and state-of-the-art LU factorisation routines.