Modified Cholesky Factorizations in Interior-Point Algorithms for Linear Programming

Stephen Wright

We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill-conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach.

Preprint ANL/MCS-P600-0596, Mathematics and Computer Science Division, Argonne National Laboratory, May, 1996. Revised June, 1998.