Sparse Matrix Ordering Methods for Interior Point Linear Programming

The main cost of solving a linear programming problem using an interior point method is usually the cost of solving a series of sparse, symmetric linear systems of equations, AΘA^Tx = b. These systems are typically solved using a sparse direct method. The first step in such a method is a reordering of the rows and columns of the matrix to reduce fill in the factor and/or reduce the required work. This article evaluates several methods for performing fill-reducing ordering on a variety of large-scale linear programming problems. We find that a new method, based on the nested dissection heuristic, provides significantly better orderings than the most commonly used ordering method, minimum degree.