Computing the Splitting Preconditioner for Interior Point Method Using an Incomplete Factorization Approach

Abstract

The splitting preconditioner is very effective, in the final iterations, when applied together with the conjugate gradient method for solving the linear systems arising from interior point methods. This preconditioner relies on an LU factorization of unknown linearly independent subset of the linear problem constraint matrix columns. However, that preconditioner is expensive to compute since a nonsingular matrix must be built from such set of columns. In this work, a new splitting preconditioner is presented which eliminates the need to obtain a nonsingular matrix. The controlled Cholesky factorization is used to compute the preconditioner from normal equations matrix from a given set of not necessarily independent columns. Such an approach is practicable since the controlled Cholesky factorization may be computed by adding suitable diagonal perturbations. Numerical experiments show that the new approach improves previous performance results in some large-scale linear programming problems.