Solving linear systems in interior-point methods

Abstract

There are two approaches to solve the linear systems in interior-point methods: the normal equation approach and the augmented system approach. We integrated the two methods by applying matrix partitioning to the augmented system approach. Specifically, we show the Schur complement method which is applied to problems with dense columns is a special case of the augmented system approach. We will use this property for the integrated approach. If we use the integrated approach, we can solve linear systems maintaining sparsity of matrices without respect of the existence of dense columns. Scope and purpose Interior-point methods require a step to solve the linear systems for computing a new direction at every iteration. Generally, we solve the linear systems by applying Cholesky factorization. When there is a dense column, we can not exploit the sparsity of matrices. The most popular way of treating such a dense column employs the Schur complement method or the augmented system approach. The Schur complement method is faster than the augmented system approach, but suffers from numerical unstability. We present a fast and numerically stable approach by integrating former approaches.