We discuss the use of preconditioned conjugate gradients (CG) method for solving the reduced KKT systems arising in interior point algorithms for linear programming. The (indefinite) augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the (positive definite) normal equations form. Therefore, we use the CG method to solve the augmented system and look for a suitable preconditioner.An explicit null space representation of linear constraints is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. This is achieved by means of recently developed efficient basis matrix factorisation techniques which exploit hyper-sparsity and are used in implementations of the revised simplex method.The approach has been implemented within the HOPDM interior point solver and applied to medium and large-scale problems from public domain test collections. Computational experience is encouraging.

In this paper we present the convergence analysis of the inexact infeasible path-following (IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix.The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of interior point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis.