Final Iterations in Interior Point Methods | Preconditioned Conjugate Gradients and Modiied Search Directions

In this article we consider modiied search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modiied search directions are computed by solving perturbed systems in which the systems may be solved ef-ciently by the preconditioned conjugate gradient solver. We prove the convergence of the interior point methods using the modiied search directions and show that each barrier problem is solved with a superlinear convergence rate. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.