Postponing the choice of the barrier parameter in Mehrotra-type predictor–corrector algorithms

Abstract

Recently Salahi et al. have considered a variant of Mehrotra’s celebrated predictor–corrector algorithm. By a numerical example they showed that this variant might make very small steps in order to keep the iterate in a certain neighborhood of the central path, that itself implies the inefficiency of the algorithm. This observation motivated them to incorporate a safeguard in their algorithmic scheme that gives a lower bound for the step size at each iteration and thus imply polynomial iteration complexity. In this paper we propose a different approach that enables us to have control on the iterates. Our new approach is based on postponing the choice of the barrier parameter and it does not require any safeguard strategy. To do so, first we fix a step size in the corrector step, then by solving a one dimensional optimization problem we calculate the barrier parameter. Finally, using the calculated barrier parameter we make the next step. It is proved for the feasible case that our new algorithm stops after at most O n 2 log n ϵ iterations without any safeguard strategy. We further modified the proposed algorithm by slightly changing the Newton system in the corrector step. This variant enjoys an even better worst case iteration complexity bound, i.e., O n log n ϵ . Superlinear convergence of both algorithms are established. Finally, our numerical results demonstrate that a theoretically rigorous IPM may have the same, or better computational efficiency than the ones enhanced by aggressive heuristics.