A Second-Order Modified Version of Mehrotra-type Predictor-Corrector Algorithm for Convex Quadratic Optimization

Abstract

Mehrotra-type predictor-corrector algorithms are the core of the interior point methods software packages. The example made by Salahi et al. recently has shown that a second-order modified version of Mehrotra-type predictor-corrector algorithm for linear optimization may be forced to take very small steps in order to remain in a certain neighborhood of the central path. This motivates them to introduce a safe strategy in order to guarantee a lower bound for the maximum feasible step in the corrector, and subsequently ensure the polynomial complexity. Based on their research, this paper extend the algorithm to convex quadratic optimization. The polynomial complexity of the new algorithm is derived, namely, $\mathcal {O}\left(n\log\frac{(x^0)^Ts^0}{\epsilon}\right)$. Since the search directions are not orthogonal, the new algorithm is different from their method by the way of computing the barrier parameter and performing the complexity analysis.