An arc-search infeasible-interior-point method for symmetric optimization in a wide neighborhood of the central path

Abstract

In this paper, we propose a new arc-search infeasible-interior-point method for symmetric optimization using a wide neighborhood of the central path. The proposed algorithm searches for optimizers along the ellipses that approximate the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. The complexity bound is $$\mathcal {O}(r^{5/4}\log \varepsilon ^{-1})$$ for the Nesterov–Todd direction, and $$\mathcal {O}(r^{7/4}\log \varepsilon ^{-1})$$ for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and $$\varepsilon $$ is the required precision. The obtained complexity bounds coincide with the currently best known theoretical complexity bounds for the short step path-following algorithm. Some limited encouraging computational results are reported.