A dynamic large-update primal‐dual interior-point method for linear optimization

Abstract

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound but work very poorly in practice, while the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this article, by restricting us to linear optimization (LO), we first exploit some interesting properties of a self-regular proximity function, proposed recently by the authors of this work and Roos, and use it to define a neighborhood of the central path. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, then the large-update IPM emerges to be the only natural choice. Then, we apply these results to design a specific self-regularity based IPM. Among others, we show that this self-regularity based IPM can also predict precisely the change of the duality gap as the standard IPM does. Therefore, we can directly apply the modified IPM to the simplified self-dual homogeneous model for LO. This provides a remedy for an implementation issue of the new self-regular IPMs. A dynamic large-update IPM in large neighborhood is proposed. Different from traditional large-update IPMs, the new dynamic IPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established.