An Improved and Simplified Full-Newton Step $O(n)$ Infeasible Interior-Point Method for Linear Optimization

Abstract

Abstract. We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few-at most three-centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the earlier bound by a factor 2 √ 2. The improvements are due to a new lemma that gives a much tighter upper bound for the proximity after the feasibility step.

Improved Full-Newton-Step Infeasible Interior-Point Method for Linear Complementarity Problems

“…This strong relationship is based on the fact that on the Lemma 3 first proposed in [18]. The algorithm is still globally convergent and iteration bound matches the best known iteration bound for these types of IP M .…”

“…The centering steps were necessary to bring the point obtained by feasibility step to the τ -neighborhood of the central path again. In the improved version of the method, with the suitable choice of the threshold parameter τ and barrier parameter θ, it is guaranteed that after one feasibility step, the new iterate is feasible and immediately in the τ -neighborhood of the central path thanks to the much tighter proximity estimate which is based on the new Lemma 3 introduced in [18]. Thus, the centering steps are eliminated.…”

“…The following function ξ and Lemma 3, introduced in [18], play a central role in the subsequent analysis.…”

“…In the earlier version, each iteration consisted of one feasibility step and few centering steps. The improved version guarantees that after one feasibility step, the new iterate is feasible and close enough to the central path thanks to the much tighter proximity estimate which is based on the new lemma introduced in [18]. Thus, the centering steps are eliminated.…”

A generalized direction in interior point method for monotone linear complementarity problems

“…Further research may extend this result to a more general family of LCPs such that P * (κ)-LCPs or P 0 -LCPs with bounded level sets. Recent developments by Roos in [26] consider an infeasible IPM with full Newton step using only one feasibility step. More investigations regarding the method presented in this paper could extend the results in [26].…”