It is known that predictor-corrector methods in a large neighborhood of the central path are among the most efficient interior point methods (IPMs) for linear optimization (LO) problems. The best iteration bound based on the classical logarithmic barrier function is $O\left(n\log{\frac{n}{\epsilon}}\right)$. In this paper we propose a family of self-regular proximity based predictor-corrector (SR-PC) IPM for LO in a large neighborhood of the central path. Like all predictor-corrector IPMs, our new SR-PC algorithms have a predictor step and a corrector step. In the predictor step we use either the affine scaling or a self-regular direction, while in the corrector step we always use a self-regular direction. Our new algorithms use a special proximity function with different search directions and thus allows us to improve the so far best theoretical iteration complexity for a family of SR-PC IPMs. An $O\left(\sqrt{n} \exp(\frac{1-q+\log n}{2})\log n\log{\frac{n}{\epsilon}}\right)$ worst-case iteration bound with quadratic convergence is established, where $q$ is the barrier degree of the SR proximity function. If $q=1+\log n,$ then we have the so far best iteration complexity for the first order predictor-corrector method in a large neighborhood, and if $q=1+2\log(\log n)$ our algorithm has an $O\left(n\log\frac{n}{\epsilon}\right)$ iteration complexity. For the case $q=1,$ the result is a factor $\log n$ worse than the exist recent results.
Citation
Advanced Optimization Lab. McMaster University, Report 2004/7, Submitted.