A primal-dual large-update interior-point algorithm for P*(κ)-LCP based on a new class of kernel functions

In this paper, we propose a large-update primal-dual interior point algorithm for P*(κ)-linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor self-regular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has o((1+2k)pn(1plogn+1)2lognε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o\left( {(1 + 2k)p\sqrt n {{\left( {\frac{1}{p}\log n + 1} \right)}^2}\log \frac{n}{\varepsilon }} \right)$$\end{document} iteration complexity which becomes o((1+2k)nlognlognε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o\left( {(1 + 2k)\sqrt n log{\kern 1pt} {\kern 1pt} n\log \frac{n}{\varepsilon }} \right)$$\end{document} with special choice of the parameter p. It is matches the currently best known iteration bound for P*(κ)-linear complementarity problem. Some computational results have been provided.