A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions

Abstract

<p style='text-indent:20px;'>In an attempt to improve theoretical complexity of large-update methods, in this paper, we propose a primal-dual interior-point method for <inline-formula><tex-math id="M2">\begin{document}$ P_{\ast}\left( \kappa \right) $\end{document}</tex-math></inline-formula>-horizontal linear complementarity problem. The method is based on a class of parametric kernel functions. We show that the corresponding algorithm has <inline-formula><tex-math id="M3">\begin{document}$ O\left( \left( 1+2\kappa \right) p^{2}n^{\frac{2+p}{2\left( 1+p\right) }}\log \frac{n}{\epsilon }\right) $\end{document}</tex-math></inline-formula> iteration complexity for large-update methods and we match the best known iteration bounds with special choice of the parameter <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ P_{\ast }\left(\kappa \right) $\end{document}</tex-math></inline-formula>-horizontal linear complementarity problem that is <inline-formula><tex-math id="M6">\begin{document}$ O\left(\left( 1+2\kappa \right) \sqrt{n}\log n\log \frac{n}{\epsilon }\right) $\end{document}</tex-math></inline-formula>. We illustrate the performance of the proposed kernel function by some comparative numerical results that are derived by applying our algorithm on five kernel functions.</p>