Summary: ``In this paper we present a new polynomial interior-point algorithm for the Cartesian $P_*(\kappa)$ second-order cone linear complementarity problem based on a finite kernel function. The symmetrization of the search directions used in this paper is based on the Nesterov and Todd scaling scheme. We derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods, namely, $O((1 + 2\kappa )\sqrt N\log N\log \frac{N}{\epsilon})$ and $O((1 + 2\kappa )\sqrt N\log\frac{N}{\epsilon})$, respectively, which are as good as the $P_*(\kappa)$ linear complementarity problem analogue. Moreover, this unifies the analysis for $P_*(\kappa)$ linear complementarity problem and second-order cone optimization.''