The purpose of this paper is to obtain new complexity results for a second-order cone optimization (SOCO) problem. We define a proximity function for the SOCO by a kernel function. Furthermore we formulate an algorithm for a large-update primal-dual interior-point method (IPM) for the SOCO by using the proximity function and give its complexity analysis, and then we show that the new worst-case iteration bound for the IPM is $O(qsqrt{N}(logN)^{frac{q+1}{q}}log{frac{N}{epsilon})$, where $q{geqq}1$.