Complexity Analysis of Primal-Dual Interior-Point Methods for Linear Optimization Based on a New Parametric Kernel Function with a Trigonometric Barrier Term

Abstract

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods, $O(n^{\mathrm{2}/\mathrm{3}}\log (n/\epsilon ))$, and small-update methods, $O(\sqrt{n}\log (n/\epsilon ))$. These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.