A new primal-dual interior-point method for semidefinite optimization based on a parameterized kernel function

Abstract

As indicated in the recent studies about primal-dual interior-point methods (IPMs) based on kernel functions, a kernel function not only serves to determine the search direction and measure the distance of the current iteration point to the $$\mu $$ -center, but also affects the iteration complexity and the practical computational efficiency of the algorithm. In this paper, we propose a new IPM for semidefinite optimization (SDO) based on a parameterized kernel function which is a generalization of the one presented by Bai et al. (Optim Methods Softw 17(6):985–1008, 2002). By using the good properties of the parameterized kernel function, we deduce that the iteration bound for large-update method is $$O(\sqrt{n}\log {n}\log {\frac{n}{\epsilon }})$$ for $$q=O(n)$$ , which is the best known complexity results for such methods. In our knowledge, this result is the first instance of primal-dual interior point method for SDO which involving the kernel function. Some numerical results have been provided.