Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term

Abstract

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 > q2 > 1, the algorithm has \(O((q_1 + 1)n^{\frac{{q_1 + 1}} {{2(q_1 - q_2 )}}} \log \tfrac{n} {e}) \) and \(O((q_1 + 1)^{\frac{{3q_1 - 2q_2 + 1}} {{2(q_1 - q_2 )}}} \sqrt n \log \tfrac{n} {e}) \) complexity results for large- and small-update methods, respectively.