Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

Abstract

Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J. Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximities for linear optimization (LO) problems. They also extended the approach for LO to SDO. In this paper we present a primal-dual interior-point algorithm for SDO problems based on a simple kernel function which was first presented at the Proceedings of Industrial Symposium and Optimization Day, Australia, November 2002; the function is not self-regular. We derive the complexity analysis for algorithms based on this kernel function, both with large- and small-updates. The complexity bounds are $\mathrm{O}(qn)\log\frac{n}{\epsilon}$ and $\mathrm{O}(q^{2}\sqrt{n})\log\frac{n}{\epsilon}$ , respectively, which are as good as those in the linear case.