Complexity analysis of interior-point methods yielding the best known iteration bound for semidefinite optimization

The purpose of this paper is to obtain new complexity results for solving the semidefinite optimization (SDO) problem. We define a new proximity function for the SDO by a new kernel function with an efficient logarithmic barrier term. Furthermore, we formulate an algorithm for the large and small-update primal-dual interior-point method (IPM) for the SDO. It is shown that the best result of iteration bounds for large-update methods and small-update methods can be achieved, namely $\mathcal{O}\left(qn^{\frac{q+1}{2q}}\log \frac{n}{\epsilon }\right) $\ for large-update and $\mathcal{O}(q^{2}\sqrt{n}\log \frac{n}{\epsilon })$ for small-update methods, where $q>1.$ The analysis in this paper is new and different from the one using for LO. Several new tools and techniques are derived in this paper. Furthermore, numerical tests to investigate the behavior of the algorithm so as to be compared with other approaches.