Douglas-Rachford splitting method for semidefinite programming

In this paper, we study the Douglas-Rachford (DR) splitting method when applied to the standard semidefinite programming, and we introduce a new variant of this method and under weak assumptions prove its global convergence. At each iteration, it needs to solve two subproblems. First, project once onto the set of all symmetric positive semidefinite matrices. Then, solve one system of linear equations whose co-efficient matrix is symmetric positive definite. This is in sharp contrast to symmetric positive semi-definiteness of the corresponding co-efficient matrix in the boundary point method recently proposed by Povh et al. More importantly, we analyze the iteration complexity of the DR splitting method and derive the currently best result. Based on rigorous analysis in theory, we suggest an implementable version. Rudimentary numerical experiments confirm its efficiency and robustness on certain test problems with more than one million equality constraints. © Korean Society for Computational and Applied Mathematics 2015.