Distributed Optimization for Rank-Constrained Semidefinite Programs

This letter develops a distributed optimization framework for solving the rank-constrained semidefinite programs (RCSPs). Since the rank constraint is non-convex and discontinuous, solving an optimization problem with rank constraints is NP-hard and notoriously time-consuming, especially for large-scale RCSPs. In the proposed approach, by decomposing an unknown matrix into a set of submatrices with much smaller sizes, the rank constraint on the original matrix is equivalently transformed into a set of constraints on the decomposed submatrices. The distributed framework allows parallel computation of subproblems while requiring coordination among them to satisfy the coupled constraints. As the scale of every subproblem solved independently is significantly reduced, the decomposition scheme and the distributed framework can be applied to large-scale RCSPs. Moreover, optimality conditions of the proposed distributed optimization algorithm for RCSPs at the converged point are analyzed. Finally, the efficiency and effectiveness of the proposed method are demonstrated via simulation examples for solving the image denoising problem.