A multilevel approach for a class of semidefinite programs

We consider a class of semidefinite programs (SDPs) that arises from combinatorial optimization problems on graphs. We propose a multilevel approach that produces a sequence of progressively coarser problems by coarsening the underlying graphs. We use the solution of each coarse problem to provide an initial approximation to the solution at a finer level. At the coarsest level we employ Newton's method for high-accuracy solutions, and at finer levels we take advantage of inexpensive coordinate descent updates. We coarsen graphs based on an algebraic distance that can be computed efficiently. Furthermore, our coarsening scheme preserves the properties of graph Laplacian matrices between the fine and coarse levels. Numerical experiments show that the hybrid multilevel approach is competitive with the state-of-the-art SDP solver on large synthetic graphs.