ADMM for SDP Relaxation of GP

Abstract

We consider the problem of partitioning the set of nodes of a graph G into k sets of given sizes in order to minimize the cut obtained after removing the k-th set. This is a variant of the well-known vertex separator problem that has applications in e.g., numerical linear algebra. This problem is well studied and there are many lower bounds such as: the standard eigenvalue bound; projected eigenvalue bounds using both the adjacency matrix and the Laplacian; quadratic programming (QP) bounds derived from imitating the (QP) bounds for the quadratic assignment problem; and semidefinite programming (SDP) bounds. For the quadratic assignment problem, a recent paper of [8] had great success from applying the ADMM (altenating direction method of multipliers) to the SDP relaxation. We consider the SDP relaxation of the vertex separator problem and the application of the ADMM method in solving the SDP. The main advantage of the ADMM method is that optimizing over the set of doubly non-negative matrices is about as difficult as optimizing over the set of positive semidefinite matrices. Enforcing the non-negativity constraint gives us a clear improvement in the quality of bounds obtained. We implement both a high rank and a nonconvex low rank ADMM method, where the difference is the choice of rank of the projection onto the semidefinite cone. As for the quadratic assignment problem, though there is no theoretical convergence guarantee, the nonconvex approach always converges to a feasible solution in practice.