Facial Reduction for Symmetry Reduced Semidefinite Doubly Nonnegative Programs

Abstract: We consider both facial reduction, \FRp, and symmetry reduction, \SRp, techniques for semidefinite programming, \SDPp. We show that the two together fit surprisingly well in an alternating direction method of multipliers, \ADMMp, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, \DNN, relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after \SRp, and that the \DNN relaxations considered here have singularity degree one, that is reduced to zero after \FRp. The combination of \FR and \SR leads to a significant improvement in both numerical stability and running time for both the \ADMM and interior point approaches. We test our method on various \DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than $n=500$. This translates to a semidefinite constraint of order $250,000$ and $625\times 10^8$ nonnegative constrained variables, before applying the reduction techniques.