Finding sparse, equivalent SDPs using minimal coordinate projections

Abstract

We present a new method for simplifying SDPs that blends aspects of symmetry reduction with sparsity exploitation. By identifying a subspace of sparse matrices that provably intersects (but doesn't necessarily contain) the set of optimal solutions, we both block-diagonalize semidefinite constraints and enhance problem sparsity for many SDPs arising in sums-of-squares optimization. The identified subspace is in analogy with the fixed-point subspace that appears in symmetry reduction, and, as we illustrate, can be found using an efficient combinatorial algorithm that searches over coordinate projections. Effectiveness of the method is illustrated on several examples.