Sparse sum-of-squares certificates on finite abelian groups

Sums-of-squares techniques have played an important role in optimization and control. One question that has attracted a lot of attention is to exploit sparsity in order to reduce the size of sum-of-squares programs. In this paper we consider the problem of finding sparse sum-of-squares certificates for functions defined on a finite abelian group G. In this setting the natural basis over which to measure sparsity is the Fourier basis of G (also called the basis of characters of G). We establish combinatorial conditions on subsets S and τ of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support τ. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S with maximal cliques related to τ. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at most ⌈n/2⌉, resolving a conjecture of Laurent [11]. They also allow us to show that any nonnegative function of degree d on G = ℤN has a sum-of-squares certificate supported on at most 3d log(N/d) Fourier basis elements. By duality this construction yields the first explicit family of polytopes in increasing dimensions that have a semidefinite programming description that is vanishingly smaller than any linear programming description.