Moment approximations for set-semidefinite polynomials

The set of polynomials which are nonnegative over a subset of the nonnegative orthant (we call them set semidefinite) have many uses in optimization. A common example of this type of set is the set of copositive matrices, where effectively we are considering nonnegativity over the entire nonnegative orthant and we limit the polynomials to be homogeneous of degree two. Lasserre in A new look at nonnegativity on closed sets and polynomial optimization, SIAM Journal of Optimization, 21 (2011), has previously provided a method using moments to provide an outer approximation to this cone for a general subset of the real space, not necessarily contained in the nonnegative orthant. In this paper we show that in the special case of considering nonnegativity over a subset of the nonnegative orthant, we can provide a new outer approximation hierarchy based on the completely positive moment matrices, which is a new idea and implies other approximations which are at least as good as that provided by Lasserre. We also provide an interesting new insight into the use of moments for constructing these approximations.