  We construct a convergent family of outer approximations for the problem of
optimizing polynomial functions over convex bodies subject to polynomial
constraints. This is achieved by generalizing the polarization hierarchy, which
has previously been introduced for the study of polynomial optimization
problems over state spaces of $C^*$-algebras, to convex cones in finite
dimensions. If the convex bodies can be characterized by linear or semidefinite
programs, then the same is true for our hierarchy. Convergence is proven by
relating the problem to a certain de Finetti theorem for general probabilistic
theories, which are studied as possible generalizations of quantum mechanics.
We apply the method to the problem of nonnegative matrix factorization, and in
particular to the nested rectangles problem. A numerical implementation of the
third level of the hierarchy is shown to give rise to a very tight
approximation for this problem.
