Bilinear optimality constraints for the cone of positive polynomials

For a proper cone K ⊂ R and its dual cone K∗ the complementary slackness condition x s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K∗ }. When K is a symmetric cone, this fact translates to a set of n bilinear optimality conditions satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we examine several well-known cones, in particular the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1. We show that there are exactly four linearly independent bilinear identities which hold for all (x, s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials.