Optimality Constraints For the Cone of Positive Polynomials

Consider a proper cone K ⊂ < and its dual cone K. It is well known that the complementary slackness condition xs = 0 defines an n-dimensional manifold C(K) = { (x, s) : x ∈ K, s ∈ K, xs = 0 } ⊂ <×<. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. This fact proves to be very useful when optimizing over such cones, therefore it is natural to look for similar optimality constraints for non-symmetric cones. In this paper we look at the cone of positive polynomials P2n+1 and its dual, the moment cone M2n+1. We show that there are exactly 4 linearly independent bilinear identities which hold for all (x, s) ∈ C(K), regardless of the dimension of the cones. Since these are not sufficient to describe C(P2n+1) we then look for more complicated constraints and present a set of 2n + 3 valid cubic conditions. We also examine the cone of positive polynomials over a finite interval and the cone of positive trigonometric polynomials. In an Appendix we give some examples of cones where our approach can be used to show that no non-trivial bilinear optimality constraints exist.