Bilinear optimality constraints for the cone of positive polynomials

Abstract

For a proper cone ${{\mathcal K}\subset\mathbb{R}^n}$ and its dual cone ${{\mathcal K}^*}$ the complementary slackness condition ${\langle{\rm {\bf x}},{\rm {\bf s}}\rangle=0}$ defines an n-dimensional manifold ${C({\mathcal K})}$ in the space ${{\mathbb R}^{2n}}$ . When ${{\mathcal K}}$ is a symmetric cone, points in ${C({\mathcal K})}$ must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore it is natural to look for similar bilinear relations for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for points in ${C({\mathcal K})}$ . We examine several well-known cones, in particular the cone of positive polynomials ${{\mathcal P}_{2n+1}}$ and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all ${({\rm {\bf x}},{\rm {\bf s}})\in C({\mathcal P}_{2n+1})}$, 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. We prove similar results for Müntz polynomials.