\title{Optimality Constraints For the Cone of Positive Polynomials\\[10pt]
}

\authornames{
G\'abor Rudolf, Nilay Noyan, Farid Alizadeh }




For a proper cone $\K\subset\mathbb{R}^n$ and its dual
cone $\K^*$ the complementary slackness condition $\vx^T\vs=0$
defines an $n$-dimensional manifold $C(\K)$ in the space
$\left\{~(\vx,\vs)~|~\vx\in\K,~\vs\in\K^*~\right\}$. 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 examine the cone of positive polynomials $\Po_{2n+1}$ and
its dual, the closure of the moment cone $\M_{2n+1}$. We show that
there are exactly $4$ linearly independent bilinear identities
which hold for all $(\vx,\vs)\in C(\K)$, regardless of the
dimension of the cones. Since these are not sufficient to describe
$C(\Po_{2n+1})$ we then look for more complicated constraints and
present a set of $2n+3$ valid cubic conditions. We then establish
similar results for 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.