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We investigate the hierarchy of conic inner approximations$\mathcal{K}^{(r)}_n$ ($r\in \mathbb{N}$) for the copositive cone$\text{COP}_n$, introduced by Parrilo (Structured Semidefinite Programs andSemialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis,California Institute of Technology, 2001). It is known that$\text{COP}_4=\mathcal{K}^{(0)}_4$ and that, while the union of the cones$\mathcal{K}^{(r)}_n$ covers the interior of $\text{COP}_n$, it does not coverthe full cone $\text{COP}_n$ if $n\geq 6$. Here we investigate the remainingcase $n=5$, where all extreme rays have been fully characterized by Hildebrand(The extreme rays of the 5 $\times$ 5 copositive cone. Linear Algebra and itsApplications, 437(7):1538--1547, 2012). We show that the Horn matrix $H$ andits positive diagonal scalings play an exceptional role among the extreme raysof $\text{COP}_5$. We show that equality $\text{COP}_5=\bigcup_{r\geq 0}\mathcal{K}^{(r)}_5$ holds if and only if any positive diagonal scaling of $H$belongs to $\mathcal{K}^{(r)}_5$ for some $r\in \mathbb{N}$. As a mainingredient for the proof, we introduce new Lasserre-type conic innerapproximations for $\text{COP}_n$, based on sums of squares of polynomials. Weshow their links to the cones $\mathcal{K}^{(r)}_n$, and we use an optimizationapproach that permits to exploit finite convergence results on Lasserrehierarchy to show membership in the new cones.

On the Exactness of Sum-of-Squares Approximations for the Cone of $5\times 5$ Copositive Matrices