On the exactness of sum-of-squares approximations for the cone of 5x5 copositive matrices

We investigate the hierarchy of conic inner approximations K^r_n for the copositive cone COP_n, introduced by Parrilo (2000) [22]. It is known that COP_4=K^0_4 and that, while the union of the cones K^r_n covers the interior of COP_n, it does not cover the full cone if n \ge 6. Here we investigate the remaining case n=5, where all extreme rays have been fully characterized by Hildebrand (2012) [12]. We show that the Horn matrix H and its positive diagonal scalings play an exceptional role among the extreme rays of COP_5. We show that equality holds if and only if every positive diagonal scaling of H belongs to for some K^r_5. As a main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for COP_n, based on sums of squares of polynomials. We show their links to the cones K^r_n, and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones.