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 and Semialgebraic 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 cover the full cone $\text{ COP}_n$ if $n\geq 6$. Here we investigate the remaining case $n=5$, where all extreme rays have been fully characterized by Hildebrand (The extreme rays of the 5 $\times$ 5 copositive cone. Linear Algebra and its Applications, 437(7):1538--1547, 2012). We show that the Horn matrix $H$ and its positive diagonal scalings play an exceptional role among the extreme rays of $\text{ COP}_5$. 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 main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for $\text{ COP}_n$, based on sums of squares of polynomials. We show their links to the cones $\mathcal{ K}^{ (r)}_n$, and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones.
Categories: math.OC