We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined … Read more
We study copositive and completely positive cones over symmetric cones of rank at least $5$, with particular emphasis on whether these cones are spectrahedral shadows and on the behavior of a sum-of-squares inner-approximation hierarchy. We examine to what extent known results for nonnegative orthants of dimension at least $5$ carry over to general symmetric cones … Read more
We classify the faces of copositive and completely positive cones over a second-order cone and investigate their dimension and exposedness properties. Then we compute two parameters related to chains of faces of both cones. At the end, we discuss some possible extensions of the results with a view toward analyzing the facial structure of general … Read more