The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique—two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible in general to identify cones using Lyapunov rank. But suppose we look … Read more
Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that … Read more
We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a … Read more