Sum of Squares Certificates for Containment of H-Polytopes in V-Polytopes

Given an H-polytope P and a V-polytope Q, the decision problem whether P is contained in Q is co-NP-complete. This hardness remains if P is restricted to be a standard cube and Q is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, there seems to be only limited progress on that problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certificates can be computed by a semidefinite hierarchy. As a main result, we show that under mild and explicitly known preconditions the semidefinite hierarchy converges in finitely many steps. In particular, this is the case if P is the standard cube and Q is the standard cross polytope.