Positivstellensatz 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. Here, we study the H-in-V containment problem from the viewpoint of a bilinear feasibility problem and in connection with linear and semidefinite relaxations. Handelman’s and Putinar’s Positivstellensatz yield hierarchies of linear programs and semidefinite programs, respectively, to decide the containment problem. We study their geometric properties and Positivstellensatz certificates. 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.