Checking if one polytope is contained in another

Question

I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.

At the moment I am checking, for each row $a \in A$, whether there exists some $x$ satisfying $ax > 0$ but $Bx \leq 0$. Is there a more efficient approach?

Thanks

Answer 1

I would like to draw your attention to the 2002 survey by Volker Kaibel and Marc Pfetsch, "Some Algorithmic Problems in Polytope Theory," arXiv:math/0202204v1, which contains this on p.6:
 
As you probably know, an $\cal{H}$-description is by halfspaces, whereas a $\cal{V}$-description is by vertices. Reference [20] is: R. M. Freund and J. B. Orlin, "On the complexity of four polyhedral set containment problems," Math. Program., 33 (1985), pp. 139–145. Reference [17] is B. C. Eaves and R. M. Freund, "Optimal scaling of balls and polyhedra," Math. Program., 23 (1982), pp. 138–147.