Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$, I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to determine that $P$ has no vertices).
I know that any method based on Fourier-Motzkin elimination would certainly not be very efficient, but I am just curious whether this could work at all.
To me, it feels like by setting each variable to the upper or lower bound during the back-substitution phase of Fourier-Motzkin, we should end up with a vertex of $P$. This is certainly the case in one dimension, and I don't see why it wouldn't work in $n > 1$ dimensions. If it turns out that a variable is unbounded in both directions, then $P$ cannot have a vertex (because it's either empty or it contains a line).
Just to clarify: I only want to compute one single vertex of $P$, not all.
Any help or pointers to relevant literature are appreciated.
