Given polyhedra $\mathcal{P}$ and $\mathcal{Q}$ with their representations in terms of their extreme points and extreme rays, I need to construct an algorithm that decides whether $\mathcal{P} \subset \mathcal{Q}$.
I think it suffices to check whether the rays that pass through the extreme points of $\mathcal{P}$ along the direction of its extreme rays, lie within $\mathcal{Q}$. That is, given $x^1, \cdots, x^k$ as extreme points of $\mathcal{P}$ and $w^1, \cdots, w^m$ as the extreme rays of $\mathcal{P}$, we can check to see whether $x^i + \theta w^j \in \mathcal{Q} \; , \; \forall i = 1, \cdots, k \; , \forall j = 1, \cdots, m$ where $\theta > 0$ is an arbitrarily large step size. If any of points constructed as such does not lie in $\mathcal{Q}$, I can conclude that $\mathcal{P} \not\subset \mathcal{Q}$.
I appreciate any ideas and alternative approaches.
Note that $\mathcal{P}$ is the convex hull of $\{ x^i + \theta w^j : \theta \geq 0, \; i=1,\ldots,k, \; j=1,\ldots,m \}$. Now, $x^i + \theta w^j \in \mathcal{Q}$ for all $\theta \geq 0$, $i=1,\ldots,k$, $j=1,\ldots,m$ if and only if $\mathcal{P} \subset \mathcal{Q}$. The "if" part is trivial while the "only if" follows from convexity of $\mathcal{Q}$. Your observation that $\mathcal{P} \not\subset \mathcal{Q}$ if $x^i + \theta w^j \not\in \mathcal{Q}$ for some $\theta \geq 0$, $i$ and $j$ is therefore correct.