Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two convex polytopes with $\mathcal{H}$-representation, i.e.,
\begin{align} \mathcal{P}_1 &= \{x \in \mathbb{R}^n\colon A_1x \leq b_1\},\\ \mathcal{P}_2 &= \{x \in \mathbb{R}^n\colon A_2x \leq b_2\}, \end{align}
where $A_1, A_2 \in \mathbb{R}^{m\times n}$ and $b_1,b_2 \in \mathbb{R}^m$. Then, how to judge if $\mathcal{P}_1 \subseteq \mathcal{P}_2$? (Can I convert this problem into an optimization problem?) Thanks!
When $\mathcal{P}_1$ happens to be bounded, then the set $\mathcal{V}_1$ of its vertices would be finite. Thus you just would have to check $x_k\in\mathcal{P}_2$ for each $x_k\in\mathcal{V}_1$.
I.e. you don't need any optimization here. But you'd need to convert the $\mathcal{H}$-representation of $\mathcal{P}_1$ into a corresponding $\mathcal{V}$-representation.
--- rk