I have two convex polytopes defined as
$$ P_L = \{x\in\mathbb{R}^L \mid A_Lx \leq b_L\}\\ P_M = \{x\in\mathbb{R}^M \mid A_Mx \leq b_M\} $$
where $L$ and $M$ are in the order of $100$ or higher. Now define $P_l$ and $P_m$ as the projections of $P_L$ and $P_M$ onto their first $l$ and $m$ dimensions, whith $l<L$ and $m<M$.
I need to check if $$ P_l \subseteq P_m, $$ and to do this, I currently first do the two projections and then check the condition, but the projections become very computationally expensive as $M$ and $L$ grow large.
Is there any way to check this condition without performing the projections explicitly?