Currently, I have a hard time to find out the solution to this problem:
Given a matrix $A \in Z^{m \times n}$, $b \in Z^m$, $c \in R^n$ and $\lambda \in R$. Is there $x \in R^n$ with $Ax \leq b$ and $c^Tx \geq \lambda$?
I was able to show that the given problem is $NP$, but now I have to show it is in $NP \cap co-NP$ by using strong duality and Farkas lemma. I thought it would be enough to show, that given a no certificate $x$, I can calculate $c^Tx$ and compare it with $\lambda$ in polynomial time.
Edit: We can assume, that all extreme points have polynomial size.
