Valid inequalities in a polyhedron

Question

Let $\alpha, \beta >0, k \in \mathbb{Z}$, $\pi \in \mathbb{R}^n, \pi_0 \in \mathbb{R}$. Let $P\subset \mathbb{R}^n$ be a polyhedron, such that the inequalities: $$\pi x \leq \pi_0 +\alpha (x_j -k) $$ $$\pi x \leq \pi_0 +\beta (k+1-x_j)$$ are valid, for all $j$. I am asked to prove that then, the inequality $$\pi x \leq \pi_0$$ is valid in $P \cap \mathbb{Z}^n$.

I have tried many ways of combining the equations, so that I can get to the result, but so far, without much luck. I don't see a way of using the fact that the points are whole numbers, but I guess it has to do with the parameter k. I have also tried working in one or two dimensions, to see if there is any pattern in the inequations, but they don't seem to give a particular structure. I also tried to use the floor function, but the vector $\pi$ doesn't let me do anything.

Please help, I just need to know where to start from.

Answer 1

Hints:

• $x_j\in\mathbb{Z}$ implies the disjunction $(x_j \le k) \lor (x_j \ge k+1)$.
• Valid for $P \cap \{x \in \mathbb{R}^n: x_j \in \mathbb{Z}\}$ for all $j$ implies valid for $P \cap \mathbb{Z}^n$.