I will consider a family of Integer Linear Programs parametrized by a positive integer $t$ Let $\mathbf{x} = (x_1, \ldots, x_n)$ be the indeterminates. Let $A$ an $m$ by $n$ matrix whose elements are in $\mathbb{Z}[t],$ $\mathbf{b}$ be an $m$-D vector whose elements are in $\mathbb{Z}[t],$ and $P_i(t), Q_i(t)$ be in $\mathbb{Z}[t]$ with positive leading coefficient for $i=1, \ldots, n.$ Let $f(t)$ be the maximum value of $\sum_{i=1}^n Q_i(t) x_i$ with constraints $0 \le x_i \le P_i(t)$ $A(t) \mathbf{x} \le \mathbf{b}(t)$ $x_i \in \mathbb{Z}$ or $0$ if no points satisfy all constraints. Is it true that $f(t)$ is eventually a quasi-polynomial? Equivalently, do there exist $m, N \in \mathbb{Z}^+$ and polynomials $R_0, \ldots, R_{m-1}$ in $\mathbb{R}[t]$ such that for all integers $t$ greater than $N,$ $f(t)=R_{t \pmod{m}}(t)$? I think this could be true because the the set satisfying the constraints seems to have a convex hull whose vertices coordinates are eventually quasi-polynomials, possibly with some redundancy. I'm having a very hard time with convex hulls in high dimensions. Note: quasi-polynomial as opposed to polynomial is necessary because maximizing $x_1$ subject to $0 \le x_1 \le t$ and $2 x_1 \le t$ gives $\lfloor t/2 \rfloor.$ Integer Linear Programming seems prominent enough that I thought I would ask this here first