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 univariate polynomials with integer coefficients, let $\mathbf{b}$ be an $m$-dimensional vector whose elements are also univariate polynomials with integer coefficients, and let $P_i$ and $Q_i$ be univariate polynomials with integer coefficients and with positive leading coefficients 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 function of $t$?
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
