  We consider a family of integer linear programs in which the coefficients of
the constraints and objective function are polynomials of an integer parameter
$t.$ For $\ell$ in $\mathbb{Z}_+,$ we define $f_\ell(t)$ to be the
$\ell^{\text{th}}$ largest value of the objective function with multiplicity
for the integer linear program at $t.$ We prove that for all $\ell,$ $f_\ell$
is eventually quasi-polynomial; that is, there exists $d$ and polynomials $P_0,
\ldots, P_{d-1}$ such that for sufficiently large $t,$ $f_\ell(t)=P_{d
\pmod{t}}(t).$ Closely related to finding the $\ell^{\text{th}}$ largest value
is describing the vertices of the convex hull of the feasible set. Calegari and
Walker showed that if $R(t)$ is the convex hull of $\mathbf{v_1}(t), \ldots,
\mathbf{v_k}(t)$ where $\mathbf{v_i}$ is a vector whose coordinates are in
$\mathbb{Q}(u)$ and of size $O(u),$ then the vertices of the convex hull of the
set of lattice points in $R(t)$ has eventually quasi-polynomial structure. We
prove this without the $O(u)$ assumption.
