1. If
$$\min x'Qx + Rx$$
$$Ax\leq b$$
$$x\in\mathbb Z^n$$
is a quadratic program with $x'Qx$ is convex is there a polynomial time algorithm for this if $A$ is totally unimodular? 

2. In particular if we ask that the smallest $x'Qx$ that touches on a vertex of $AX\leq b$ is that in $\mathcal P$?

We know that it is in $\mathcal P$ if $Q=0$. We also know for trivial reasons it will always first touch a vertex point.

>Is it the same for convex case? 

>If not in what cases can be find minimum $x'Qx$ that first touch a vertex point in $\mathcal P$?