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 $\mathcal P$?