I want to find an upper bound for the minimization problem
$\min x+y$ subject to $x\geq 0, y \geq 0, -Ax+By=C, (x,y)\in \mathbb{Z}^2$ where $A, B$ are non-negative constants.
I've seen that there are posts regarding finding the minimal solution of a certain example of the above problem and moreover, I know that for the solution, we just have to follow the slop created by A,B and take the first point on the grid. But does there exist any known bounds? Our professor told us that Hilbert Bases might be of help, but I don't see how.