Given a solvable integer programming problem on the form
$$ \bar{a}_i\cdot\bar{x} + y \geq 1\ \forall\ i\in\{1,\dots,n\}\\ \bar{x}\in\mathbb{Z}^d,\ y\in\mathbb{Z} $$
where $\bar{a}_i\in\{-L,-L+1,\dots,L-1,L\}^d\ \forall\ i\in\{1,\dots,n\}$ are known vectors, is it possible to say anything about the magnitude of the solution, i.e., $\sqrt{\bar{x}\cdot\bar{x}+y^2}$, that is the closest to the origin? Is it possible to find the maximum value, expressed using big $O$ notation, that this magnitude can ever assume, as a function of $L$ and $d$, and possibly also $n$ (otherwise we can consider $n$ to be very large)?
Note that $y$ and the elements of $\bar{x}$ can be negative.