Suppose we have a point $p = (p_1, p_2, ..., p_n)^T$. I'd like to find an integer point $x$:
$$\min_{x}{||x-p||}$$ $$s.t. Ax<b$$
This is a general description of my question. And in my specific case, $b=0$. Actually, in half of the cases I met, simply rounding $p$ to the nearest integer will satisfy $Ax<b$. However, the other half causes severe errors in my following data processing.
Currently I'm doing with a very specific method: check if $A[p]<b$, and if not, I check which rows of $A[p]$ doesn't satisfy the restriction and modify the corresponding entries of $[p]$. However, this method results in my code full of if-statements and is quite slow. I wonder if any faster methods, e.g., a few matrix operations, will find the solution.