Given a $d$ dimensional vector $\bar{x} = [x_0,...,x_d]^t$,

how do I minimize $||\bar{x}-\bar{y}||_p$ such that $A\bar{y}=0$, for $p= 0$ i.e.,Minimize $L_0$ norm.

I also have the constraints that $\bar{x},\bar{y} \in Z^d$ (and not in $R^d$). and the components of the vectors are bounded. $-N \le y_i \le N$.

The exhaustive search is too complex for me to evaluate since it is exponential complexity. The matrix $A$ contains only entries -1,0,1.

Thanks in advance for any help.