How to reformulate my problem as a mixed-integer quadratic problem

Question

I have an unknown $n$-dimensional vector $x$ whose analytical expression depends on the following sum $x = z + Ba$ where the vector $z$ and the matrix $B\in \mathbb{R}^{n\times s}$ are given. So the $s$-dimensional vector $a$ is to be computed to find $x$.

The only assumption that we have is $x=0$ when we project $x$ onto the space spanned by $s$ different rows (that we don’t know their indices) of the matrix $B$ which has $n$ rows. To do this projection we can use $P_s\in \mathbb{R}^{n\times n}$ which is $1$ on the diagonal entries that correspond to the $s$ selected rows of $B$ and $0$ elsewhere. Hence, $P_s x= P_s z + P_s Ba=0 \implies a=-(P_sB)^{-1}P_sz$.

The main issue is that we don’t know the positions of these $s$ rows, so the problem is combinatorial and we need to go through all possible $n\choose s$ projections to find the exact $x$ which corresponds to the least cost $f(x)=\|y-Ax\|_2$ where $\|v\|_2=\big(\sum_iv_i^2\big)^{1/2}$, $y\in \mathbb{R}^{m\times 1}$ and the matrix $A\in \mathbb{R}^{m\times n}$ are given.

So my question is how I can reformulate my problem as a mixed-integer quadratic programming to go through all possible $n\choose s$ submatrices of $B$ formed by the $s$ selected rows and finally find the set of rows which corresponds to the least $f(x)$.