I have the following Boolean least-squares problem.
$$ \underset{x \in \{0,1\}^n}{\text{minimize}} \quad \| A x - b \|_2^2 $$
where $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^n$. A solution does not necessarily exist, i.e., there does not exist $x'$ such that $A x' = b$.
I have tried looking online for a solution but could not find one that suits my problem. From what I understood, algorithms like gradient descent only work with real valued vectors. Then, when I tried looking into MIP problems (for example Minimize $\| x \|_1$ subject to $Ax=b$ where $x$ is a binary vector) the goal was to minimize the norm of $x$ while supposing that a solution exists, which is different than my case.
I would like to know if there exists any way to optimize this value, and if there is any efficient way to do it. I am expecting this problem to be NP-hard from what I saw with MIP, but I am not sure yet.
Thank you for your help !
