Given an optimization problem $$ \text{min} \space x^T M x \\ \text{s.t.} \space Ax = b $$ where an analytical solution called weighted general inverse exists. However, we now know that Ax = b some times cannot be satisfied because A becomes singular. In this case, how can I still write the above problem as a solvable convex optimization (QP)?
Currently, I am thinking of $$ \text{min} \space w_1x^T M x +w_2(Ax - b)^T(Ax - b)\\ \text{where } w_2 >> w_1 $$ However, this will cause the solution to be less accurate when A is not Singular. Is there any way to re-write this?
Assume $x = x_r+x_n$ where $x_r\in \text{Range}\left(A\right)$ and $x_n\in \text{Null}\left(A\right)$. Then we have
$$ \text{min} \space \left(x_r+x_n\right)^T M \left(x_r+x_n\right) \\ \text{s.t.} \space Ax_r = b\\ \quad \space Ax_n = 0 $$
Is this okay?! You can write Lagrangian now.