For example the total variation regularization formulation is given as $$ \min_x \frac{1}{2} || Ax -y ||_2^2 + \alpha ||\nabla x||_1. $$
Now I am wondering, would it make sense to ask for a solution of the minimization problem given as $$ \min_x ||\nabla x||_1 + \frac{\alpha}{2} || Ax -y ||_2^2, $$ where now the data fit term acts as a regularization term? Since I have never seen something like this in literature until now, I would guess this is not feasible (for maybe obvious reasons), but why is that? If yes, what are suitable algorithms to solve problems of this form?
Thank you in advance.
Under optimization, we have that $$ ||\nabla x||_1 + \frac{\alpha}{2}||Ax-y||^2_2 \equiv \frac{1}{2}||Ax-y||^2_2+\frac{1}{\alpha}||\nabla x||_1, $$ since it is just a scaling by the positive constant $\alpha^{-1}$. Hence your problem is also a TV-problem.