How can I formulate joint cost functions if Lagrangians are involved? For example, if I have
$J_1 = \|\mathbf{Ax} - \mathbf{b}\|^2_2 + \lambda f$ and $J_2 = \|\mathbf{Cx} - \mathbf{d}\|^2_2$,
would it be correct to do something like
$J_{12} = \| \frac{1}{2}(\mathbf{Ax} - \mathbf{b}) + \frac{1}{2}(\mathbf{Cx} - \mathbf{d})\|^2_2 + \lambda (\frac{f}{2})$?
I'm not sure if I can just effectively shift the Lagrange multiplier term.