So, I've got the standard $L_1$ minimization problem: $\arg\min_x \{\|Ax-b\|_2^2+\lambda|Wx|_1\}$. I use Split-Bregman to solve this problem and it becomes $\arg\min_{x,d}\{\|Ax-b\|_2^2+\beta\|Wx-d\|^2+\lambda|d|\}$. It's convenient because now we're solving separately a quadratic problem in $x$ and another much simpler one with the $L_1$ norm in $d$.
So, in my original problem $\lambda$ is the strength of the regularization term of the problem. It's always difficult to justify setting a free parameter but somehow we'll manage. In the second problem instead I've got two free parameters!! The regularization strength and a parameter that enforces the equality $d=Wx$. How am I supposed to set $\beta$? Does it have any physical interpretation? As it is just enforcing a constraint it would look like any value would lead to the same results, doesn't it? But obviously this is not the case. What am I missing?