Usually I've seen the basis pursuit problem formulated as
$$\min_{x\in R^N}{\left\lVert x\right\rVert}_1 \quad \text{subject to} \; b=Ax$$
but in this paper, they formulate it as
$$\min_{x \in R^N} \mu||x||_1 + \frac{1}{2}||Ax-b||_2^2$$
(with $u=x$ and $f=b$, equation (1) in the paper).
I can't find the reference that proves these two formulations are equal. Is it based on a general rule similar to a Lagrangian? Or is there a proof for this specific instance (in which case, please provide a reference).
Update: It looks like this is actually the formulation for basis pursuit denoising.
In the wikipedia article, it says this problem becomes basis pursuit as $\mu \rightarrow \infty$ ($\mu=\lambda$ in their form), but I don't think this is true, as then the constraint term falls off. Is there a way to find $\mu$ such that the formulations are equal?