We are given $w \in \mathbb{R}^n$.
We wish to compute the proximal map $$\arg\min_u \Big[\frac{1}{2\eta}||u-w||_2^2 + \frac{\lambda}{2}u^Tu + \mu \sum|u_i|\Big]$$
I tried getting the derivative, but I ended up with something like $$\frac{1}{\eta}(u-w) + \lambda u + \mu \cdot\mathrm{sign}(u)$$ and I don't know how to deal with the $\mathrm{sign}$, so I can't isolate $u$. Am I going in the right direction here? Or is there another way to compute this?