I'm a little new to this and not sure how to evaluate the proximal operator in this context, assuming a closed form exists.
Calculate $\text{prox}_{c,f+g}(v)$, where
$f(x) = \frac{1}{2}\|Ax-b\|_2^2$ $~~~$ and $g(x) = \|x\|_1$.
If somebody here could help me a bit that would be great. Thanks!
I could help if it were $(f+g)(x) = \lambda \|x\|_1 + \tfrac{\mu}{2}\|x\|^2$ - that is the elastic net function and has a known closed-form proximal operator.
Typically, even when $\mathrm{prox}_{\lambda \phi(\cdot)}$ is easy to compute, $\mathrm{prox}_{\gamma \phi(A\cdot\cdot)}$ is not necessarily easy to compute (at least not with a closed-form formula).
By the way, assuming that there is a closed-form for the proximal operator of $f+g$ implies that there is a closed-form solution of the $\ell_1$-regularised least squares problem.
If you are interested in applying a proximal point algorithm for the solution of a LASSO problem, then you will have to solve the minimisation problem for the computation of the prox-operator numerically, i.e., at every iteration you will have to solve
$$ x^{\nu+1} = \operatorname*{argmin}_{z\in\mathbb{R}^n} \{\tfrac{1}{2}\|Az-b\|^2 + \|z\|_1 - \frac{1}{2\lambda}\|x^\nu - z\|^2\}$$
But this may not be very practical. However, there are many other ways to solve this problem (e.g., you can use a proximal gradient method and take the prox on $\|\cdot\|_1$ only)
