Proximal Operator for an $ {L}_{1} $ Optimization Problem

Question

I am encountering an unconstrained minimization problem and I would like to solve it with an iterative proximal gradient method. The problem is of the form $$ \min_{x} ||f(x)-y||_2^2 + \lambda||x||_1 + \gamma||x-\beta||_p^p $$ where $x$, $\beta \in \mathbb{R}^n$ and $\beta$ is fixed and $f$ is differentiable.

I can easily write the proximal operator in the case where $p=2$. However I am struggling for the case where $p = 1$ and can't find any solution online.

Hope anyone helps. Thanks a lot !

Answer 1

Have you tried checking http://proximity-operator.net/multivariatefunctions.html ? It also may appear in Bauschke & Combettes' 2017 book.