I am currently in the situation where I have to find the $\arg\min$ of a convex objective function with a non-smooth convex constraint. More formally, this takes the following form
$$\beta^* := \arg \min_{g(\beta) \leq R} \frac{1}{2} \|\beta-z\|_2^2$$
where $g(\beta)$ is convex but non-smooth.
I looked around for a while but could not find an algorithm able to solve this kind of question. All algorithms I found assumed that $g(\beta)$ is smooth. Does anyone of you know an algorithm able to solve such problems?
The constraint $g(\beta)\leq R$ is equivalent to saying that $\beta$ is in the sublevel set of $g$. Since $g$ is convex, this is a convex set. If it easy to project onto this sublevel set then you can use projected gradient descent to solve this problem. If the sublevel set is not easy to project onto but is also compact then you can use Frank-Wolfe to solve this problem.
There is also a relationship/equivalence between a constraint of the form $g(\beta)\leq R$ and an unconstrained problem with a regularizer $\lambda\tilde{g}(\beta)$. If you write the problem in this way then you can use a number of splitting algorithms, for example forward-backward.