With a non-differentiable and convex functions $f$ and $g$, I am solving a problem of the form $\arg \min_y f(A - By) + g(y)$ using ADMM. That is, I formulate the problem as $$\min f(x) + g(y)$$ subject to $$ x = A - By$$ where $y \in \mathbb{R}^{p}$ is the primal variable, and $x \in \mathbb{R}^m, A\in\mathbb{R}^M, B \in \mathbb{R}^{M \times p}$ with $M \gg p$.
However, in implementing the ADMM algorithm to solve this problem, I am finding that the algorithm is extremely slow to converge. I suppose this is not surprising given that a small change in $y$ can have a dramatic effect on many of the constraints $x = A - By$.
I am guessing this is a known problem: I am hoping someone can point my in the direction of work which deals with this issue, or suggest alternative approaches to solving this type of problem.