Consider the following convex problem, where $M,N$ are integers and $a_n\ge0$ is a constant.$$(P_0)\min_{\{x_n\},\{y_m\}}\sum_{n=1}^Nf(x_n)\\ s.t. a_nx_n\le0\\ x_n+\sum_{m=1}^My_m\le0\\ x_n\in[0,1],n=1,2,\cdots,N\\ y_m\in[0,1], m=1,2,\cdots,M $$
In Problem $P_0$, $f(\cdot)$ is smooth and convex, and the constraints are linear (notes: $f(\cdot) $ depends only on $\{x_n\}$). I know $P_0$ can be solved by CVX efficiently when $N,M$ are relatively small. But I still have some questions about solving this problem.
(1) Can $P_0$ be solved by block coordinate descent method with guaranteeing convergence? (I noticed that the block $\{y_m\}$ is coupled with block $\{x_n\}$)
(2) When $N$ or $M$ is very large, this large-scale convex optimization problem $P_0$ may not be solved efficiently by CVX and other solvers. So, is there some method to handle this type of problem?
Any help is appreciated.