Recently, I try to learn Forward backward splitting algorithm.
I find it was proposed in 1988 'Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities'.
It is to minimize $$f(x)+g(x)$$
by two steps, $f(x) $ and $g(x)$ are convex functions, but not necessarily smooth.
$$x_{t+1}=arg\min_u\left\{\frac{1}{2}\|u-(w_t-\nabla t \partial f(w_t)\|^2+\nabla tg(u)) \right\}$$
I want to know the best choice for $\nabla t$ so far. Thank you.
As I know the forward-backward algorithm, you need at least one of your summands to be smooth (appears to be $g$ in your case?) and $L$-Lipschitz continuous. I'll assume this holds.
The forward-backward iteration $$x_{n+1}=\textrm{prox}_{(\Delta t) f}(x_n-(\Delta t) \nabla g x_n)$$ is proven to converge whenever a solution exists and $\Delta t\in]0,2/L[$. There is usually no way to know ``a priori'' which choice of $\Delta t$ is best, since this varies from problem to problem. Based on heuristics and numerical considerations, I usually avoid very small values of $\Delta t$. I usually try out $\Delta t=1/L$ first.
