There are some methods originally from Nesterov, which accelerates optimization methods, e.g. Nesterov 1983, Nesterov 2003, Nesterov 2005 ( smooth minimization of non-smooth functions), Nesterov 2013 ( Gradient Methods for minimizing composite functions) and some more recent methods.
I want to use one of these methods for minimizing a composite function (some of a smooth and a non-smooth ( max ) function. Although some of these paper, especially considers this type of problems, it is not clear which of these methods is better for this problem. Is there any comprehensive source which compares these methods for such a function?, or can you provide some information for this?
EDIT: I want to minimize a problem with the structure $f(x)=\Psi(x)+\phi(x)$. Where $\Psi(x)$ is nonsmooth function ( it is a max function over another variable) and $\phi(x)$ is a smooth function. There is a paper, HPE-convex concave saddle-point which uses exactly the method of nesterov 2013,"Gradient methods for minimizing composite functions" for this type of problem. But I want to know what is the best method for solving a composite of non-smooth + smooth function?
