I have a function called f(x) which is convex and I can have access to its first order derivative , my objective function is $$\ J(\bf{x}) = f(\bf{x}) + \lambda |\bf{x}|_0 $$$$\  \bigtriangledown f(\bf{x}) \space is \space available$$Could you please provide a list of state of the art methods in the literature that can solve this sparse regularization problem?
an over view of sparsity promoting inversion techniques
sparse;inverse problem
The short answer is that in general this problem is intractable (NP-Hard) with the $\| x \|_{0}$ regularization.  Are you willing to consider minimizing $ f(x) + \lambda \| x \|_{1}$instead?  There are many methods for the 1-norm regularized problem.  There are also lots of theoretical results that give conditions under which 1-norm minimization produces solutions that are very close to the 0-norm regularized solution.