Let $f(x):\Re^n\rightarrow \Re$ be a proper and closed convex function. Its Moreau-Yosida regularization is defined as
Lots of literature say $F(x)$ is Lipschtiz continuous and give explicitly the expression of $\nabla F(x)$ involving $Prox_f(x)$. But I have no idea how to calculate $\nabla F(x)$. Can anyone provide a straightforward method? I know Rockafellar's book gives a proof. But it assumes too much prior knowledge. I am wondering if there is a more elementary method to prove the Lipschtiz continuity and calculate its gradient.