  We aim to solve a structured convex optimization problem, where a nonsmooth
function is composed with a linear operator. When opting for full splitting
schemes, usually, primal-dual type methods are employed as they are effective
and also well studied. However, under the additional assumption of Lipschitz
continuity of the nonsmooth function which is composed with the linear operator
we can derive novel algorithms through regularization via the Moreau envelope.
Furthermore, we tackle large scale problems by means of stochastic oracle
calls, very similar to stochastic gradient techniques. Applications to total
variational denoising and deblurring are provided.
