  We tackle highly nonconvex, nonsmooth composite optimization problems whose
objectives comprise a Moreau-Yosida regularized term. Classical nonconvex
proximal splitting algorithms, such as nonconvex ADMM, suffer from lack of
convergence for such a problem class. To overcome this difficulty, in this work
we consider a lifted variant of the Moreau-Yosida regularized model and propose
a novel multiblock primal-dual algorithm that intrinsically stabilizes the dual
block. We provide a complete convergence analysis of our algorithm and identify
respective optimality qualifications under which stationarity of the original
model is retrieved at convergence. Numerically, we demonstrate the relevance of
Moreau-Yosida regularized models and the efficiency of our algorithm on robust
regression as well as joint feature selection and semi-supervised learning.

Optimization and Control