  In the lines of our approach in \cite{Ouorou2019}, where we exploit Nesterov
fast gradient concept \cite{Nesterov1983} to the Moreau-Yosida regularization
of a convex function, we devise new proximal algorithms for nonsmooth convex
optimization. These algorithms need no bundling mechanism to update the
stability center while preserving the complexity estimates established in
\cite{Ouorou2019}. We report some preliminary computational results on some
academic test problem to give a first estimate of their performance in relation
with the classical proximal bundle algorithm.
