  We propose new proximal bundle algorithms for minimizing a nonsmooth convex
function. These algorithms are derived from the application of Nesterov fast
gradient methods for smooth convex minimization to the so-called Moreau-Yosida
regularization $F_\mu$ of $f$ w.r.t. some $\mu>0$. Since the exact values and
gradients of $F_\mu$ are difficult to evaluate, we use approximate proximal
points thanks to a bundle strategy to get implementable algorithms. One of
these algorithms appears as an implementable version of a special case of
inertial proximal algorithm. We give their complexity estimates in terms of the
original function values, and report some preliminary numerical results.
