  When considering the minimization of a quadratic or strongly convex function,
it is well known that first-order methods involving an inertial term weighted
by a constant-in-time parameter are particularly efficient (see Polyak [32],
Nesterov [28], and references therein). By setting the inertial parameter
according to the condition number of the objective function, these methods
guarantee a fast exponential decay of the error. We prove that this type of
schemes (which are later called Heavy Ball schemes) is relevant in a relaxed
setting, i.e. for composite functions satisfying a quadratic growth condition.
In particular, we adapt V-FISTA, introduced by Beck in [10] for strongly convex
functions, to this broader class of functions. To the authors' knowledge, the
resulting worst-case convergence rates are faster than any other in the
literature, including those of FISTA restart schemes. No assumption on the set
of minimizers is required and guarantees are also given in the non-optimal
case, i.e. when the condition number is not exactly known. This analysis
follows the study of the corresponding continuous-time dynamical system (Heavy
Ball with friction system), for which new convergence results of the trajectory
are shown.
