  Several decades ago the Proximal Point Algorithm (PPA) started to gain a
long-lasting attraction for both abstract operator theory and numerical
optimization communities. Even in modern applications, researchers still use
proximal minimization theory to design scalable algorithms that overcome
nonsmoothness. Remarkable works as
\cite{Fer:91,Ber:82constrained,Ber:89parallel,Tom:11} established tight
relations between the convergence behaviour of PPA and the regularity of the
objective function. In this manuscript we derive nonasymptotic iteration
complexity of exact and inexact PPA to minimize convex functions under
$\gamma-$Holderian growth: $\BigO{\log(1/\epsilon)}$ (for $\gamma \in [1,2]$)
and $\BigO{1/\epsilon^{\gamma - 2}}$ (for $\gamma > 2$). In particular, we
recover well-known results on PPA: finite convergence for sharp minima and
linear convergence for quadratic growth, even under presence of deterministic
noise. Moreover, when a simple Proximal Subgradient Method is recurrently
called as an inner routine for computing each IPPA iterate, novel computational
complexity bounds are obtained for Restarting Inexact PPA. Our numerical tests
show improvements over existing restarting versions of the Subgradient Method.
