Computational complexity of Inexact Proximal Point Algorithm for Convex Optimization under Holderian Growth

Several decades ago the Proximal Point Algorithm (PPA) started to gain much attraction for both abstract operator theory and the numerical optimization communities. Even in modern applications, researchers still use proximal minimization theory to design scalable algorithms that overcome nonsmoothness in high dimensional models. Several remarkable references as \cite{Fer:91,Ber:82constrained,Ber:89parallel,Tom:11} analyzed the tight local relations between the convergence rate of PPA and the regularity of the objective function. However, without taking into account the concrete computational effort paid for computing each PPA iteration, any iteration complexity remains abstract and purely informative. In this manuscript we aim to evaluate the computational complexity of practical PPA in terms of (proximal) gradient/subgradient iterations, which might allow a fair positioning of the famous PPA numerical performance in the class of first order methods. First, we derive nonasymptotic iteration complexity estimates 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 exact PPA: finite convergence for sharp minima and linear convergence for quadratic growth, even under presence of inexactness. Second, assuming that an usual (proximal) gradient/subgradient method subroutine is employed to compute inexact PPA iteration, we show novel computational complexity bounds on a restarted variant of the inexact PPA, available when no information on the growth of the objective function is known. In the numerical experiments we confirm the practical performance and implementability of our schemes.