\noindent Several decades ago the Proximal Point Algorithm (PPA) stated 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 inexactness. However, without taking into account the concrete computational effort paid for computing each PPA iteration, any iteration complexity remains abstract and purely informative. Therefore, using an inner (proximal) gradient/subgradient method subroutine that computes inexact PPA iteration, we secondly show novel computational complexity bounds on a restarted 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 framework.

翻译：数十年前Proximal Point Alogorithm (PPA) 指出, Proximal Point Algorithm (PPA) 旨在为抽象操作者理论和数字优化社区获得长期吸引。 即使在现代应用中, 研究人员仍然使用最接近最小化理论来设计可缩放的算法, 以克服非移动性。 值得注意的工作有\ cite{Fer: 91, Ber: 82 control, Ber: 89parallel, Tom: 11} 在 PPPA 的趋同行为与目标功能的常规性之间建立起了密切的关系。 特别是, 我们在这个手稿中, 我们获得了精确和不精确的离线性 PPPPA 的不匹配复杂性复杂性。 使用精确的粘结 IMO (1/\ eepslational) grealization a procial developation a commocial decremodia decal demogration ex decal decal demogration.