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.

翻译：几十年前，邻近点算法（PPA）开始对抽象算子理论和数值优化领域产生持久的吸引力。即使在现代应用中，研究者仍利用邻近最小化理论来设计可扩展算法以克服非光滑性。如\cite{Fer:91,Ber:82constrained,Ber:89parallel,Tom:11}等杰出工作建立了PPA收敛行为与目标函数正则性之间的紧密联系。本文推导了在$\gamma-$Hölder增长条件下最小化凸函数时，精确与非精确PPA的非渐近迭代复杂度：$\BigO{\log(1/\epsilon)}$（当$\gamma \in [1,2]$）和$\BigO{1/\epsilon^{\gamma - 2}}$（当$\gamma > 2$）。特别地，我们恢复了PPA的经典结论：尖锐极小值情形下的有限收敛性及二次增长情形下的线性收敛性，即使在存在确定性噪声时亦然。此外，当采用简单的邻近次梯度方法作为计算每个IPPA迭代的内层例行程序时，我们获得了重启非精确PPA的新计算复杂度边界。数值实验表明，该方法较现有的重启版次梯度方法有所改进。