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 behavior 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:几十年前, Proximal Point Alogorithm (PPA) 声明要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, 研究人员仍然使用准最小化理论来设计可缩放的算法, 以克服非移动性。 值得注意的作品为\ cite{Fer: 91, Ber: 82 constrept, Ber: 89parallel, Tom: 11} 在 PPPA 的趋同行为与目标功能的规律性之间建立了密切的关系 。 在这份手稿中, 我们得出准确和不精确的 PPPPA 的不简化复杂性, 在 $\ gamma- $ Holderian 增长中最小最小最小化最小化最小化最小化最小化最小化最小化的最小化最小化函数: $\ bigO log( 1/\ epsilonlation) $\\\\\\\\\\\\\\\\\\\ excial macal cloadal deal deal deal dealation exmodealation exmodeal decal dealation) a macal demode demotion ex fal develgistration coal deal deal deal deal deal devel ex ex ex ex code ex exportment ex ex 任何已知化的计算法的计算法的精确化数据, 。 。 任何已知化的计算法的精确化的计算法的精确化的精确化的精确化的精确化,,, 。 。 任何已知化的精确化的精确化的精确化的精确化的精确化的精确化的计算法的计算法的计算法的计算法的精确化, 。