The presence of sharp minima in nondifferentiable optimization models has been exploited, in the last decades, in the benefit of various subgradient or proximal methods. One of the long-lasting general proximal schemes of choice used to minimize nonsmooth functions is the Proximal Point Algorithm (PPA). Regarding the basic PPA, several well-known works proved finite convergence towards weak sharp minima, when supposedly each iteration is computed exactly. However, in this letter we show finite convergence of a common Inexact version of PPA (IPPA), under sufficiently low but persistent perturbations of the proximal operator. Moreover, when a simple Subgradient Method is recurrently called as an inner routine for computing each IPPA iterate, a suboptimal minimizer of the original problem lying at $ϵ$ distance from the optimal set is obtained after a total $\mathcal{O}\left(log(1/ϵ)\right)$ subgradient evaluations. Our preliminary numerical tests show improvements over existing restartation versions of Subgradient Method.

在过去的几十年中，由于各种次梯度或近似方法的好处，已经利用了不可微优化模型中存在的尖锐最小值。用于最小化非光滑函数的一种持久的一般近端方案选择是近点算法 (PPA)。关于基本 PPA，一些著名的工作证明了向弱尖锐最小值的有限收敛，假设每次迭代都是精确计算的。然而，在这封信中，我们展示了 PPA 的常见不精确版本（IPPA）的有限收敛，在近端算子的足够低但持续的扰动下。此外，当一个简单的次梯度方法被反复调用作为计算每个 IPPA 迭代的内部例程时，原始问题的次优最小化器位于$\epsilon$与最优集的距离是在总$\mathcal{○}\left(日志(1/\epsilon )\right)$次梯度评价。我们的初步数值测试显示了对 Subgradient Method 的现有重新启动版本的改进。