In this paper we study the convergence of an Inertial Forward–Backward algorithm, with a particular choice of an over-relaxation term. In particular we show that for a sequence of over-relaxation parameters, that do not satisfy Nesterov’s rule, one can still expect some relatively fast convergence properties for the objective function. In addition we complement this work by studying the convergence of the algorithm in the case where the proximal operator is inexactly computed with the presence of some errors and we give sufficient conditions over these errors in order to obtain some convergence properties for the objective function.

中文翻译：

---

惯性Forward-Backward算法超越Nesterov规则的收敛率

在本文中，我们研究了惯性前向后向算法的收敛性，并特别选择了过松弛项。我们特别表明，对于一系列不满足 Nesterov 规则的过松弛参数，我们仍然可以期望目标函数具有一些相对快速的收敛特性。此外，我们通过研究算法在存在一些错误的情况下不精确计算近端算子的情况下的收敛性来补充这项工作，并且我们给出了这些错误的充分条件以获得目标函数的一些收敛特性。