Over the past decades, operator splitting methods have become ubiquitous for non-smooth optimization owing to their simplicity and efficiency. In this paper, we consider the Forward–Douglas–Rachford splitting method and study both global and local convergence rates of this method. For the global rate, we establish a sublinear convergence rate in terms of a Bregman divergence suitably designed for the objective function. Moreover, when specializing to the Forward–Backward splitting, we prove a stronger convergence rate result for the objective function value. Then locally, based on the assumption that the non-smooth part of the optimization problem is partly smooth, we establish local linear convergence of the method. More precisely, we show that the sequence generated by Forward–Douglas–Rachford first (i) identifies a smooth manifold in a finite number of iteration and then (ii) enters a local linear convergence regime, which is for instance characterized in terms of the structure of the underlying active smooth manifold. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from applicative fields including, for instance, signal/image processing, inverse problems and machine learning.

中文翻译：

---

Forward-Douglas-Rachford 分裂方法的收敛率

在过去的几十年中，算子分裂方法由于其简单性和效率而在非平滑优化中无处不在。在本文中，我们考虑了 Forward-Douglas-Rachford 分裂方法并研究了该方法的全局和局部收敛速度。对于全局速率，我们根据为目标函数适当设计的 Bregman 散度建立了一个次线性收敛速率。此外，当专门用于前向后向分裂时，我们证明了目标函数值的收敛速度更强。然后在局部，基于优化问题的非光滑部分是部分光滑的假设，我们建立了该方法的局部线性收敛。更确切地说，我们表明，由 Forward-Douglas-Rachford 生成的序列首先 (i) 在有限次数的迭代中识别出一个平滑的流形，然后 (ii) 进入局部线性收敛状态，例如，其特征在于底层主动光滑流形。为了举例说明所得结果的有用性，我们考虑了一些来自应用领域的具体数值实验，包括例如信号/图像处理、逆问题和机器学习。