SIAM Review, Volume 65, Issue 2, Page 375-435, May 2023.
Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we easily derive new variants of the algorithms and revisit existing convergence results, extending the parameter ranges in several cases. In particular, we emphasize that when the smooth term in the objective function is quadratic, e.g., for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial for the convergence speed in practice.

中文翻译：

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用于凸优化的近端分裂算法：最新进展之旅，新曲折

SIAM Review，第 65 卷，第 2 期，第 375-435 页，2023 年 5 月。
凸非光滑优化问题，其解决方案存在于非常高的维度空间中，已经变得无处不在。为了解决它们，被称为近端分裂算法的一阶算法特别合适：它们由简单的操作组成，分别处理目标函数中的项。在此概述中，我们揭开了最近选择的近端分裂算法的神秘面纱：我们将它们呈现在一个统一的框架中，该框架包括将分裂方法应用于原始对偶产品空间中的单调包含，以及精心选择的指标。在此过程中，我们很容易推导出算法的新变体并重新审视现有的收敛结果，在几种情况下扩展了参数范围。特别地，我们强调当目标函数中的平滑项是二次项时，例如，对于最小二乘问题，使用比以前已知的更大的松弛参数值可以保证收敛。这种较大的值通常有利于实践中的收敛速度。