Many applications arising from machine learning, statistics and image processing can be formulated as a convex minimization model with separable structures both in objective function and constraints. The Peaceman–Rachford splitting method is very efficient for solving these problems, but it is not convergent in the absence of some restrictive assumptions. In this paper, we propose a linearized Peaceman–Rachford splitting method by linearizing one subproblem. We analyze its convergence by proving the global convergence and establishing its worst-case convergence rate in the ergodic sense. Some randomly generated stable principal component pursuit problems are tested to illustrate the efficiency of the new algorithm.

中文翻译：

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一种用于结构化凸优化的线性化 Peaceman-Rachford 分裂方法，适用于稳定主成分追踪

机器学习、统计学和图像处理产生的许多应用都可以表述为一个凸最小化模型，在目标函数和约束条件上都具有可分离的结构。Peaceman-Rachford 分裂方法对于解决这些问题非常有效，但在没有一些限制性假设的情况下它是不收敛的。在本文中，我们通过线性化一个子问题，提出了一种线性化的 Peaceman-Rachford 分裂方法。我们通过证明全局收敛并建立遍历意义上的最坏情况收敛率来分析其收敛性。测试了一些随机生成的稳定主成分追踪问题，以说明新算法的效率。