Many contemporary applications in signal processing and machine learning give rise to structured nonconvex nonsmooth optimization problems that can often be tackled by simple iterative methods quite effectively. One of the keys to understanding such a phenomenon-and, in fact, a very difficult conundrum even for experts-lies in the study of "stationary points" of the problem in question. Unlike smooth optimization, for which the definition of a stationary point is rather standard, there are myriad definitions of stationarity in nonsmooth optimization. In this article, we provide an introduction to different stationarity concepts for several important classes of nonconvex nonsmooth functions, discuss the geometric interpretations of these concepts, and further clarify their relationships. We then demonstrate the relevance of these constructions in some representative applications and indicate how they could affect the performance of iterative methods for addressing these applications.

中文翻译：

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理解非光滑优化中的平稳性概念：非光滑函数次微分的各种构造导览

信号处理和机器学习中的许多当代应用产生了结构化的非凸非光滑优化问题，这些问题通常可以通过简单的迭代方法非常有效地解决。理解这种现象的关键之一——事实上，即使对专家来说也是一个非常困难的难题——在于研究问题的“静止点”。与平稳优化不同，平稳点的定义相当标准，非平滑优化中有无数的平稳性定义。在本文中，我们介绍了几类重要的非凸非光滑函数的不同平稳性概念，讨论了这些概念的几何解释，并进一步阐明了它们之间的关系。