Understanding Notions of Stationarity in Non-Smooth Optimization

Many contemporary applications in signal processing and machine learning give rise to structured non-convex non-smooth 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, one of the very difficult conundrums even for experts---lie 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 is a myriad of definitions of stationarity in non-smooth optimization. In this article, we give an introduction to different stationarity concepts for several important classes of non-convex non-smooth functions and discuss the geometric interpretations and further clarify the relationship among these different concepts. We then demonstrate the relevance of these constructions in some representative applications and how they could affect the performance of iterative methods for tackling these applications.