Understanding Notions of Stationarity in Nonsmooth Optimization: A Guided Tour of Various Constructions of Subdifferential for Nonsmooth Functions

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.