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Understanding Notions of Stationarity in Non-Smooth Optimization

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Document pages: 13 pages

Abstract: Many contemporary applications in signal processing and machine learning giverise to structured non-convex non-smooth optimization problems that can oftenbe tackled by simple iterative methods quite effectively. One of the keys tounderstanding such a phenomenon---and, in fact, one of the very difficultconundrums even for experts---lie in the study of "stationary points " of theproblem in question. Unlike smooth optimization, for which the definition of astationary point is rather standard, there is a myriad of definitions ofstationarity in non-smooth optimization. In this article, we give anintroduction to different stationarity concepts for several important classesof non-convex non-smooth functions and discuss the geometric interpretationsand further clarify the relationship among these different concepts. We thendemonstrate the relevance of these constructions in some representativeapplications and how they could affect the performance of iterative methods fortackling these applications.

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