摘要：In this talk, we shall explain why nonsmooth analysis plays a critical role in solving large scale sparse optimization problems.  We start with introducing some basic concepts such as Rademacher’theorem and the Moreau-Yosida regularization for convex functions. Then we talk about semismooth analysis including inverse and implicit functions theorems to demonstrate why nonsmooth systems are indispensable for solving constrained optimization problems and why smooth systems inevitably lead to singularity. Finally, we shall illustrate how we can employ nonsmooth analysis to design highly efficient sparse nonsmooth secant and Newton methods for solving several important machine learning models including convex clustering, lasso, and exclusive lasso of sparse solutions.