Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

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In this paper, we introduce the notion of generalized $\epsilon$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space.

• To find a generalized $\epsilon$-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods.
• We prove that the Riemannian smoothing gradient method has the iteration complexity of $\mathcal{O}(\epsilon^{-3})$ for driving a generalized $\epsilon$-stationary point.
• To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds.