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In this paper, we introduce the notion of generalized $\epsilon$-stationarityfor a class of nonconvex and nonsmooth composite minimization problems oncompact Riemannian submanifold embedded in Euclidean space. To find ageneralized $\epsilon$-stationarity point, we develop a family of Riemanniangradient-type methods based on the Moreau envelope technique with a decreasingsequence of smoothing parameters, namely Riemannian smoothing gradient andRiemannian smoothing stochastic gradient methods. We prove that the Riemanniansmoothing gradient method has the iteration complexity of$\mathcal{O}(\epsilon^{-3})$ for driving a generalized $\epsilon$-stationarypoint. To our knowledge, this is the best-known iteration complexity result forthe nonconvex and nonsmooth composite problem on manifolds. For the Riemanniansmoothing stochastic gradient method, one can achieve the iteration complexityof $\mathcal{O}(\epsilon^{-5})$ for driving a generalized $\epsilon$-stationarypoint. Numerical experiments are conducted to validate the superiority of ouralgorithms.

applied mathematics and optimization/applied mathematics and optimization

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