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We consider a class of nonsmooth and nonconvex optimization problems over theStiefel manifold where the objective function is the summation of a nonconvexsmooth function and a nonsmooth Lipschitz continuous convex function composedwith an linear mapping. We propose three numerical algorithms for solving thisproblem, by combining smoothing methods and some existing algorithms for smoothoptimization over the Stiefel manifold. In particular, we approximate theaforementioned nonsmooth convex function by its Moreau envelope in oursmoothing methods, and prove that the Moreau envelope has many favorableproperties. Thanks to this and the scheme for updating the smoothing parameter,we show that any accumulation point of the solution sequence generated by theproposed algorithms is a stationary point of the original optimization problem.Numerical experiments on building graph Fourier basis are conducted todemonstrate the efficiency of the proposed algorithms.

Smoothing algorithms for nonsmooth and nonconvex minimization over the stiefel manifold