This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method. 

　　Publication: 

　　Mathematical Programming, pp: 1-61, 2022, DOI:10.1007/s10107-022-01898-1. 

　　Author: 

　　Yuhao Zhou 

　　Department of Computer Science and Technology, Tsinghua University, Beijing, China 

　　Chenglong Bao 

　　Yau Mathematical Sciences Center, Tsinghua University, China and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, China 

　　Chao Ding 

　　Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China 

　　Email: dingchao@amss.ac.cn 

　　Jun Zhu 

　　Department of Computer Science and Technology, Tsinghua University, Beijing, China