This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establishing their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.  

　　Publication:  

　　IMA Journal of Numerical Analysis, drad098, 21 December 2023  

　　https://doi.org/10.1093/imanum/drad098  

　　Author:  

　　Xiaoyin Hu  

　　School of Computer and Computing Science, Hangzhou City University, Hangzhou, 310015, China  

　　Nachuan Xiao  

　　The Institute of Operations Research and Analytics, National University of Singapore, Singapore, 117602  

　　Xin Liu  

　　State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, 100190, China  

　　Email: liuxin@lsec.cc.ac.cn  

　　Kim-Chuan Toh  

　　Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, Singapore, 119076