A semismooth Newton based augmented Lagrangian method for nonsmooth optimization on matrix manifolds

• 成果类型: Article
• 署名作者: Zhou, Yuhao; Bao, Chenglong; Ding, Chao; Zhu, Jun
• 署名单位: Tsinghua University; Tsinghua University; Yanqi Lake Beijing Institute of Mathematical Sciences & Applications; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01898-1
• 发表日期: 2023
• 页码: 1-61
• 关键词: alternating minimization; optimality conditions; nonconvex; algorithm; MODEL

摘要

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.