A Riemannian Smoothing Steepest Descent Method for Non-Lipschitz Optimization on Embedded Submanifolds of Rn

• 成果类型: Article
• 署名作者: Zhang, Chao; Chen, Xiaojun; Ma, Shiqian
• 署名单位: Beijing Jiaotong University; Hong Kong Polytechnic University; Rice University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.0286
• 发表日期: 2024
• 关键词: Augmented Lagrangian method; worst-case complexity; nonsmooth optimization; optimality conditions; image-restoration; gradient-method; line-search; nonconvex; algorithms; MODEL

摘要

In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded sub manifolds of R-n. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of Rn. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. We also prove that any accumulation point of the sequence generated by our method that satisfies the Riemannian gradient subconsistency is a limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the advantages of Riemannian europ (0 < p < 1) optimization over Riemannian euro1 optimization for finding sparse solutions and the effectiveness of the proposed method.