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

Authors :
Zhang, Chao
Chen, Xiaojun
Ma, Shiqian
Source :
Mathematics of Operations Research; Aug2024, Vol. 49 Issue 3, p1710-1733, 24p
Publication Year :
2024

Abstract

In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded submanifolds of Rn. 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 ℓp (0<p<1) optimization over Riemannian ℓ1 optimization for finding sparse solutions and the effectiveness of the proposed method. Funding: C. Zhang was supported in part by the National Natural Science Foundation of China [Grant 12171027] and the Natural Science Foundation of Beijing [Grant 1202021]. X. Chen was supported in part by the Hong Kong Research Council [Grant PolyU15300219]. S. Ma was supported in part by the National Science Foundation [Grants DMS-2243650 and CCF-2308597], the UC Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program, and a startup fund from Rice University. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
0364765X
Volume :
49
Issue :
3
Database :
Complementary Index
Journal :
Mathematics of Operations Research
Publication Type :
Academic Journal
Accession number :
179020562
Full Text :
https://doi.org/10.1287/moor.2022.0286