A Riemannian Smoothing Steepest Descent Method for Non-Lipschitz Optimization on Embedded Submanifolds of ${\mathbb{R}}^n$

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 ${\mathbb{R}}^n$. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of ${\mathbb{R}}^n$. 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 ${\mathcal{\ell }}_p$$(0<p<1)$ optimization over Riemannian ${\mathcal{\ell }}_1$ 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.