A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds
This work addresses optimization challenges in non-smooth settings on manifolds, which is incremental as it extends smoothing techniques to Riemannian contexts.
The paper tackles the problem of minimizing nonconvex and non-Lipschitz functions on submanifolds by proposing a Riemannian smoothing steepest descent method, proving that accumulation points are stationary points for the smoothing function and, under certain conditions, Riemannian limiting stationary points for the original problem, with numerical experiments demonstrating efficiency.
In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. 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. Under the Riemannian gradient sub-consistency condition, we also prove that any accumulation point is a Riemannian limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the efficiency of the proposed method.