A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

Understand

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.