  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.
