Strong variational sufficiency of nonsmooth optimization problems on Riemannian manifolds

The Riemannian augmented Lagrangian method (RALM) is proposed to solve the nonsmooth optimization problems on Riemannian manifolds. However, the local convergence rate of this algorithm still remains unknown without imposing any constraint qualifications. In this paper, we introduce the manifold variational sufficient condition and show that its strong version is equivalent to the manifold strong second-order sufficient condition (M-SSOSC) in some cases. More importantly, we formulate a local dual problem based on this condition, consequently establishing the R-linear convergence rate of RALM. Furthermore, the validity of the semismooth Newton method for solving the RALM subproblem is demonstrated under the M-SSOSC.