Riemannian Trust Region Methods for SC1 Minimization

Manifold optimization has recently gained significant attention due to its wide range of appli-cations in various areas. This paper introduces the first Riemannian trust region method forminimizing an SC1 function, which is a differentiable function that has asemismoothgradientvector field, on manifolds with convergence guarantee. We provide proof of both global andlocal convergence results, along with demonstrating the local superlinear convergence rateof our proposed method. As an application and to demonstrate our motivation, we utilizeour trust region method as a subproblem solver within an augmented Lagrangian methodfor minimizing nonsmooth nonconvex functions over manifolds. This represents the firstapproach that fully explores the second-order information of the subproblem in the contextof augmented Lagrangian methods on manifolds. Numerical experiments confirm that ourmethod outperforms existing methods