Riemannian Trust Region Methods for SC$^1$ Minimization

TL;DR

This paper introduces the first Riemannian trust region method for SC^1 functions on manifolds, providing convergence guarantees and demonstrating superior performance in nonsmooth, nonconvex optimization problems.

Contribution

It develops a novel trust region algorithm for SC^1 functions on manifolds with proven convergence and applies it within an augmented Lagrangian framework for nonsmooth optimization.

Findings

Method achieves global and local convergence.

Demonstrates local superlinear convergence rate.

Outperforms existing methods in numerical experiments.

Abstract

Manifold optimization has recently gained significant attention due to its wide range of applications in various areas. This paper introduces the first Riemannian trust region method for minimizing an SC$^1$ function, which is a differentiable function that has a semismooth gradient vector field, on manifolds with convergence guarantee. We provide proof of both global and local convergence results, along with demonstrating the local superlinear convergence rate of our proposed method. As an application and to demonstrate our motivation, we utilize our trust region method as a subproblem solver within an augmented Lagrangian method for minimizing nonsmooth nonconvex functions over manifolds. This represents the first approach that fully explores the second-order information of the subproblem in the context of augmented Lagrangian methods on manifolds. Numerical experiments confirm that our method outperforms existing methods.