Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds

Abstract

This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ:TM→ℝ on the tangent bundle TM, and at the kth iteration, using the restricted function Φ|TxkM, where TxkM is the tangent space at xk, a local model function Qk that carries both first- and second-order information for the locally Lipschitz objective function f:M→ℝ on a Riemannian manifold M, is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian ε-subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.