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 $\Phi :TM\rightarrow \mathbb {R}$ on the tangent bundle $TM$ , and at the $k$ th iteration, using the restricted function $\Phi |_{T_{x_k}M}$ , where $T_{x_k}M$ is the tangent space at $x_k$ , a local model function $Q_k$ that carries both first- and second-order information for the locally Lipschitz objective function $f:M\rightarrow \mathbb {R}$ 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 $\varepsilon $ -subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.