Line search algorithms for locally Lipschitz functions on Riemannian manifolds

Abstract

This paper presents line search algorithms for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we generalize the so-called Wolfe conditions for nonsmooth functions on Riemannian manifolds. Using <em>ε</em>-subgradient-oriented descent directions and the Wolfe conditions, we propose a nonsmooth Riemannian line search algorithm and establish the convergence of our algorithm to a stationary point. Moreover, we extend the classical BFGS algorithm to nonsmooth functions on Riemannian manifolds. Numerical experiments illustrate the effectiveness and efficiency of the proposed algorithm.