Convergence of nonsmooth descent methods via Kurdyka–Lojasiewicz inequality on Riemannian manifolds

Abstract

We develop a subgradient-oriented descent method in nonsmooth optimization on Riemannian manifolds and prove convergence of the method in the sense of subsequences for nonsmooth functions whose standard models are strict. Moreover, we present a nonsmooth version of the Kurdyka-Lojasiewicz inequality and show that a locally Lipschitz <em>C</em>-function defined on an analytic manifold satisfies this inequality. Finally, we prove that if the objective function satisfies the Kurdyka-Lojasiewicz inequality and its standard model is strict, then the sequence of iterates of the subgradient-oriented descent algorithm converges to a singular critical point.