Proximal Point Methods for Lipschitz Functions on Hadamard Manifolds: Scalar and Vectorial Cases

Abstract

We study the convergence of exact and inexact versions of the proximal point method with a generalized regularization function in Hadamard manifolds for solving scalar and vectorial optimization problems involving Lipschitz functions. We consider a local dominance property of the directional derivative of the objective function over the regularization term in order to obtain that cluster points of the sequence are stationary points. Under an additional assumption, we prove that every cluster point of the sequence is a minimizer in the scalar case and a weak efficient point in the vectorial case. Our results extend some of the existing ones in the literature about optimization on manifolds.