Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds

Abstract

A proximal point method for nonsmooth multiobjective optimization in the Riemannian context is proposed, and an optimality condition for multiobjective problems is introduced. This allowed replacing the classic approach, via “scalarization,” by a purely vectorial and considering the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Moreover, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.