Local Convergence of the Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds

TL;DR

This paper analyzes the local convergence of the proximal point method for a specific class of nonconvex functions on Hadamard manifolds, establishing conditions for well-definedness and convergence to minimizers.

Contribution

It provides the first local convergence analysis of the proximal point method for nonconvex functions on Hadamard manifolds, including conditions for convergence to minimizers.

Findings

Sequence generated by the method is well-defined.

Cluster points satisfy necessary optimality conditions.

Convergence to a minimizer is established under additional assumptions.

Abstract

Local convergence analysis of the proximal point method for special class of nonconvex function on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.