Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport

TL;DR

This paper develops nonlinear conjugate gradient algorithms for vector optimization on Riemannian manifolds, extending Wolfe and Zoutendjik conditions, analyzing convergence, and demonstrating practical effectiveness through numerical experiments.

Contribution

It introduces Riemannian vector conjugate gradient methods with extended Wolfe conditions and convergence analysis, including new parameter choices.

Findings

Algorithms converge to Pareto stationary points.

Existence of Wolfe-satisfying step sizes is established.

Numerical experiments confirm practical effectiveness.

Abstract

In this paper, we propose nonlinear conjugate gradient methods for vector optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik conditions are extended for Riemannian manifolds. Specifically, we establish the existence of intervals of step sizes that satisfy the Wolfe conditions. The convergence analysis covers the vector extensions of the Fletcher--Reeves, conjugate descent, and Dai--Yuan parameters. Under some assumptions, we prove that the sequence obtained by the algorithm can converge to a Pareto stationary point. Moreover, we also discuss several other choices of the parameter. Numerical experiments illustrating the practical behavior of the methods are presented.