Unconstrained steepest descent method for multicriteria optimization on Riemmanian manifolds

TL;DR

This paper introduces a steepest descent method with Armijo's rule for multicriteria optimization on Riemannian manifolds, guaranteeing convergence to Pareto critical points under mild assumptions.

Contribution

It extends steepest descent methods to the Riemannian setting for multicriteria problems, providing convergence guarantees and conditions for Pareto optimality.

Findings

Sequence well-definedness is guaranteed.

Accumulation points satisfy first-order necessary conditions.

Full convergence to Pareto critical points under additional assumptions.

Abstract

In this paper we present a steepest descent method with Armijo's rule for multicriteria optimization in the Riemannian context. The well definedness of the sequence generated by the method is guaranteed. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if they exist) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasi-convexity of the multicriteria function and non-negative curvature of the Riemannian manifold, we prove full convergence of the sequence to a Pareto critical.