Convergence analysis of inexact descent algorithm for multiobjective optimizations on Riemannian manifolds without curvature constraints

TL;DR

This paper analyzes the convergence of an inexact descent algorithm for multiobjective optimization on Riemannian manifolds, establishing new local and global convergence results under various convexity and Kurdyka-{\L}ojasiewicz-like conditions, with improvements over existing Euclidean space results.

Contribution

It provides new convergence analysis for inexact descent algorithms on Riemannian manifolds without curvature constraints, including under Kurdyka-{\L}ojasiewicz-like conditions, extending and sharpening previous results.

Findings

Established local and global convergence under convexity assumptions.

Proved linear convergence under Kurdyka-{\L}ojasiewicz-like conditions.

Extended results for algorithms using Armijo rule.

Abstract

We study the convergence issue for inexact descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. On the other hand, without the assumption of the local convexity/quasi-convexity, but under a Kurdyka-{\L}ojasiewicz-like condition, local/global linear convergence results are presented, which seem new even in Euclidean spaces setting and improve sharply the corresponding results in [24] in the case when the multiobjective optimization is reduced to the scalar case. Finally, for the special case when the inexact descent algorithm employing Armijo rule, our results improve sharply/extend the corresponding ones in [3,2,38].