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

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].