# Iteration-complexity and asymptotic analysis of steepest descent method for multiobjective optimization on Riemannian manifolds

## TL;DR

This paper analyzes the convergence and complexity of the steepest descent method for multiobjective optimization on Riemannian manifolds, providing theoretical bounds and numerical validation for various stepsize strategies.

## Contribution

It introduces iteration-complexity bounds and asymptotic analysis for the steepest descent method on Riemannian manifolds with multiple stepsize rules, a novel extension in this context.

## Key findings

- The method converges under different stepsize strategies.
- Complexity bounds are established for each stepsize rule.
- Numerical experiments confirm theoretical results.

## Abstract

The steepest descent method for multiobjective optimization on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The aim of the paper is twofold. Firstly, an asymptotic analysis of the method is presented with three different finite procedures for determining the stepsize, namely, Lipschitz stepsize, adaptive stepsize and Armijo-type stepsize. The second aim is to present, by assuming that the Jacobian of the objective function is componentwise Lipschitz continuous, iteration-complexity bounds for the method with these three stepsizes strategies. In addition, some examples are presented to emphasize the importance of working in this new context. Numerical experiments are provided to illustrate the effectiveness of the method in this new setting and certify the obtained theoretical results.