Multiobjective Accelerated Gradient-like Flow with Asymptotic Vanishing Normalized Gradient

TL;DR

This paper introduces a novel multiobjective gradient flow with asymptotic vanishing normalized gradient, achieving faster convergence rates and convergence to weak Pareto solutions, supported by theoretical analysis and numerical validation.

Contribution

It generalizes a known gradient flow to multiobjective optimization, providing new convergence rates and a discretized algorithm with practical advantages.

Findings

Achieves $O(1/t^2)$ and $O( ext{ln}^2 t / t^2)$ convergence rates.

Proves convergence to weak Pareto solutions under convexity.

Numerical experiments show superior convergence speed.

Abstract

This paper generalizes the dynamical system proposed by Wang et al. [Siam. J. Sci. Comput., 2021] to multiobjective optimization by investigating a multiobjective accelerated gradient-like flow with asymptotically vanishing normalized gradient. Using Lyapunov analysis, we obtain convergence rates of $O(1/t^2)$ and $O(\ln^2 t / t^2)$ for the trajectory solution under two distinct parameter selections. Under certain assumptions, we further prove that the trajectory solution of this gradient flow converges to a weak Pareto solution for convex multiobjective optimization problems. Through corresponding discretization, we derive a new class of multiobjective gradient methods achieving a convergence rate of $O(\ln^2 k / k^2)$. Additionally, numerical experiments validate the theoretical results, demonstrating that this gradient flow outperforms other existing dynamical systems in the literature regarding convergence speed, and our algorithm exhibits corresponding advantages.