Multiobjective Accelerated Gradient-like Flow with Asymptotic Vanishing Normalized Gradient

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 $O(1\mathrm{/}{t}^2)$ and $O({\ln }^{2}t\mathrm{/}{t}^2)$ 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 $O({\ln }^{2}k\mathrm{/}{k}^2)$. 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.