Point Convergence Analysis of the Accelerated Gradient Method for Multiobjective Optimization: Continuous and Discrete
Yingdong Yin
TL;DR
The paper proves point convergence for a continuous multiobjective inertial gradient system at the critical damping value $\alpha=3$ and for a discrete multiobjective accelerated gradient method with generalized momentum. Using Lyapunov and energy arguments, it derives a sublinear rate $u_0(x(t))=O\left(t^{-2\alpha/3}\right)$ for the continuous dynamics and, in the discrete setting, achieves $u_0(x_k)=O\left(1/k^2\right)$ along with bounded step-norm growth. Under standard convexity and level-set assumptions, both the continuous trajectory $x(t)$ and the discrete iterates $\{x_k\}$ converge to a weak Pareto optimal point $x^*$, with $u_0(x^*)=0$, providing rigorous convergence guarantees for accelerated multiobjective methods. The results extend Nesterov-type acceleration to multiobjective optimization, offering theoretically grounded design principles for continuous and discrete accelerated algorithms with Pareto-optimal guarantees.
Abstract
This paper investigates the point convergence of accelerated gradient methods for multiobjective optimization, in both continuous and discrete settings. We address the open problems of whether the solution trajectory of the multiobjective inertial gradient-like dynamical system (MAVD) with asymptotic vanishing damping converges when $α= 3$, and whether the sequence generated by the multiobjective Nesterov accelerated method (MAG) converges to a weakly Pareto optimal solution. For the continuous system (MAVD) with $α= 3$, we prove that the trajectory $x(t)$ converges to a weakly Pareto optimal solution. For the discrete case, we propose a multiobjective accelerated gradient method with a generalized momentum factor (MAG-GM), and prove that the generated sequence $\{x_k\}$ converges to a weakly Pareto optimal solution.