Point Convergence Analysis of the Accelerated Gradient Method for Multiobjective Optimization: Continuous and Discrete

TL;DR

This paper analyzes the convergence of accelerated gradient methods for multiobjective optimization in both continuous and discrete settings, establishing convergence to weakly Pareto optimal solutions.

Contribution

It proves convergence of the MAVD system with $ ext{α} = 3$ and introduces MAG-GM, a new discrete method that also converges to weakly Pareto optimal solutions.

Findings

Trajectory of MAVD with α=3 converges to a weakly Pareto optimal solution.

Proposed MAG-GM method's sequence converges to a weakly Pareto optimal solution.

Addresses open problems in multiobjective accelerated gradient methods.

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 $\alpha = 3$, 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 $\alpha = 3$, 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.