Fast Convergence of Multiobjective Inertial Gradient Systems with Time Scaling

Abstract

In multiobjective optimization, inertial gradient systems accelerate convergence toward weakly Pareto optimal solutions. To achieve even faster convergence, we introduce a multiobjective inertial gradient system with time scaling (MITS), formulated as a second-order differential equation comprising an inertial term, asymptotically vanishing damping, and a time-scaled gradient term. We first establish the existence of solution trajectories for MITS. Through Lyapunov analysis, we show that with suitable parameters, the trajectory attains a convergence rate of $O(1/t^{2}\beta(t))$ with respect to a merit function, where $\beta(t)$ is a time-scaling function. Specifically, choosing $\beta(t)=t^{p}$ for $0\leq p<\alpha-3$ yields the rate $O(1/t^{2+p})$, enabling arbitrarily fast sublinear convergence by tuning $p$. We also prove that the trajectory converges to a weakly Pareto optimal solution. Furthermore, an implicit discretization of MITS leads to a multiobjective inertial proximal point method (MIPP), whose iterates share the $O(1/k^{2}\beta_{k})$ rate and converge to a weakly Pareto optimum under appropriate conditions. Numerical experiments support the theoretical findings.