Fast Convergence of Multiobjective Inertial Gradient Systems with Time Scaling

TL;DR

This paper introduces a novel multiobjective inertial gradient system with time scaling (MITS) that achieves faster convergence rates toward weakly Pareto optimal solutions, supported by theoretical analysis and numerical experiments.

Contribution

It proposes a new second-order differential system with time scaling for multiobjective optimization, achieving arbitrarily fast sublinear convergence rates.

Findings

Convergence rate of $O(1/t^{2}eta(t))$ established for MITS.

Choosing $eta(t)=t^{p}$ yields adjustable convergence rates up to $O(1/t^{2+p})$.

Numerical experiments confirm theoretical convergence rates and effectiveness.

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.