Fast Convergence of Multiobjective Inertial Gradient Systems with Time Scaling

TL;DR

This work introduces a fast-converging multiobjective inertial gradient framework with time scaling (MITS) and its implicit-discretized counterpart, the Multiobjective Inertial Proximal Point Method (MIPP). It establishes existence of solution trajectories and proves arbitrarily fast sublinear convergence of the merit function $u_0(x)$, with rates $O\left(1/t^{2}\beta(t)\right)$ and, for $\beta(t)=t^{p}$, $O\left(1/t^{2+p}\right)$ for $0\le p<\alpha-3$, alongside convergence to a weakly Pareto optimum. The discrete method MIPP inherits a rate of $O\left(1/k^{2}\beta_k\right)$ and converges under suitable assumptions, supported by a discrete Lyapunov analysis. Numerical experiments on smooth and nonsmooth problems corroborate the theoretical rates and demonstrate accelerated convergence and Pareto front behavior. These results extend inertial dynamics and proximal schemes to the multiobjective setting with provable fast convergence.

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 $O(1/t^{2}β(t))$ with respect to a merit function, where $β(t)$ is a time-scaling function. Specifically, choosing $β(t)=t^{p}$ for $0\leq p<α-3$ 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 $O(1/k^{2}β_{k})$ rate and converge to a weakly Pareto optimum under appropriate conditions. Numerical experiments support the theoretical findings.

Figures (3)

• Figure 1: Smooth test problems.
• Figure 2: Non-smooth test problems.
• Figure 3: Performance evaluation

Theorems & Definitions (52)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Theorem 2.1: tanabe2024new
• Theorem 2.2: sonntag2024fastSIAMonOptimization
• Definition 3.1
• Theorem 3.1
• Lemma 3.1
• proof
• Example 1