Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization

TL;DR

This paper introduces a second-order dynamical system with vanishing damping and Tikhonov regularization for multiobjective convex optimization, proving convergence properties and validating results through numerical experiments.

Contribution

It extends convex single-objective optimization results to multiobjective settings with convergence guarantees and numerical validation.

Findings

Fast convergence of function values demonstrated

Weak and strong convergence towards Pareto optimal points proven

Numerical experiments support theoretical results

Abstract

In this paper, we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions towards a weak Pareto optimal point. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. Furthermore, we conduct numerical experiments to validate the theoretical results and investigate the qualitative behavior of the dynamical system. This work extends results from convex single objective optimization into the multiobjective setting. The results presented in this paper lay the groundwork for the development of fast gradient and proximal point methods in multiobjective optimization, offering strong convergence guarantees.