Inertial dynamics with vanishing Tikhonov regularization for multiobjective optimization

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.