In a Hilbert space setting, we consider new continuous gradient-like dynamical systems for constrained multiobjective optimization. This type of dynamics was first investigated by Cl. Henry, and B. Cornet, as a model of allocation of resources in economics. Based on the Yosida regularization of the discontinuous part of the vector field which governs the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and in the quasi-convex case, convergence of the trajectories to Pareto critical points. We give an interpretation of the dynamic in terms of Pareto equilibration for cooperative games. By time discretization, we make a link to recent studies of Svaiter et al. on the algorithm of steepest descent for multiobjective optimization.
Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier – UMR CNRS 5149 34095 Montpellier, France April 26, 2013