A Continuous Gradient-like Dynamical Approach to Pareto-Optimization in Hilbert Spaces

• Published in: Set-valued and variational analysis Vol. 22; no. 1; pp. 189 - 219
• Main Authors: Attouch, Hédy, Goudou, Xavier
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2014; Springer
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Optimization; Optimization and Control; 34E10; 37L05; Asymptotic behavior; Pareto optimization; 90C29; 91B06; Pareto critical; Cooperative games; Continuous gradient systems; 37L65; 90C31; 91A12; 91B55; 91A35; Multiobjective optimization; Steepest descent; 90B50
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-013-0245-4

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.