A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

TL;DR

This paper introduces a novel dynamical system approach for multiobjective optimization with nonsmooth convex functions, ensuring convergence to weak Pareto minima within a Hilbert space framework.

Contribution

It extends previous work by handling nonsmooth convex objectives using Yosida regularization, proving existence, descent properties, and convergence of trajectories.

Findings

Existence of strong global trajectories in the system

Trajectories converge to weak Pareto minima

Applicable to cooperative games and inverse problems

Abstract

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions. Based on the Yosida regularization of the subdi erential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions. Applications are given to cooperative games, inverse problems, and numerical multiobjective optimization.