Approaching the solving of constrained variational inequalities via penalty term-based dynamical systems

TL;DR

This paper studies a penalty term-based dynamical system for solving constrained variational inequalities, proving convergence of trajectories to solutions using Lyapunov analysis and ergodic methods, with stronger results under enhanced monotonicity.

Contribution

It introduces a novel dynamical system approach for constrained variational inequalities and establishes convergence results using Lyapunov and ergodic techniques.

Findings

Weak ergodic convergence of trajectories to solutions

Strong convergence under stronger monotonicity conditions

Use of Lyapunov analysis and Opial Lemma for proofs

Abstract

We investigate the existence and uniqueness of (locally) absolutely continuous trajectories of a penalty term-based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem. Relying on Lyapunov analysis and on the ergodic continuous version of the celebrated Opial Lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation. If one of the operators involved satisfies stronger monotonicity properties, then strong convergence of the trajectories can be shown.