A second order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

TL;DR

This paper introduces a second order dynamical system with Hessian-driven damping and penalty terms to solve convex variational inequality problems, proving convergence properties using Lyapunov analysis.

Contribution

It presents a novel continuous dynamical system approach for convex variational inequalities with convergence guarantees, including strong convergence under strong convexity.

Findings

Trajectories weakly converge to a minimizer

Objective function values converge along trajectories

Strong convergence achieved in the strongly convex case

Abstract

We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories.