Second order dynamical systems associated to variational inequalities

TL;DR

This paper studies the convergence behavior of second order dynamical systems linked to variational inequalities, showing weak and strong convergence results under certain convexity and penalty conditions, extending previous first order results.

Contribution

It introduces new convergence results for second order dynamical systems related to variational inequalities, addressing an open problem and generalizing prior first order findings.

Findings

Weak convergence to a minimizer under certain conditions

Strong convergence when the function is strongly convex

Extension of first order results to second order systems

Abstract

We investigate the asymptotic convergence of the trajectories generated by the second order dynamical system $\ddot x(t) + \gamma\dot x(t) + \nabla \phi(x(t))+\beta(t)\nabla \psi(x(t))=0$, where $\phi,\psi:{\cal H}\rightarrow \R$ are convex and smooth functions defined on a real Hilbert space ${\cal H}$, $\gamma>0$ and $\beta$ is a function of time which controls the penalty term. We show weak convergence of the trajectories to a minimizer of the function $\phi$ over the (nonempty) set of minima of $\psi$ as well as convergence for the objective function values along the trajectories, provided a condition expressed via the Fenchel conjugate of $\psi$ is fulfilled. When the function $\phi$ is assumed to be strongly convex, we can even show strong convergence of the trajectories. The results can be seen as the second order counterparts of the ones given by Attouch and Czarnecki (Journal of Differential Equations 248(6), 1315--1344, 2010) for first order dynamical systems associated to constrained variational inequalities. At the same time we give a positive answer to an open problem posed in \cite{att-cza-16} by the same authors.