Third-Order Dynamical Systems for Generalized Inverse Mixed Variational Inequality Problems

TL;DR

This paper introduces a third-order dynamical system and a corresponding algorithm for solving generalized inverse mixed variational inequality problems in Hilbert spaces, demonstrating exponential and linear convergence under certain conditions.

Contribution

It proposes a novel third-order dynamical system and discretization method with inertial effects for efficient convergence in variational inequality problems.

Findings

Existence and uniqueness of trajectories established.

Exponential convergence of the dynamical system proven.

Linear convergence of the discretized algorithm shown.

Abstract

In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the system under suitable continuity assumptions, and prove their exponential convergence to the unique solution under strong monotonicity and Lipschitz continuity conditions. Furthermore, we derive an explicit discretization of the proposed dynamical system, leading to a forward -backward algorithm with double inertial effects. We then establish the linear convergence of the generated iterates to the unique solution.