Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

TL;DR

This paper introduces a new fixed-time stable dynamical system for solving mixed variational inequality problems, ensuring convergence within a uniform finite time regardless of initial conditions.

Contribution

It proposes a modified proximal dynamical system that guarantees fixed-time convergence to the solution of MVIPs under certain conditions, extending stability even to pseudomonotonic cases.

Findings

Convergence within a fixed time for the proposed system.

Unique solution exists under strong monotonicity.

Discretized solutions approximate the true solution within fixed steps.

Abstract

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined, and converges to the unique solution of the associated MVIP within a fixed time. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions.