Regularized neural network for general variational inequalities involving monotone couples of operators in Hilbert spaces

TL;DR

This paper introduces a neural network-based approach for solving general variational inequalities involving monotone operators in Hilbert spaces, utilizing Tikhonov regularization and demonstrating convergence through numerical experiments.

Contribution

It develops a neural network method for regularized variational inequalities, extending previous results to more general settings with proven convergence and numerical validation.

Findings

Neural network solutions converge strongly to the true solution.

The proposed algorithms are effective and computationally feasible.

Numerical tests confirm the method's efficiency.

Abstract

In this paper, based on the Tikhonov regularization technique, we study a monotone general variational inequality (GVI) by considering an associated strongly monotone GVI, depending on a regularization parameter $\alpha,$ such that the latter admits a unique solution $x_\alpha$ which tends to some solution of the initial GVI, as $\alpha \to 0.$ However, instead of solving the regularized GVI for each $\alpha$, which may be very expensive, we consider a neural network (also known as a dynamical system) associated with the regularized GVI and establish the existence and the uniqueness of the strong global solution to the corresponding Cauchy problem. An explicit discretization of this neural network leads to strongly convergent iterative regularization algorithms for monotone general variational inequality. Numerical tests are performed to show the effectiveness of the proposed methods. This work extends our recent results in [Anh, Hai, Optim. Eng. 25 (2024) 2295-2313] to more general setting.