Regularized neural network for general variational inequalities involving monotone couples of operators in Hilbert spaces
Pham Ky Anh, Trinh Ngoc Hai, Nguyen Van Manh
TL;DR
This work addresses solving generalized variational inequalities $GVI(A,F,C)$ driven by monotone couples $(A,F)$ in real Hilbert spaces by marrying Tikhonov regularization with a neural-network dynamical systems framework. It proves existence and uniqueness results for strongly monotone GVIs and shows that the regularized problem with $F_\alpha = F+\alpha I$ has a unique solution $x_\alpha$ that converges to a bona fide solution $x^\dagger$ of the original GVI as $\alpha\to 0$, with $x^\dagger$ expressible via an auxiliary bilevel VI. The core contribution is a regularized neural network given by $\dot x(t) = P(F_{\alpha(t)}x(t) - \mu(t)Ax(t)) - F_{\alpha(t)}x(t)$ (and its non-regularized variant) whose trajectories converge strongly to $x^\dagger$ under suitable schedules for $\alpha(t)$ and $\mu(t)$; a discretized Euler scheme yields an iterative regularization algorithm with proven convergence to $x^\dagger$, provided standard step-size and parameter-update conditions. Theoretical results are complemented by numerical experiments in both infinite- and finite-dimensional settings, demonstrating robust convergence of the continuous dynamics and the discrete scheme to the unique GVI solution, thereby offering a scalable approach to GVIs in Hilbert spaces.
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 $α,$ such that the latter admits a unique solution $x_α$ which tends to some solution of the initial GVI, as $α\to 0.$ However, instead of solving the regularized GVI for each $α$, 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.