A novel neural network with predefined-time stability for solving generalized monotone inclusion problems with applications

TL;DR

This paper introduces a new neural network framework with predefined-time stability for solving generalized monotone inclusion problems, extending classical methods and providing convergence guarantees with practical applications.

Contribution

It develops a novel dynamical system approach with fixed-time and predefined-time stability for generalized monotone inclusions, including a new discretization algorithm and convergence analysis.

Findings

The proposed method guarantees convergence within a user-defined time horizon.

The discretized algorithm converges rigorously under mild conditions.

Numerical experiments demonstrate the approach's effectiveness across various problem classes.

Abstract

We propose a novel dynamical framework for solving inclusion problems of the form \(0 \in F(x) + G(x)\) in Hilbert spaces, where \(F\) is a maximal set-valued operator and \(G\) is a single-valued mapping. The analysis is conducted under a generalized monotonicity assumption, which relaxes the classical monotonicity conditions commonly imposed in the literature and thereby extends the applicability of the proposed approach. Under mild conditions on the system parameters, we establish both fixed-time and predefined-time stability of the resulting dynamical system. The fixed-time stability guarantees a uniform upper bound on the settling time that is independent of the initial condition, whereas the predefined-time stability framework allows the system parameters to be selected \emph{a priori} in order to ensure convergence within a user-specified time horizon. Moreover, we investigate an explicit forward Euler discretization of the continuous-time dynamics, leading to a novel forward--backward iterative algorithm. A rigorous convergence analysis of the resulting discrete scheme is provided. Finally, the effectiveness and versatility of the proposed method are illustrated through several classes of problems, including constrained optimization problems, mixed variational inequalities, and variational inequalities, together with numerical experiments that corroborate the theoretical results.