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

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.

💡 Research Summary

This paper proposes a novel continuous-time dynamical system framework, conceptualized as a neural network, for solving generalized monotone inclusion problems of the form 0 ∈ F(x) + G(x) in Hilbert spaces, where F is a maximal set-valued operator and G is a single-valued mapping. The core innovation lies in establishing not only fixed-time stability but, more significantly, predefined-time stability for the proposed system under a generalized monotonicity assumption. This assumption relaxes the classical requirement by allowing the monotonicity modulus η to be negative (weak monotonicity), thereby significantly extending the method’s applicability to a broader class of operators.

The dynamical system is formulated as ż(t) = -μ Φ(z(t)), where Φ(z) = z - J_γF(z - γG(z)), μ > 0 is a gain parameter, and J_γF is the resolvent of F. The equilibrium points of this system are shown to be exactly the solutions of the original inclusion problem. The primary theoretical contribution is Theorem 3.2, which proves that under mild conditions on parameters γ, μ, and the operators’ properties (Lipschitz constant L, monotonicity moduli η_F, η_G), the system is predefined-time stable. This means that by appropriately selecting system parameters (specifically μ and an auxiliary parameter α), the convergence time can be guaranteed not to exceed any user-specified upper bound T_p, regardless of the initial state. This is a stronger guarantee than fixed-time stability, which only ensures a uniform bound exists but does not allow for its arbitrary pre-specification. The proof leverages a quadratic Lyapunov function and applies an advanced Lyapunov condition for predefined-time stability.

Furthermore, the paper provides a rigorous discretization of the continuous-time dynamics using an explicit forward Euler scheme, leading to a novel relaxed forward-backward iterative algorithm: z_{k+1} = (1-λ)z_k + λ J_γF(z_k - γG(z_k)), where λ ∈ (0,1) is a relaxation parameter. Theorem 4.1 establishes the linear convergence of this discrete-time algorithm to the unique solution of the inclusion problem, bridging the gap between continuous-time theory and practical numerical implementation.

To demonstrate versatility, the authors show that several important problem classes can be cast into the form of (1), including constrained optimization problems (via their KKT conditions), mixed variational inequalities (MVIs), and standard variational inequalities (VIs). Consequently, the proposed framework and its convergence guarantees directly apply to these problems. Numerical experiments on illustrative examples in both low and higher dimensions corroborate the theoretical findings, confirming convergence within the predefined time horizon T_p for the continuous system and linear convergence for the discrete algorithm.

In summary, this work makes substantial contributions by: 1) introducing and analyzing predefined-time stability for generalized monotone inclusions, a concept not previously studied in this context; 2) conducting the analysis under weaker generalized monotonicity conditions; 3) deriving a corresponding discrete-time algorithm with proven linear convergence; and 4) demonstrating broad applicability across optimization and variational inequality problems. The results offer a powerful tool for time-critical applications where convergence deadlines must be strictly met.