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

Nam Van Tran

TL;DR

The paper addresses solving generalized monotone inclusion problems of the form $0 \in F(z) + G(z)$ in Hilbert spaces by designing a forward–backward splitting dynamical system with predefined-time stability. It proves the existence, uniqueness, and convergence of trajectories to the unique solution under a relaxed generalized monotonicity framework, and provides a discrete-time counterpart via forward Euler discretization with convergence guarantees. The work extends the stability toolbox by incorporating predefined-time convergence, yields explicit bounds and parameter choices, and demonstrates applicability to constrained optimization, mixed variational inequalities, and variational inequalities. The results offer practical convergence-time guarantees and a versatile algorithmic approach for a broad class of equilibrium and optimization problems.

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.

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

TL;DR

The paper addresses solving generalized monotone inclusion problems of the form in Hilbert spaces by designing a forward–backward splitting dynamical system with predefined-time stability. It proves the existence, uniqueness, and convergence of trajectories to the unique solution under a relaxed generalized monotonicity framework, and provides a discrete-time counterpart via forward Euler discretization with convergence guarantees. The work extends the stability toolbox by incorporating predefined-time convergence, yields explicit bounds and parameter choices, and demonstrates applicability to constrained optimization, mixed variational inequalities, and variational inequalities. The results offer practical convergence-time guarantees and a versatile algorithmic approach for a broad class of equilibrium and optimization problems.

Abstract

We propose a novel dynamical framework for solving inclusion problems of the form \(0 \in F(x) + G(x)\) in Hilbert spaces, where is a maximal set-valued operator and 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.
Paper Structure (14 sections, 13 theorems, 69 equations, 3 figures)

This paper contains 14 sections, 13 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminary Notions on functions
  4. Preliminary Notions on Operators
  5. Definitions of stability
  6. Predefined-time stability analysis
  7. Discretization of the differential equation
  8. Applications
  9. Application to constrained optimization problems
  10. Application to mixed variational inequality problems
  11. Application to variational inequality problems
  12. Numerical Illustration
  13. Conclusion
  14. Introduction

Key Result

Lemma 2.5

Minh Consider an $\eta_F$-monotone set-valued mapping $F : \mathscr{H} \to 2^\mathscr{H}$ and a single-valued operator $G : \mathscr{H} \to \mathscr{H}$, along with a positive parameter $\gamma$ such that $1 + \gamma \eta_F > 0$. Then the following hold:

Figures (3)

  • Figure 1: Transient responses and error decay of the dynamical system \ref{['newydynamicalsystem']} for Example \ref{['ex1']}.
  • Figure 2: Convergence rate with different values of parameters of dynamic system \ref{['newydynamicalsystem']} for Example \ref{['ex1']} .
  • Figure 3: Convergence rate with different values of $p_3$ of dynamic system \ref{['newydynamicalsystem']} for Example \ref{['ex1']}.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 18 more