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Forward-Backward-Forward Dynamical System for Solving Mixed Variational Inequality Problems

Chidi Elijah Nwakpa, Chinedu Izuchukwu, Chibueze Christian Okeke

TL;DR

The paper develops a continuous Tseng-style forward-backward-forward dynamical system to solve mixed variational inequality problems in real Hilbert spaces. It proves weak convergence of trajectories under a Lipschitz, monotone operator and convex, lower semicontinuous function, and establishes global exponential stability under $h$-strong pseudomonotonicity. The analysis hinges on Lyapunov methods and shows that a time discretization recovers Tseng's proximal forward-backward-forward algorithm. Numerical experiments on simple 1D, higher-dimensional, and logistic regression problems illustrate convergence behavior and align with the theoretical results.

Abstract

We study in this paper a forward-backward-forward dynamical system for solving a mixed variational inequality problem in a real Hilbert space. For the convergence analysis of our proposed system, we apply the Lyapunov analysis to obtain the weak convergence of the generated trajectories when the associated operator is Lipschitz continuous and satisfies the general monotonicity condition. We also assume that the involved real-valued convex function satisfies some mild assumptions. Furthermore, the Lipschitz continuous operator is taken to be $h-$strongly pseudomonotone to establish the global exponential stability of the equilibrium point of the system for all the orbits generated. Finally, we present some numerical examples which illustrate how the trajectories of the proposed system converge to the equilibrium point of the proposed dynamical system.

Forward-Backward-Forward Dynamical System for Solving Mixed Variational Inequality Problems

TL;DR

The paper develops a continuous Tseng-style forward-backward-forward dynamical system to solve mixed variational inequality problems in real Hilbert spaces. It proves weak convergence of trajectories under a Lipschitz, monotone operator and convex, lower semicontinuous function, and establishes global exponential stability under -strong pseudomonotonicity. The analysis hinges on Lyapunov methods and shows that a time discretization recovers Tseng's proximal forward-backward-forward algorithm. Numerical experiments on simple 1D, higher-dimensional, and logistic regression problems illustrate convergence behavior and align with the theoretical results.

Abstract

We study in this paper a forward-backward-forward dynamical system for solving a mixed variational inequality problem in a real Hilbert space. For the convergence analysis of our proposed system, we apply the Lyapunov analysis to obtain the weak convergence of the generated trajectories when the associated operator is Lipschitz continuous and satisfies the general monotonicity condition. We also assume that the involved real-valued convex function satisfies some mild assumptions. Furthermore, the Lipschitz continuous operator is taken to be strongly pseudomonotone to establish the global exponential stability of the equilibrium point of the system for all the orbits generated. Finally, we present some numerical examples which illustrate how the trajectories of the proposed system converge to the equilibrium point of the proposed dynamical system.

Paper Structure

This paper contains 5 sections, 9 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Numerical Experiments
  5. Conclusion

Key Result

Lemma 2.7

BAUS For each $\underline{u}\in\mathcal{H},$$\underline{v}\in\mathcal{C}$ and a positive constant $\lambda,$ the following inequality is true where $prox_{\lambda h}(\underline{u}) := \underset{\underline{v}\in\mathcal{C}}{argmin}\left\{\lambda h(\underline{v})+\frac{\|\underline{u}-\underline{v}\|^2}{2}\right\}.$

Figures (4)

  • Figure 1: Convergence of $x(t)$ generated \ref{['DS MVI']} to the unique solution $\bar{x}=3$ over time for Example \ref{['ex1']} with $x_0=0.1$: Upper Left: $\lambda=\frac{0.99}{2}$; Upper Right: $\lambda=\frac{0.8}{2}$; Bottom: $\lambda=\frac{0.5}{2}$.
  • Figure 2: Trajectories generated by the dynamical system \ref{['DS MVI']} with $x_0=(-4, 3, 5)^T$ for Example \ref{['ex2']}: Upper Left: $\lambda=\frac{0.99}{1+\beta^2}$; Upper Right: $\lambda=\frac{0.8}{1+\beta^2}$; Bottom: $\lambda=\frac{0.5}{1+\beta^2}$.
  • Figure 3: 3D visualization of the trajectories generated by the dynamical system \ref{['DS MVI']} for Example \ref{['ex2']}: $\lambda=\frac{0.99}{1+\beta^2}$.
  • Figure 4: Dynamical system \ref{['DS MVI']} for Example \ref{['ex3']}: Left: Shows the convergence of the loss function over iterations; Right: Convergence of the trajectories over time.

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 16 more