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Third-Order Dynamical Systems for Generalized Inverse Mixed Variational Inequality Problems

Nam Van Tran

Abstract

In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the system under suitable continuity assumptions, and prove their exponential convergence to the unique solution under strong monotonicity and Lipschitz continuity conditions. Furthermore, we derive an explicit discretization of the proposed dynamical system, leading to a forward -backward algorithm with double inertial effects. We then establish the linear convergence of the generated iterates to the unique solution.

Third-Order Dynamical Systems for Generalized Inverse Mixed Variational Inequality Problems

Abstract

In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the system under suitable continuity assumptions, and prove their exponential convergence to the unique solution under strong monotonicity and Lipschitz continuity conditions. Furthermore, we derive an explicit discretization of the proposed dynamical system, leading to a forward -backward algorithm with double inertial effects. We then establish the linear convergence of the generated iterates to the unique solution.
Paper Structure (15 sections, 12 theorems, 128 equations)

This paper contains 15 sections, 12 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Background on Inverse Variational Inequalities and Dynamical Systems
  3. Some Historical Aspects
  4. Our Contributions
  5. Definitions and Preliminaries
  6. Background on Monotone Operators and Generalized Projection Operators
  7. Absolutely Continuous Functions
  8. A Third Order Dynamical System
  9. Difference Operators
  10. Continuous Time Dynamical System
  11. Global Exponential Convergence
  12. On the Selection of Parameter Values
  13. Discrete Time Dynamical System
  14. Global Linear Convergence
  15. On Seclection of Parameter Values

Key Result

Lemma 2.4

35 Let $\Omega$ be a nonempty closed convex subset of $H$. Then:

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 24 more