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Continuous and Discrete Systems for Quasi Variational Inequalities with Application to Game Theory

Oday Hazaimah

TL;DR

This work introduces a third-order projected dynamical system for parametric quasi-variational inequalities with moving constraints and develops both continuous-time and discrete-time inertial proximal algorithms. Under mild conditions on the moving constraint and the operator $T$, the authors show exponential-type convergence of the continuous system and convergence of discrete schemes to QVI solutions, establishing a principled link between dynamics and fixed-point formulations. The framework encompasses applications to obstacle problems and generalized Nash equilibrium problems, demonstrating how moving-constraint QVIs capture complex equilibrium behavior. The results offer new high-order methods for nonconvex and parametric equilibrium problems, with potential extensions to stochastic settings and advanced merit-function formulations for GNEP.

Abstract

A new class of projected dynamical systems of third order is investigated for quasi (parametric) variational inequalities in which the convex set in the classical variational inequality also depends upon the solution explicitly or implicitly. We study the stability of a continuous method of a gradient type. Some iterative implicit and explicit schemes are suggested as counterparts of the continuous case by inertial proximal methods. The convergence analysis of these proposed methods is established under sufficient mild conditions. Moreover, some applications dealing with the generalized Nash equilibrium problems are presented.

Continuous and Discrete Systems for Quasi Variational Inequalities with Application to Game Theory

TL;DR

This work introduces a third-order projected dynamical system for parametric quasi-variational inequalities with moving constraints and develops both continuous-time and discrete-time inertial proximal algorithms. Under mild conditions on the moving constraint and the operator , the authors show exponential-type convergence of the continuous system and convergence of discrete schemes to QVI solutions, establishing a principled link between dynamics and fixed-point formulations. The framework encompasses applications to obstacle problems and generalized Nash equilibrium problems, demonstrating how moving-constraint QVIs capture complex equilibrium behavior. The results offer new high-order methods for nonconvex and parametric equilibrium problems, with potential extensions to stochastic settings and advanced merit-function formulations for GNEP.

Abstract

A new class of projected dynamical systems of third order is investigated for quasi (parametric) variational inequalities in which the convex set in the classical variational inequality also depends upon the solution explicitly or implicitly. We study the stability of a continuous method of a gradient type. Some iterative implicit and explicit schemes are suggested as counterparts of the continuous case by inertial proximal methods. The convergence analysis of these proposed methods is established under sufficient mild conditions. Moreover, some applications dealing with the generalized Nash equilibrium problems are presented.
Paper Structure (10 sections, 8 theorems, 78 equations, 1 algorithm)

This paper contains 10 sections, 8 theorems, 78 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Main Results
  4. Continuous gradient method
  5. Discrete methods
  6. Convergence of a discrete system
  7. Applications
  8. The obstacle boundary value
  9. Generalized Nash equilibrium problems
  10. Conclusion

Key Result

Theorem 1

Let $\mathcal{K}$ be a nonempty, convex and closed subset of $\mathcal{H}$. Let $T$ be a smooth convex function. Then $x\in\mathcal{K}$ is the minimum of the smooth convex $T(x)$ if and only if, $x\in\mathcal{K}$ satisfies where $T'$ is the Frechet derivative of $T$ at $x\in\mathcal{K}$.

Theorems & Definitions (16)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 2
  • Proposition 3: Jabeen-Noor
  • Definition 3
  • Theorem 4: Vasiliev
  • Theorem 5
  • proof
  • Remark 1
  • ...and 6 more