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Second order continuous and discrete dynamical systems for solving inverse quasi-variational inequalities

Pham Viet Hai, Thanh Quoc Trinh, Phan Tu Vuong

TL;DR

The paper addresses solving inverse quasi-variational inequalities (IQVIP) in finite dimensions by introducing a second-order dynamical system whose trajectory converges exponentially to the IQVIP solution under Lipschitz and strong monotonicity assumptions. By discretizing the dynamics, it yields a projection-type algorithm with inertial effects that achieves linear convergence under explicit parameter conditions. Theoretical results are complemented by numerical experiments in low dimensions and a traffic assignment application, demonstrating faster convergence than first-order approaches and validating practical performance. Overall, the work advances efficient, provably convergent methods for IQVIPs with potential impact on applications like traffic systems and related equilibrium problems.

Abstract

In this paper, we investigate the inverse quasi-variational inequality problem in finite-dimensional spaces. First, we introduce a second-order dynamical system whose trajectory converges exponentially to the solution of the inverse quasi-variational inequality, under the assumptions of Lipschitz continuity and strong monotonicity. Next, we discretize the proposed dynamical system to develop an algorithm, and prove that the iterations converge linearly to the unique solution of the inverse quasi-variational inequality. Finally, we present numerical experiments and applications to validate the theoretical results and compare the performance with existing methods.

Second order continuous and discrete dynamical systems for solving inverse quasi-variational inequalities

TL;DR

The paper addresses solving inverse quasi-variational inequalities (IQVIP) in finite dimensions by introducing a second-order dynamical system whose trajectory converges exponentially to the IQVIP solution under Lipschitz and strong monotonicity assumptions. By discretizing the dynamics, it yields a projection-type algorithm with inertial effects that achieves linear convergence under explicit parameter conditions. Theoretical results are complemented by numerical experiments in low dimensions and a traffic assignment application, demonstrating faster convergence than first-order approaches and validating practical performance. Overall, the work advances efficient, provably convergent methods for IQVIPs with potential impact on applications like traffic systems and related equilibrium problems.

Abstract

In this paper, we investigate the inverse quasi-variational inequality problem in finite-dimensional spaces. First, we introduce a second-order dynamical system whose trajectory converges exponentially to the solution of the inverse quasi-variational inequality, under the assumptions of Lipschitz continuity and strong monotonicity. Next, we discretize the proposed dynamical system to develop an algorithm, and prove that the iterations converge linearly to the unique solution of the inverse quasi-variational inequality. Finally, we present numerical experiments and applications to validate the theoretical results and compare the performance with existing methods.
Paper Structure (8 sections, 7 theorems, 103 equations, 7 figures, 2 tables)

This paper contains 8 sections, 7 theorems, 103 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Second-order dynamical system
  4. Existence and uniqueness of solution
  5. Exponential convergence
  6. Discretization of the dynamical system
  7. Numerical Experiment
  8. Application in Traffic Assignment

Key Result

Proposition 2.1

For any $y, z\in \mathbb{R}^n$ and $a\in C$ we have

Figures (7)

  • Figure 1: Convergence of sequences generated by algorithm \ref{['202403251017']} with different starting points and different inertial parameter $1-\sigma$.
  • Figure 2: Comparing norm of $x_n$ generated by corresponding parameters of the algorithm \ref{['202403251017']}.
  • Figure 3: Performance of sequences generated by scheme \ref{['202403251017']} and \ref{['1stAlg']}.
  • Figure 4: Road pricing problem with four bridge network- Source from Thanh2024
  • Figure 5: Link flows and the residual of the projection algorithm \ref{['202403251017']} with $\sigma=0.6, \tau = 0.02$ and $\mu =0.5$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Proposition 2.1: GR84
  • Remark 2.1
  • Proposition 2.2: Dey2023
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • proof
  • Proposition 3.1
  • proof
  • ...and 7 more