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A Douglas-Rachford Splitting Method for Solving Monotone Variational Inequalities in Linear-quadratic Dynamic Games

Reza Rahimi Baghbadorani, Emilio Benenati, Sergio Grammatico

TL;DR

The paper addresses constrained linear-quadratic dynamic games and shows that open-loop Nash equilibria (OL-NE) can be formulated as an affine variational inequality (AVI) $\mathrm{AVI}(\mathcal{C}, M, q)$, enabling structured solution techniques. It develops a Douglas-Rachford splitting algorithm tailored to the AVI, using a matrix-splitting $M=M_1+M_2$ with a positive-definite and a semidefinite component to guarantee linear convergence. Near the equilibrium attractor, the OL-NE admits a closed-form solution that reduces computation, which supports fast receding-horizon control for multi-agent systems. Numerical experiments in automated driving demonstrate faster online convergence and scalability compared with standard VI solvers, validating the method's potential for real-time multi-vehicle coordination.

Abstract

This paper considers constrained linear dynamic games with quadratic objective functions, which can be cast as affine variational inequalities. By leveraging the problem structure, we apply the Douglas-Rachford splitting, which generates a solution algorithm with linear convergence rate. The fast convergence of the method enables receding-horizon control architectures. Furthermore, we demonstrate that the associated VI admits a closed-form solution within a neighborhood of the attractor, thus allowing for a further reduction in computation time. Finally, we benchmark the proposed method via numerical experiments in an automated driving application.

A Douglas-Rachford Splitting Method for Solving Monotone Variational Inequalities in Linear-quadratic Dynamic Games

TL;DR

The paper addresses constrained linear-quadratic dynamic games and shows that open-loop Nash equilibria (OL-NE) can be formulated as an affine variational inequality (AVI) , enabling structured solution techniques. It develops a Douglas-Rachford splitting algorithm tailored to the AVI, using a matrix-splitting with a positive-definite and a semidefinite component to guarantee linear convergence. Near the equilibrium attractor, the OL-NE admits a closed-form solution that reduces computation, which supports fast receding-horizon control for multi-agent systems. Numerical experiments in automated driving demonstrate faster online convergence and scalability compared with standard VI solvers, validating the method's potential for real-time multi-vehicle coordination.

Abstract

This paper considers constrained linear dynamic games with quadratic objective functions, which can be cast as affine variational inequalities. By leveraging the problem structure, we apply the Douglas-Rachford splitting, which generates a solution algorithm with linear convergence rate. The fast convergence of the method enables receding-horizon control architectures. Furthermore, we demonstrate that the associated VI admits a closed-form solution within a neighborhood of the attractor, thus allowing for a further reduction in computation time. Finally, we benchmark the proposed method via numerical experiments in an automated driving application.

Paper Structure

This paper contains 11 sections, 4 theorems, 46 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Matrices
  3. Euclidean spaces and operators
  4. Real sequences
  5. Game theory
  6. Open-Loop Nash Equilibria as Solutions of a Variational Inequality
  7. Algorithm and Convergence
  8. Numerical Experiments
  9. Conclusion
  10. Appendix
  11. State-of-the-art algorithms for solving (strongly) monotone variational inequalities

Key Result

Lemma 1

benenati2024linear Let $\boldsymbol{u}^*$ solve $\mathcal{P}_1(x_0)$, defined in eq:finite_hor_problem. Let $x_T:=\phi(T, x_0, \boldsymbol{u}^*)$ and let $x_T\in \mathbb{X}_f$. Then, the sequence defined as is an OL-NE for the initial state $x_0$.

Figures (5)

  • Figure 1: Comparison of residuals for different state-of-the-art VI solution methods.
  • Figure 2: Block scheme of the closed-loop dynamics with receding-horizon open-loop Nash equilibrium controller.
  • Figure 3: Simulated scenario. The full animation is available at https://github.com/bemilio/CDC2025_DR_OLNE/blob/aa6a7633c8d4964b2ec475ff0e964d75041d4d2f/Simulations/vehicle_animation.mp4
  • Figure 4: (a): Distance between $\chi(i)$ and $i$. (b): Velocity of each agent. The dotted lines denote the reference values, and the red dashed lines denote the constraints.
  • Figure 5: Number of iterations to achieve convergence of the VI solution algorithm.

Theorems & Definitions (7)

  • Definition 1: OL-NE
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • proof
  • Lemma 3: Linear convergence ferris1996operator
  • proof