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The explicit game-theoretic linear quadratic regulator for constrained multi-agent systems

Emilio Benenati, Giuseppe Belgioioso

TL;DR

This paper addresses the computational bottleneck of online solving for constrained multi-agent linear-quadratic dynamic games by deriving an explicit, piecewise-affine solution mapping (mpAVI) from initial state to open-loop Nash equilibria. It develops an efficient offline algorithm that exploits active-set structure to construct a global state-dependent policy, extending explicit MPC concepts to non-cooperative game settings and enabling real-time GT-MPC at high sampling rates. Infinite-horizon NE are handled via finite-horizon problems with carefully designed terminal costs, preserving equivalence to the infinite-horizon solution under suitable assumptions. Numerical studies show orders-of-magnitude online speedups and improved accuracy over state-of-the-art solvers, with a compelling autonomous-driving demonstration that completes a multi-phase overtaking maneuver at 10 Hz.

Abstract

We present an efficient algorithm to compute the explicit open-loop solution to both finite and infinite-horizon dynamic games subject to state and input constraints. Our approach relies on a multiparametric affine variational inequality characterization of the open-loop Nash equilibria and extends the classical explicit constrained LQR and MPC frameworks to multi-agent non-cooperative settings. A key practical implication is that linear-quadratic game-theoretic MPC becomes viable even at very high sampling rates for multi-agent systems of moderate size. Extensive numerical experiments demonstrate order-of-magnitude improvements in online computation time and solution accuracy compared with state-of-the-art game-theoretic solvers.

The explicit game-theoretic linear quadratic regulator for constrained multi-agent systems

TL;DR

This paper addresses the computational bottleneck of online solving for constrained multi-agent linear-quadratic dynamic games by deriving an explicit, piecewise-affine solution mapping (mpAVI) from initial state to open-loop Nash equilibria. It develops an efficient offline algorithm that exploits active-set structure to construct a global state-dependent policy, extending explicit MPC concepts to non-cooperative game settings and enabling real-time GT-MPC at high sampling rates. Infinite-horizon NE are handled via finite-horizon problems with carefully designed terminal costs, preserving equivalence to the infinite-horizon solution under suitable assumptions. Numerical studies show orders-of-magnitude online speedups and improved accuracy over state-of-the-art solvers, with a compelling autonomous-driving demonstration that completes a multi-phase overtaking maneuver at 10 Hz.

Abstract

We present an efficient algorithm to compute the explicit open-loop solution to both finite and infinite-horizon dynamic games subject to state and input constraints. Our approach relies on a multiparametric affine variational inequality characterization of the open-loop Nash equilibria and extends the classical explicit constrained LQR and MPC frameworks to multi-agent non-cooperative settings. A key practical implication is that linear-quadratic game-theoretic MPC becomes viable even at very high sampling rates for multi-agent systems of moderate size. Extensive numerical experiments demonstrate order-of-magnitude improvements in online computation time and solution accuracy compared with state-of-the-art game-theoretic solvers.

Paper Structure

This paper contains 22 sections, 3 theorems, 42 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Problem formulation
  4. Game Theoretic Model Predictive Control
  5. The Game-theoretic Linear Quadratic Regulator
  6. Multiparametric Affine VIs
  7. Efficient Construction of the Explicit Solution Mapping
  8. Numerical results
  9. Comparison against state-of-the-art iterative solvers
  10. Illustrative example: autonomous driving
  11. Platooning
  12. Initiate overtake
  13. Perform overtake
  14. Complete overtake
  15. Clear road driving
  16. ...and 7 more sections

Key Result

Proposition 1

For each $x^0\in\mathcal{X}$, either of the following holds:

Figures (5)

  • Figure 1: In game-theoretic MPC, control actions are generated by solving a finite-horizon dynamic game and applying the first element of the ol-NE control sequence to the system.
  • Figure 2: Left: combinatorial tree of constraint in ahmadi-moshkenani_combinatorial_2018. Right: state space $\mathcal{X} = [-1.5,1.5]^2$ (drawn twice for clarity), partitioned in the critical regions found via Alg. \ref{['alg:mpVI_sol']}. The constraints corresponding to $\mathcal{A}=\{1,2,3,4\}$ are not linearly independent, thus the corresponding (non-empty) critical region is excluded from the state-space partition. For $\mathcal{A}=\{1,2\}$ and $\{3,4\}$ we find a one-dimensional critical region: the colored lines segment. Any path on the graph is a combinatorially valid sequence as for Definition \ref{['def:comb_valid']}. There is no edge leading to $\{\}$, as it exhibits an empty critical region.
  • Figure 3: Online evaluation time of the explicit solution mapping \ref{['eq:pwa_function']} constructed with Alg. 1, compared against the computation time for a high-quality solution ($r(u)=10^{-6}$) required by the Douglas--Rachford (DR) eckstein_operator-splitting_1998, ADMM min_admm-iclqg_2025, and DGSQP zhu_sequential_2023 solvers. The left plot shows the results in function of the horizon length, while the right one in function of the state dimension.
  • Figure 5: Safety distance constraint in the maneuvre phases.
  • Figure 6: Finite-state machine that switches between the overtake controller cases. We denote $\Delta p = p_1-p_2$.

Theorems & Definitions (8)

  • Definition 1
  • Remark 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Definition 3
  • Definition 4
  • Lemma 1