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\texttt{DR-DAQP}: An Hybrid Operator Splitting and Active-Set Solver for Affine Variational Inequalities

Daniel Arnström, Emilio Benenati, Giuseppe Belgioioso

Abstract

We present \texttt{DR-DAQP}, an open-source solver for strongly monotone affine variational inequaliries that combines Douglas-Rachford operator splitting with an active-set acceleration strategy. The key idea is to estimate the active set along the iterations to attempt a Newton-type correction. This step yields the exact AVI solution when the active set is correctly estimated, thus overcoming the asymptotic convergence limitation inherent in first-order methods. Moreover, we exploit warm-starting and pre-factorization of relevant matrices to further accelerate evaluation of the algorithm iterations. We prove convergence and establish conditions under which the algorithm terminates in finite time with the exact solution. Numerical experiments on randomly generated AVIs show that \texttt{DR-DAQP} is up to two orders of magnitude faster than the state-of-the-art solver \texttt{PATH}. On a game-theoretic MPC benchmark, \texttt{DR-DAQP} achieves solve times several orders of magnitude below those of the mixed-integer solver \texttt{NashOpt}. A high-performing C implementation is available at \textt{https://github.com/darnstrom/daqp}, with easily-accessible interfaces to Julia, MATLAB, and Python.

\texttt{DR-DAQP}: An Hybrid Operator Splitting and Active-Set Solver for Affine Variational Inequalities

Abstract

We present \texttt{DR-DAQP}, an open-source solver for strongly monotone affine variational inequaliries that combines Douglas-Rachford operator splitting with an active-set acceleration strategy. The key idea is to estimate the active set along the iterations to attempt a Newton-type correction. This step yields the exact AVI solution when the active set is correctly estimated, thus overcoming the asymptotic convergence limitation inherent in first-order methods. Moreover, we exploit warm-starting and pre-factorization of relevant matrices to further accelerate evaluation of the algorithm iterations. We prove convergence and establish conditions under which the algorithm terminates in finite time with the exact solution. Numerical experiments on randomly generated AVIs show that \texttt{DR-DAQP} is up to two orders of magnitude faster than the state-of-the-art solver \texttt{PATH}. On a game-theoretic MPC benchmark, \texttt{DR-DAQP} achieves solve times several orders of magnitude below those of the mixed-integer solver \texttt{NashOpt}. A high-performing C implementation is available at \textt{https://github.com/darnstrom/daqp}, with easily-accessible interfaces to Julia, MATLAB, and Python.

Paper Structure

This paper contains 20 sections, 5 theorems, 19 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Affine variational inequalities
  4. Active-set structure
  5. The DR-DAQP Algorithm
  6. Douglas-Rachford splitting for AVIs
  7. Active-set acceleration: the DR-DAQP algorithm
  8. Merit function and Newton-step acceptance
  9. Convergence Analysis
  10. Numerical Experiments
  11. Randomly generated AVIs
  12. Effect of the QP solver
  13. Comparison with PATH
  14. Convergence behavior
  15. Game-theoretic MPC
  16. ...and 5 more sections

Key Result

Proposition 1

For any $Q \succ 0$, a point $z^*$ solves the AVI in eq:avi if and only if $R_{\mathcal{X},Q}(z^*) = 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure E1: Total solve time when solving the subproblems \ref{['eq:dr-qp']} in Step \ref{['step:qp-solve']} using DAQP (either warm or cold started) compared to Clarabel.
  • Figure E2: Solve time comparison of DR-DAQP, PATH, and a lifted QP formulation on random AVIs with $\gamma^{\text{asym}}=0.5$ and $m=10n$. Solid lines show the median; shaded regions span the min-max range over 100 instances.
  • Figure E3: Reduction of the residual $\|x_k - x^*\|$ over iterations for a representative AVI instance. DR-DAQP with Newton steps achieves finite termination, while first-order methods converge only asymptotically.
  • Figure E4: Solve time for the game theoretic MPC controller in bemporad2025nashopt over the system evolution. Due to active constraints, the solution time is higher at the beginning of the simulation, and stabilizes when the state approaches an attractor.
  • Figure E5: Simulated driving scenario. The safety distance constraint is drawn in red. When $\Delta p$ is below a threshold, $\bar{l}_2$ is set to the left lane (b). When $\Delta p<0$, $\bar{l}_2$ is set again to the right lane to complete the overtake (c). Video:https://youtu.be/xoxPtcbNL3w

Theorems & Definitions (12)

  • Definition 1: Natural residual
  • Proposition 1: facchinei2003finite
  • Remark 1: Selection of $\rho$
  • Remark 2: Warm-starting the QP subproblems
  • Remark 3: Stabilization of the active set
  • Theorem 1: Convergence
  • Theorem 2: Finite-time termination
  • Lemma 1: Transfer of regularity
  • proof
  • Lemma 2
  • ...and 2 more