Equivariant Deep Learning of Mixed-Integer Optimal Control Solutions for Vehicle Decision Making and Motion Planning

TL;DR

This work tackles real-time decision-making and motion planning for autonomous driving by accelerating MIQP-based formulations with learning. It introduces REDS, a permutation-equivariant recurrent deep-set network that predicts obstacle-related binary variables, coupled with a soft-QP that yields candidate trajectories and a feasibility projector (FP) that enforces safety via a smooth NLP solved with SQP. An ensemble of REDS models, combined with soft-QP and FP, achieves substantial speedups while maintaining close-to-expert performance against a benchmark MIQP (expert MIP-DM) in SUMO/CommonRoad simulations, and demonstrates robustness under distribution shift. The approach enables real-time, safe planning on embedded automotive hardware and generalizes to varying numbers of obstacles and horizon lengths, offering significant practical impact for MIQP-based motion planning in autonomous driving. The combination of structural architectural priors (equivariance, recurrence), multi-hypothesis predictions, and a safety-oriented projection pipeline provides a scalable pathway to real-time, reliable mixed-integer planning on resource-constrained platforms.

Abstract

Mixed-integer quadratic programs (MIQPs) are a versatile way of formulating vehicle decision making and motion planning problems, where the prediction model is a hybrid dynamical system that involves both discrete and continuous decision variables. However, even the most advanced MIQP solvers can hardly account for the challenging requirements of automotive embedded platforms. Thus, we use machine learning to simplify and hence speed up optimization. Our work builds on recent ideas for solving MIQPs in real-time by training a neural network to predict the optimal values of integer variables and solving the remaining problem by online quadratic programming. Specifically, we propose a recurrent permutation equivariant deep set that is particularly suited for imitating MIQPs that involve many obstacles, which is often the major source of computational burden in motion planning problems. Our framework comprises also a feasibility projector that corrects infeasible predictions of integer variables and considerably increases the likelihood of computing a collision-free trajectory. We evaluate the performance, safety and real-time feasibility of decision-making for autonomous driving using the proposed approach on realistic multi-lane traffic scenarios with interactive agents in SUMO simulations.

Figures (12)

• Figure 1: Categorical overview of related work. The research domain of the proposed approach is located at the intersection of three areas.
• Figure 2: Planning and closed-loop simulation architecture. The expert MIP-DM is imitated by the REDS planner, which uses an ensemble of $n_\mathrm{e}$NN to predict values of the binary variables $\{\hat{B}^{1},\ldots,\hat{B}^{n_\mathrm{e}}\}$. A soft-QP solves a formulation of the expert MIP-DM with binary variables fixed to the prediction. The lowest cost solution $X^\mathrm{p }$ is chosen by a Selector and corrected by the FP. A reference tracking NMPC with obstacle avoidance tracks the corrected solution $X^\mathrm{s}$.
• Figure 3: Vehicle over-approximations. All trajectories outside of $\mathcal{O}^\mathrm{safe}$ are considered free of collision. The expert MIP-DM plans with the most conservative obstacle set $\mathcal{O}^\mathrm{out}$. The ellipsoidal smooth over-approximation $\mathcal{O}^\mathrm{safe}$ is used within the FP and the NMPC. The four colored regions are uniquely determined by the binary variables $\gamma$, where in each region exactly one binary variable is equal to one.
• Figure 4: REDS network. The blue blocks show the propagation of equivariant features, whereas the orange blocks show the propagation of unstructured features. An invariant connection couples both hidden states.
• Figure 5: Performance evaluation for infeasibility rate, misclassification rate and suboptimality of different network architectures, depending on the number of obstacles and horizon length. The REDS network outperforms the other architectures, particularly for a larger number of obstacles and a longer horizon. Suboptimality is shown in a logarithmic scale.

Theorems & Definitions (4)

• Definition 1.1
• Definition 1.2
• Proposition 6.1
• proof