SHIELD: Scalable Optimal Control with Certification using Duality and Convexity

Abstract: We present SHIELD, a hierarchical algorithm that reduces both the decision-variable dimension and the constraint set in $\ell_1$-regularized convex programs. From strong convexity and Lagrangian duality, we derive certificates that \emph{safely} discard constraints and decision variables while guaranteeing that all removed constraints remain satisfied and all removed variables are null. To further accelerate the proposed algorithm, we propose a transformer-based deep neural network to guide the dual certificate inference. We validate SHIELD on stochastic model predictive control (SMPC) in complex, multi-modal traffic scenarios, comparing against a full-dimensional SMPC policy. Numerical simulations demonstrate order-of-magnitude computational speedups while preserving feasibility and closed-loop safety, highlighting the practicality of certifiably safe, lightweight MPC in complex driving scenes.

• The paper introduces a duality-based safe screening method that prunes variables and constraints while certifying optimality in convex MPC.
• It leverages a transformer-based classifier to predict active constraints, achieving up to 35x speedups in high-dimensional stochastic MPC applications.
• The approach guarantees safety and feasibility even under distribution shifts, making it a robust solution for real-time autonomous control.

Scalable and Certifiable Optimal Control via SHIELD

Introduction and Problem Context

The SHIELD framework introduces a scalable approach to real-time optimal control, targeting high-dimensional convex programs with large constraint sets such as those prevalent in stochastic MPC (SMPC) for autonomous driving and robotics (2605.09171). It leverages hierarchical duality-based constraint and variable screening, which is certifiably safe by construction, to reduce computational complexity at run-time. SHIELD particularly addresses scenarios with time-varying constraints and high-dimensional decision spaces, where traditional convex MPC solvers face substantial latency due to the presence of hundreds or thousands of constraints, especially in multi-agent and multi-modal predictive control problems.

Theoretical Contributions

SHIELD builds on the strong convexity of the cost and Lagrangian duality properties. It makes two principal technical innovations:

1. Duality-Based Safe Screening: SHIELD derives conservative certificates for safely discarding variables and constraints in an $l_1$-regularized convex program. By tightening the screenable constraints and leveraging global sensitivity results and dual optimality gap bounds, SHIELD guarantees that all pruned constraints remain satisfied and all removed variables are provably zero in any optimal solution of the original (untightened) problem.
2. Hierarchical Dual Screening with Data-Driven Guidance: To avoid the overhead of solving high-dimensional dual problems, SHIELD utilizes a transformer-based neural network to predict the dual support pattern — indicating which variables and constraints are likely inactive. This classifier's predictions induce a reduced dual problem whose solution is subsequently certified for safety using the aforementioned duality gap bounds. Certification holds irrespective of any model mismatch or train-to-test distribution shift due to the online, optimization-based verification step.

Crucially, these certificates do not rely on statistical generalization bounds for the classifier, nor do they assume access to high-quality dual solutions at all points, but only require feasible dual points.

Algorithmic Framework

The SHIELD pipeline integrates four major components:

• Dual Classifier: A permutation-equivariant Transformer model trained on offline MPC solutions predicts which dual variables/constraints are potentially nonzero/active for the current instance.
• Selection and Lifting: The classifier outputs select a reduced subset of dual variables for a lower-dimensional unconstrained dual QP, which is then lifted into the full dual space via zero and $\lambda$ insertion for non-selected indices.
• Certificate Construction: The projected gradient of the dual is used to compute an upper bound on the duality gap, and robust screening certificates are applied to remove any constraint or variable whose (certified) optimal value cannot affect feasibility or cost to within the prescribed safety buffer.
• Reduced Primal Solve: The certified reduced problem, with only necessary variables and constraints, is solved to determine the control action with strong guarantees on constraint satisfaction.

This pipeline ensures that, even in the event of classifier misprediction or atypical scenarios, feasibility and safety are always preserved, and in the worst case, no speedup is realized and the method defaults to the original MPC problem.

Empirical Evaluation

The authors integrate SHIELD into an SMPC framework for closed-loop planning of autonomous vehicles in multi-modal, multi-agent traffic scenes:

• Benchmark Setting: SMPC instances with 2 modes, 3 vehicles, and 14-step horizons, resulting in 312 collision-avoidance constraints and 234 decision variables per instance.
• Data-Driven Classifier: Using a transformer-style architecture, the classifier achieves normalized binary cross-entropy (BCE) superior to strong MLP and RAID-Net baselines, with recall 0.99, precision 0.97, and accuracy 0.96, demonstrating robustness in predicting inactive sets under distribution shift.
• Computation Results: Over 25,000 simulation steps and 150+ nuPlan traffic scenarios, the Reduced MPC formulation enforces on average only 6.8% of collision-avoidance constraints and retains 3.19% of disturbance feedback gains, achieving a 35x speedup (0.498 s vs. 19.53 s including rare conservative steps; up to 119x excluding outliers). Full feasibility and safety (0 collisions) are maintained throughout, validating the strength of the duality-based certificate.
• Closed-Loop Similarity: Despite the drastic reduction in dimensionality, displacement metrics (ADE, FDE) between full and reduced plans are sub-decimeter, confirming near-identical closed-loop behavior.
• Sensitivity to Parameters: Empirical tuning of certification tolerances and sparsity weight demonstrates that more aggressive certification increases prune rates, but feasibility is retained even under parameter variation, showcasing robustness.

Practical and Theoretical Implications

SHIELD makes several impactful theoretical and practical advances:

• Scalability with Guarantee: It is among the first frameworks to achieve order-of-magnitude speedups in MPC computation while providing deterministic (not empirical) safety guarantees in the presence of dynamic, time-varying constraints and distribution shifts.
• Applicability: The framework is directly applicable to general convex programs with $l_1$-regularization, strong convexity, and Slater's condition, encompassing a large class of MPC and optimal control problems in robotics, automated driving, power systems, and process control.
• Extensibility: While the theoretical guarantees presently require strong convexity and convex constraints, the certificate-based approach suggests possible future extension to non-convex, mixed-integer, or sequential convexification regimes.
• Robustness: Safety does not degrade under classifier or dual inference failure; conservatism (i.e., failure to prune) is the only consequence, making the method highly robust for safety-critical applications.

Future Directions

Natural extensions for SHIELD include:

• Broader Problem Classes: Applicability to domains including network flow, sparse portfolio optimization, and sensor selection as future validation targets.
• Beyond Convexity: Investigation of relaxation-based certificates or sequential convex programming to extend safety guarantees to nonconvex or hybrid MPC.
• Adaptive Certification: Online or instance-adaptive tuning of certification tolerances ($\epsilon$), safety margin ($\delta$), and sparsity regularization ($\lambda$) to optimize the tradeoff between conservatism and computation time for varying scenario difficulty.
• Classifier Generalization: Further work to evaluate classifier-induced screening under high rates of distribution or task shift, and the use of meta-learning or continual learning strategies for classifier adaptation.

Conclusion

SHIELD delivers a formal framework for certifiable, aggressive dimensionality and constraint reduction in convex optimal control problems, with demonstrated utility for real-time stochastic MPC under high-dimensional, multi-agent uncertainty. Its dual certificate-based screening guarantees safety and feasibility, and its data-driven, hierarchical approach unlocks order-of-magnitude computation accelerations. The approach bridges optimization theory and learning-based acceleration, charting a rigorous path toward scalable, certifiable real-time solvers for complex, safety-critical autonomous systems.