DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables

2026-06-03

AI Summary

This work addresses the challenge of efficiently enforcing a large number of coupled nonlinear inequality constraints in neural networks, a task for which existing hard-constraint methods are either limited by constraint structure or incur prohibitive computational costs. The authors propose DiffSlack, an end-to-end differentiable projection layer that introduces learnable slack variables to convert inequalities into equalities and employs a damped Gauss–Newton method to achieve accurate projection. Innovatively, the slack variables are jointly optimized as part of the network output, and a two-stage curriculum training strategy is designed to enhance both constraint satisfaction rates and training stability. Evaluated on a vehicle routing task involving 200 nonlinear constraints, DiffSlack significantly outperforms baseline approaches, demonstrating feasible trajectories and high constraint satisfaction in both CARLA simulations and real-world vehicle experiments.

Abstract

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

Problem

• nonlinear inequality constraints
• neural networks
• constraint satisfaction
• hard constraints
• feasible manifold

Innovation

• differentiable projection
• learnable slack variables
• nonlinear inequality constraints
• Gauss-Newton optimization
• constraint satisfaction