# DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables

arXiv: https://arxiv.org/abs/2606.05247 · HTML: https://mlsanity.com/paper/2606.05247/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CARLA: An Open Urban Driving Simulator](https://mlsanity.com/paper/1711.03938/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [A predictive safety filter for learning-based control of constrained nonlinear dynamical systems](https://mlsanity.com/paper/1812.05506/) (2018)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [Lagrangian Duality for Constrained Deep Learning](https://mlsanity.com/paper/2001.09394/) (2020)
- [Self-Supervised Primal-Dual Learning for Constrained Optimization](https://mlsanity.com/paper/2208.09046/) (2022)
- [Informed RRT*: Optimal Sampling-based Path Planning Focused via Direct Sampling of an Admissible Ellipsoidal Heuristic](https://mlsanity.com/paper/1404.2334/) (2014)
- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)
- [Learning to Solve Optimization Problems with Hard Linear Constraints](https://mlsanity.com/paper/2208.10611/) (2022)
- [Implicit differentiation for fast hyperparameter selection in non-smooth convex learning](https://mlsanity.com/paper/2105.01637/) (2021)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [LinSATNet: The Positive Linear Satisfiability Neural Networks](https://mlsanity.com/paper/2407.13917/) (2024)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [PythonRobotics: a Python code collection of robotics algorithms](https://mlsanity.com/paper/1808.10703/) (2018)
- [GLinSAT: The General Linear Satisfiability Neural Network Layer By Accelerated Gradient Descent](https://mlsanity.com/paper/2409.17500/) (2024)
- [Physics-informed neural particle flow for the Bayesian update step](https://mlsanity.com/paper/2602.23089/) (2026)

## Built on by (most cited first, 1 of 1)

- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)

## Abstract (arXiv)

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.