HardNet++: Nonlinear Constraint Enforcement in Neural Networks

2026-04-21

AI Summary

Neural networks often struggle to strictly satisfy nonlinear constraints during inference, which hinders their deployment in safety-critical applications. This work proposes HardNet++, the first method capable of enforcing hard satisfaction of general nonlinear equality and inequality constraints, overcoming the limitation of existing approaches that are restricted to specific constraint forms. By integrating damped local linearization, differentiable projection layers, and end-to-end training, HardNet++ guarantees constraint compliance simultaneously during both training and inference. Evaluated on model predictive control tasks, HardNet++ achieves high-precision constraint adherence while preserving solution optimality.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization
• Machine Learning: Hardware-aware ML

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Enforcing constraint satisfaction in neural network outputs is critical for safety, reliability, and physical fidelity in many control and decision-making applications. While soft-constrained methods penalize constraint violations during training, they do not guarantee constraint adherence during inference. Other approaches guarantee constraint satisfaction via specific parameterizations or a projection layer, but are tailored to specific forms (e.g., linear constraints), limiting their utility in other general problem settings. Many real-world problems of interest are nonlinear, motivating the development of methods that can enforce general nonlinear constraints. To this end, we introduce HardNet++, a constraint-enforcement method that simultaneously satisfies linear and nonlinear equality and inequality constraints. Our approach iteratively adjusts the network output via damped local linearizations. Each iteration is differentiable, admitting an end-to-end training framework, where the constraint satisfaction layer is active during training. We show that under certain regularity conditions, this procedure can enforce nonlinear constraint satisfaction to arbitrary tolerance. Finally, we demonstrate tight constraint adherence without loss of optimality in a learning-for-optimization context, where we apply this method to a model predictive control problem with nonlinear state constraints.

Problem

• nonlinear constraints
• constraint satisfaction
• neural networks
• hard constraints
• safety-critical systems

Innovation

• nonlinear constraints
• constraint enforcement
• differentiable optimization
• HardNet++
• model predictive control