HardNet++: Nonlinear Constraint Enforcement in Neural Networks

TL;DR

HardNet++ is a novel neural network method that enforces both linear and nonlinear constraints during training and inference, ensuring safety and physical fidelity in control applications.

Contribution

It introduces an iterative, differentiable constraint enforcement layer that guarantees nonlinear constraint satisfaction, extending beyond linear or soft methods.

Findings

Successfully enforces nonlinear constraints with arbitrary tolerance

Maintains optimality while ensuring tight constraint adherence

Effective in a model predictive control scenario with nonlinear state constraints

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.