LMI-Net: Linear Matrix Inequality--Constrained Neural Networks via Differentiable Projection Layers

TL;DR

LMI-Net introduces a differentiable projection layer that enforces linear matrix inequality constraints in neural networks, ensuring formal guarantees for control systems while maintaining efficiency.

Contribution

The paper presents a novel modular projection layer that guarantees LMI constraint satisfaction in neural networks using implicit differentiation and Douglas-Rachford splitting.

Findings

LMI-Net improves feasibility over soft-constrained models under distribution shift.

The approach maintains fast inference speed.

It effectively bridges semidefinite programming with modern learning techniques.

Abstract

Linear matrix inequalities (LMIs) have played a central role in certifying stability, robustness, and forward invariance of dynamical systems. Despite rapid development in learning-based methods for control design and certificate synthesis, existing approaches often fail to preserve the hard matrix inequality constraints required for formal guarantees. We propose LMI-Net, an efficient and modular differentiable projection layer that enforces LMI constraints by construction. Our approach lifts the set defined by LMI constraints into the intersection of an affine equality constraint and the positive semidefinite cone, performs the forward pass via Douglas-Rachford splitting, and supports efficient backward propagation through implicit differentiation. We establish theoretical guarantees that the projection layer converges to a feasible point, certifying that LMI-Net transforms a generic neural network into a reliable model satisfying LMI constraints. Evaluated on experiments including invariant ellipsoid synthesis and joint controller-and-certificate design for a family of disturbed linear systems, LMI-Net substantially improves feasibility over soft-constrained models under distribution shift while retaining fast inference speed, bridging semidefinite-program-based certification and modern learning techniques.