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

arXiv:2604.05374v1 Announce Type: new 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 c

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Sunbochen Tang, Andrea Goertzen, Navid Azizan · · 1 min read · 25 views
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arXiv:2604.05374v1 Announce Type: new 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.

Executive Summary

The paper introduces LMI-Net, a novel differentiable projection layer designed to enforce Linear Matrix Inequality (LMI) constraints in neural networks, addressing a critical gap in learning-based control systems where formal guarantees are often sacrificed for flexibility. By leveraging Douglas-Rachford splitting and implicit differentiation, LMI-Net ensures that neural network outputs satisfy hard LMI constraints, thereby preserving formal certifications for stability, robustness, and invariance. Empirical evaluations demonstrate that LMI-Net significantly outperforms soft-constrained models in feasibility under distribution shifts while maintaining computational efficiency. This work bridges the divide between semidefinite programming-based certification and modern machine learning techniques, offering a robust framework for synthesizing reliable control systems.

Key Points

  • ▸ LMI constraints are essential for formal guarantees in control systems but are often violated in learning-based approaches due to soft constraints or approximations.
  • ▸ LMI-Net enforces LMI constraints by construction through a differentiable projection layer that leverages Douglas-Rachford splitting for forward passes and implicit differentiation for backward propagation.
  • ▸ Theoretical guarantees ensure convergence to feasible solutions, and empirical results show LMI-Net improves feasibility over soft-constrained models while retaining fast inference speeds.
  • ▸ Applications include invariant ellipsoid synthesis and joint controller-and-certificate design for disturbed linear systems.

Merits

Theoretical Rigor

The paper provides robust theoretical guarantees, including convergence to feasible LMI solutions and rigorous treatment of the projection layer via Douglas-Rachford splitting and implicit differentiation.

Practical Relevance

LMI-Net addresses a pressing need in control systems by bridging the gap between formal certification (via LMIs) and modern learning-based methods, ensuring hard constraints are preserved in neural network outputs.

Modularity and Efficiency

The approach is modular, allowing integration with generic neural networks, and computationally efficient due to its forward pass via Douglas-Rachford splitting and backward pass via implicit differentiation.

Demerits

Complexity of Implementation

The reliance on Douglas-Rachford splitting and implicit differentiation may introduce non-trivial implementation challenges, particularly for practitioners unfamiliar with convex optimization or operator splitting methods.

Limited Generalization to Non-LMI Constraints

While the method excels for LMI constraints, its applicability to other types of hard constraints (e.g., nonlinear inequalities) remains unexplored, potentially limiting its broader utility.

Dependence on Problem Structure

The effectiveness of LMI-Net may depend on the specific structure of the LMI problem, such as the dimensionality of the matrices involved or the convexity of the feasible set, which could pose challenges for highly non-convex or large-scale problems.

Expert Commentary

LMI-Net represents a significant advancement in the intersection of machine learning and control theory, addressing a longstanding challenge in enforcing hard constraints in neural networks. The paper’s theoretical contributions are particularly noteworthy, as they provide rigorous guarantees for convergence and feasibility, which are often lacking in learning-based methods for control. The use of Douglas-Rachford splitting and implicit differentiation is innovative, offering a computationally efficient way to integrate LMI constraints without resorting to approximations that compromise formal guarantees. However, the practical deployment of LMI-Net may face hurdles, particularly in terms of implementation complexity and scalability for large-scale problems. The paper also raises important questions about the broader applicability of this approach to other types of constraints, which could be an avenue for future research. Overall, LMI-Net sets a new benchmark for reliable control system design using neural networks and paves the way for further exploration at the intersection of optimization and learning.

Recommendations

  • ✓ Researchers should explore extensions of LMI-Net to other types of hard constraints beyond LMIs, such as nonlinear inequalities or mixed-integer constraints, to broaden its applicability.
  • ✓ Practitioners should conduct further empirical studies to validate the scalability and robustness of LMI-Net across diverse control systems, including real-world applications with high-dimensional state spaces.
  • ✓ Educational and professional development programs should incorporate training on differentiable optimization layers and operator splitting methods to equip engineers and researchers with the skills needed to implement and extend LMI-Net-like approaches.
  • ✓ Policy-makers in safety-critical industries should consider developing guidelines or standards that encourage the adoption of methods like LMI-Net to ensure formal certification in learning-based control systems.

Sources

Original: arXiv - cs.LG

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