# Stability Verification of Neural Network Controllers using Mixed-Integer Programming

arXiv: https://arxiv.org/abs/2206.13374 · HTML: https://mlsanity.com/paper/2206.13374/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 61 references that are in the map.
- First posted on arXiv 27 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [Learning an Approximate Model Predictive Controller with Guarantees](https://mlsanity.com/paper/1806.04167/) (2018)
- [Scaling Polyhedral Neural Network Verification on GPUs](https://mlsanity.com/paper/2007.10868/) (2020)
- [Learning Lyapunov Functions for Hybrid Systems](https://mlsanity.com/paper/2012.12015/) (2020)
- [Whole-Body Nonlinear Model Predictive Control Through Contacts for Quadrupeds](https://mlsanity.com/paper/1712.02889/) (2017)
- [Near-Optimal Rapid MPC using Neural Networks: A Primal-Dual Policy Learning Framework](https://mlsanity.com/paper/1912.04744/) (2019)
- [A Neural Network Architecture to Learn Explicit MPC Controllers from Data](https://mlsanity.com/paper/1911.10789/) (2019)
- [Reliably-stabilizing piecewise-affine neural network controllers](https://mlsanity.com/paper/2111.07183/) (2021)
- [Stability and feasibility of neural network-based controllers via output range analysis](https://mlsanity.com/paper/2004.00521/) (2020)
- [Stability analysis of Model Predictive Controllers using Mixed Integer Linear Programming](https://mlsanity.com/paper/1604.00863/) (2016)

## Built on by (most cited first, 11 of 11)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Synthesis of Model Predictive Control and Reinforcement Learning: Survey and Classification](https://mlsanity.com/paper/2502.02133/) (2025)
- [Imitation Learning from Nonlinear MPC via the Exact Q-Loss and its Gauss-Newton Approximation](https://mlsanity.com/paper/2304.01782/) (2023)
- [MPC-Inspired Reinforcement Learning for Verifiable Model-Free Control](https://mlsanity.com/paper/2312.05332/) (2023)
- [YANNs: Y-wise Affine Neural Networks for Exact and Efficient Representations of Piecewise Linear Functions](https://mlsanity.com/paper/2505.07054/) (2025)
- [Correctness Verification of Neural Networks Approximating Differential Equations](https://mlsanity.com/paper/2402.07621/) (2024)
- [Mapping back and forth between model predictive control and neural networks](https://mlsanity.com/paper/2404.12030/) (2024)
- [Stochastic Sample Approximations of (Local) Moduli of Continuity](https://mlsanity.com/paper/2509.15368/) (2025)
- [ReLU Networks for Model Predictive Control: Network Complexity and Performance Guarantees](https://mlsanity.com/paper/2601.16764/) (2026)
- [Toward AI Trustworthiness: Finding Analytically Proven Forward-Invariant Sets for AI-Controlled Systems](https://mlsanity.com/paper/2610.05689/) (2026)

## Abstract (arXiv)

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity.