# Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimization

arXiv: https://arxiv.org/abs/2012.05981 · HTML: https://mlsanity.com/paper/2012.05981/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2020.

## Built on by (most cited first, 18 of 18)

- [Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularization](https://mlsanity.com/paper/2310.00116/) (2023)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
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- [Compositional Curvature Bounds for Deep Neural Networks](https://mlsanity.com/paper/2406.05119/) (2024)
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- [A Neurosymbolic Approach to the Verification of Temporal Logic Properties of Learning enabled Control Systems](https://mlsanity.com/paper/2303.05394/) (2023)
- [Provable Bounds on the Hessian of Neural Networks: Derivative-Preserving Reachability Analysis](https://mlsanity.com/paper/2406.04476/) (2024)
- [Identification For Control Based on Neural Networks: Approximately Linearizable Models](https://mlsanity.com/paper/2409.15858/) (2024)
- [Performance Bounds for Neural Network Estimators: Applications in Fault Detection](https://mlsanity.com/paper/2103.12141/) (2021)
- [A Domain-Theoretic Framework for Robustness Analysis of Neural Networks](https://mlsanity.com/paper/2203.00295/) (2022)
- [Scalable Incremental Robustness Analysis of Neural Network Feedback Systems](https://mlsanity.com/paper/2609.22576/) (2026)
- [AC Power Flow Contingency Analysis Using a Single Deep Neural Network](https://mlsanity.com/paper/2609.30859/) (2026)
- [CalCErt: Bin-wise Certification of Confidence Calibration in Medical Image Classification](https://mlsanity.com/paper/2610.03142/) (2026)

## Abstract (arXiv)

Abstracting neural networks with constraints they impose on their inputs and outputs can be very useful in the analysis of neural network classifiers and to derive optimization-based algorithms for certification of stability and robustness of feedback systems involving neural networks. In this paper, we propose a convex program, in the form of a Linear Matrix Inequality (LMI), to certify incremental quadratic constraints on the map of neural networks over a region of interest. These certificates can capture several useful properties such as (local) Lipschitz continuity, one-sided Lipschitz continuity, invertibility, and contraction. We illustrate the utility of our approach in two different settings. First, we develop a semidefinite program to compute guaranteed and sharp upper bounds on the local Lipschitz constant of neural networks and illustrate the results on random networks as well as networks trained on MNIST. Second, we consider a linear time-invariant system in feedback with an approximate model predictive controller parameterized by a neural network. We then turn the stability analysis into a semidefinite feasibility program and estimate an ellipsoidal invariant set for the closed-loop system.