# Lagrangian Duality for Constrained Deep Learning

arXiv: https://arxiv.org/abs/2001.09394 · HTML: https://mlsanity.com/paper/2001.09394/

## Key facts

- Cited by 74 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jan 2020.

## Built on by (most cited first, 30 of 74)

- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Differential Privacy and Fairness in Decisions and Learning Tasks: A Survey](https://mlsanity.com/paper/2202.08187/) (2022)
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- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Enforcing fairness in private federated learning via the modified method of differential multipliers](https://mlsanity.com/paper/2109.08604/) (2021)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Controlled Sparsity via Constrained Optimization or: How I Learned to Stop Tuning Penalties and Love Constraints](https://mlsanity.com/paper/2208.04425/) (2022)
- [A Lagrangian Duality Approach to Active Learning](https://mlsanity.com/paper/2202.04108/) (2022)
- [Learning Hard Optimization Problems: A Data Generation Perspective](https://mlsanity.com/paper/2106.02601/) (2021)
- [FSNet: Feasibility-Seeking Neural Network for Constrained Optimization with Guarantees](https://mlsanity.com/paper/2506.00362/) (2025)
- [Fast Approximations for Job Shop Scheduling: A Lagrangian Dual Deep Learning Method](https://mlsanity.com/paper/2110.06365/) (2021)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [Competitive Physics Informed Networks](https://mlsanity.com/paper/2204.11144/) (2022)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [End-to-end Learning for Fair Ranking Systems](https://mlsanity.com/paper/2111.10723/) (2021)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [Constrained Discrete Diffusion](https://mlsanity.com/paper/2503.09790/) (2025)
- [Controlling Neural Networks with Rule Representations](https://mlsanity.com/paper/2106.07804/) (2021)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [Resilient Constrained Learning](https://mlsanity.com/paper/2306.02426/) (2023)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [PGLearn -- An Open-Source Learning Toolkit for Optimal Power Flow](https://mlsanity.com/paper/2505.22825/) (2025)
- [Fairness-Accuracy Trade-Offs: A Causal Perspective](https://mlsanity.com/paper/2405.15443/) (2024)
- [Neuro-Symbolic Generative Diffusion Models for Physically Grounded, Robust, and Safe Generation](https://mlsanity.com/paper/2506.01121/) (2025)
- [Near-Optimal Solutions of Constrained Learning Problems](https://mlsanity.com/paper/2403.11844/) (2024)
- [SF-PATE: Scalable, Fair, and Private Aggregation of Teacher Ensembles](https://mlsanity.com/paper/2204.05157/) (2022)
- [Learning with Explanation Constraints](https://mlsanity.com/paper/2303.14496/) (2023)
- [Generalized Disparate Impact for Configurable Fairness Solutions in ML](https://mlsanity.com/paper/2305.18504/) (2023)
- [Guaranteed Conformance of Neurosymbolic Models to Natural Constraints](https://mlsanity.com/paper/2212.01346/) (2022)

## Abstract (arXiv)

This paper explores the potential of Lagrangian duality for learning applications that feature complex constraints. Such constraints arise in many science and engineering domains, where the task amounts to learning optimization problems which must be solved repeatedly and include hard physical and operational constraints. The paper also considers applications where the learning task must enforce constraints on the predictor itself, either because they are natural properties of the function to learn or because it is desirable from a societal standpoint to impose them. This paper demonstrates experimentally that Lagrangian duality brings significant benefits for these applications. In energy domains, the combination of Lagrangian duality and deep learning can be used to obtain state-of-the-art results to predict optimal power flows, in energy systems, and optimal compressor settings, in gas networks. In transprecision computing, Lagrangian duality can complement deep learning to impose monotonicity constraints on the predictor without sacrificing accuracy. Finally, Lagrangian duality can be used to enforce fairness constraints on a predictor and obtain state-of-the-art results when minimizing disparate treatments.