# Learning to Solve the AC Optimal Power Flow via a Lagrangian Approach

arXiv: https://arxiv.org/abs/2110.01653 · HTML: https://mlsanity.com/paper/2110.01653/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Learning Optimal Solutions for Extremely Fast AC Optimal Power Flow](https://mlsanity.com/paper/1910.01213/) (2019)
- [Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/1910.09658/) (2019)
- [DeepOPF: Deep Neural Network for DC Optimal Power Flow](https://mlsanity.com/paper/1905.04479/) (2019)
- [Learning to Solve the AC-OPF using Sensitivity-Informed Deep Neural Networks](https://mlsanity.com/paper/2103.14779/) (2021)
- [Machine Learning for AC Optimal Power Flow](https://mlsanity.com/paper/1910.08842/) (2019)
- [Emulating AC OPF solvers for Obtaining Sub-second Feasible, Near-Optimal Solutions](https://mlsanity.com/paper/2012.10031/) (2020)
- [Learning to Solve AC Optimal Power Flow by Differentiating through Holomorphic Embeddings](https://mlsanity.com/paper/2012.09622/) (2020)
- [An Iterative Approach to Improving Solution Quality for AC Optimal Power Flow Problems](https://mlsanity.com/paper/2109.06356/) (2021)

## Built on by (most cited first, 5 of 5)

- [QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints](https://mlsanity.com/paper/2401.06820/) (2024)
- [Physics-Informed Graph Neural Networks for Robust AC-Optimal Power Flow](https://mlsanity.com/paper/2410.04818/) (2024)
- [Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement](https://mlsanity.com/paper/2502.10330/) (2025)
- [Enriching Neural Network Training Dataset to Improve Worst-Case Performance Guarantees](https://mlsanity.com/paper/2303.13228/) (2023)
- [Homotopy-Guided Self-Supervised Learning of Parametric Solutions for AC Optimal Power Flow](https://mlsanity.com/paper/2511.11677/) (2025)

## Abstract (arXiv)

Using deep neural networks to predict the solutions of AC optimal power flow (ACOPF) problems has been an active direction of research. However, because the ACOPF is nonconvex, it is difficult to construct a good data set that contains mostly globally optimal solutions. To overcome the challenge that the training data may contain suboptimal solutions, we propose a Lagrangian-based approach. First, we use a neural network to learn the dual variables of the ACOPF problem. Then we use a second neural network to predict solutions of the partial Lagrangian from the predicted dual variables. Since the partial Lagrangian has a much better optimization landscape, we use the predicted solutions from the neural network as a warm start for the ACOPF problem. Using standard and modified IEEE 22-bus, 39-bus, and 118-bus networks, we show that our approach is able to obtain the globally optimal cost even when the training data is mostly comprised of suboptimal solutions.