# Convex Relaxation of Optimal Power Flow, Part II: Exactness

arXiv: https://arxiv.org/abs/1405.0814 · HTML: https://mlsanity.com/paper/1405.0814/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2014.

## Built on by (most cited first, 10 of 10)

- [Learning to Solve the AC-OPF using Sensitivity-Informed Deep Neural Networks](https://mlsanity.com/paper/2103.14779/) (2021)
- [Unsupervised Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/2210.09277/) (2022)
- [Dual Conic Proxies for AC Optimal Power Flow](https://mlsanity.com/paper/2310.02969/) (2023)
- [Controlling Smart Inverters using Proxies: A Chance-Constrained DNN-based Approach](https://mlsanity.com/paper/2105.00429/) (2021)
- [Unsupervised Deep Learning for AC Optimal Power Flow via Lagrangian Duality](https://mlsanity.com/paper/2212.03977/) (2022)
- [Optimal Power Flow Based on Physical-Model-Integrated Neural Network with Worth-Learning Data Generation](https://mlsanity.com/paper/2301.03766/) (2023)
- [Physics-Informed Graph Neural Networks for Robust AC-Optimal Power Flow](https://mlsanity.com/paper/2410.04818/) (2024)
- [Dynamic Domain Adaptation-Driven Physics-Informed Graph Representation Learning for AC-OPF](https://mlsanity.com/paper/2506.00478/) (2025)
- [Tree-Preconditioned Differentiable Optimization and Axioms as Layers](https://mlsanity.com/paper/2601.06036/) (2025)
- [Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning](https://mlsanity.com/paper/2609.36310/) (2026)

## Abstract (arXiv)

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.