# Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example

arXiv: https://arxiv.org/abs/1510.04330 · HTML: https://mlsanity.com/paper/1510.04330/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2015.

## Built on by (most cited first, 3 of 3)

- [Unsupervised Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/2210.09277/) (2022)
- [Reinforcement Learning to Initialize Newton-Raphson for AC Power Flow with Quantum Annealing-Based Environment Updates](https://mlsanity.com/paper/2511.20237/) (2025)
- [Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning](https://mlsanity.com/paper/2609.36310/) (2026)

## Abstract (arXiv)

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. We use the example OPF problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related "mixed semidefinite/second-order cone programming" hierarchy.