# ADMM for the SDP relaxation of the QAP

arXiv: https://arxiv.org/abs/1512.05448 · HTML: https://mlsanity.com/paper/1512.05448/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2015.

## Built on by (most cited first, 4 of 4)

- [A Semidefinite Programming-Based Branch-and-Cut Algorithm for Biclustering](https://mlsanity.com/paper/2403.11351/) (2024)
- [Convex Joint Graph Matching and Clustering via Semidefinite Relaxations](https://mlsanity.com/paper/2110.11335/) (2021)
- [On the Expressive Power of GNNs to Solve Linear SDPs](https://mlsanity.com/paper/2604.27786/) (2026)
- [Robust Correlation Clustering with Asymmetric Noise](https://mlsanity.com/paper/2110.08385/) (2021)

## Abstract (arXiv)

The semidefinite programming (SDP) relaxation has proven to be extremely strong for many hard discrete optimization problems. This is in particular true for the quadratic assignment problem (QAP), arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that arise in efficiently solving the SDP relaxation, e.g.,~increased dimension; inefficiency of the current primal-dual interior point solvers in terms of both time and accuracy; and difficulty and high expense in adding cutting plane constraints. We propose using the alternating direction method of multipliers (ADMM) to solve the SDP relaxation. This first order approach allows for inexpensive iterations, a method of cheaply obtaining low rank solutions, as well a trivial way of adding cutting plane inequalities. When compared to current approaches and current best available bounds we obtain remarkable robustness, efficiency and improved bounds.