# Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting

arXiv: https://arxiv.org/abs/1810.05231 · HTML: https://mlsanity.com/paper/1810.05231/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2018.

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

- [Provably Faster Gradient Descent via Long Steps](https://mlsanity.com/paper/2307.06324/) (2023)
- [Fast, Scalable, Warm-Start Semidefinite Programming with Spectral Bundling and Sketching](https://mlsanity.com/paper/2312.11801/) (2023)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)

## Abstract (arXiv)

In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The proposed methodology, based on the primal-dual hybrid gradient method, allows the presence of linear inequalities without the need of adding extra slack variables and avoids solving a linear system at each iteration. More importantly, it does simultaneously compute the dual variables associated with the linear constraints. The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the low-rank property inherent to several semidefinite programming problems. This proposed modification is the key element that allows the operator splitting method to efficiently scale to larger instances. Convergence guarantees are presented along with an intuitive interpretation of the algorithm. Additionally, an open source semidefinite programming solver, called ProxSDP, is made available and implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.