# Accelerated first-order methods for a class of semidefinite programs

arXiv: https://arxiv.org/abs/2206.00224 · HTML: https://mlsanity.com/paper/2206.00224/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2022.

## Built on by (most cited first, 2 of 2)

- [Provably Faster Gradient Descent via Long Steps](https://mlsanity.com/paper/2307.06324/) (2023)
- [A Randomized Algorithm for Sparse PCA based on the Basic SDP Relaxation](https://mlsanity.com/paper/2507.09148/) (2025)

## Abstract (arXiv)

This paper introduces a new storage-optimal first-order method (FOM), CertSDP, for solving a special class of semidefinite programs (SDPs) to high accuracy. The class of SDPs that we consider, the exact QMP-like SDPs, is characterized by low-rank solutions, a priori knowledge of the restriction of the SDP solution to a small subspace, and standard regularity assumptions such as strict complementarity. Crucially, we show how to use a certificate of strict complementarity to construct a low-dimensional strongly convex minimax problem whose optimizer coincides with a factorization of the SDP optimizer. From an algorithmic standpoint, we show how to construct the necessary certificate and how to solve the minimax problem efficiently. We accompany our theoretical results with preliminary numerical experiments suggesting that CertSDP significantly outperforms current state-of-the-art methods on large sparse exact QMP-like SDPs.