# An Optimal-Storage Approach to Semidefinite Programming using Approximate Complementarity

arXiv: https://arxiv.org/abs/1902.03373 · HTML: https://mlsanity.com/paper/1902.03373/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2019.

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

- [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)
- [Learning Transition Operators From Sparse Space-Time Samples](https://mlsanity.com/paper/2212.00746/) (2022)
- [Suboptimality bounds for trace-bounded SDPs enable a faster and scalable low-rank SDP solver SDPLR+](https://mlsanity.com/paper/2406.10407/) (2024)

## Abstract (arXiv)

This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the eigenspace with small eigenvalues of the dual slack matrix. For weakly constrained SDPs, this eigenspace has very low dimension, so this observation significantly reduces the search space for the primal solution. This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the eigenspace with small eigenvalues of the dual slack matrix; (3) solve the compressed primal SDP. The paper also provides numerical experiments showing that this approach is successful for a range of interesting large-scale SDPs.