Accelerated first-order methods for a class of semidefinite programs

TL;DR

This paper presents CertSDP, a storage-efficient first-order method tailored for a specific class of low-rank semidefinite programs, leveraging certificates of strict complementarity for high-accuracy solutions.

Contribution

The paper introduces CertSDP, a novel first-order method that exploits problem structure and certificates of strict complementarity to solve certain SDPs more efficiently.

Findings

CertSDP outperforms existing methods on large sparse SDPs.

The method achieves high accuracy with low storage requirements.

Numerical experiments validate the theoretical advantages.

Abstract

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.