This paper introduces a new storage-optimal first-order method, 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. Our algorithms for strongly convex minimax problems with inexact prox maps may be of independent interest. 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.

本文介绍了一种新的存储优化一阶方法 CertSDP，用于高精度求解一类特殊的半定规划 (SDP)。我们考虑的 SDP 类别，即精确的类似 QMP 的 SDP，其特征是低秩解、SDP 解限制在小子空间的先验知识以及标准正则性假设（例如严格互补性）。至关重要的是，我们展示了如何使用严格互补性证书来构造低维强凸极小极大问题，其优化器与 SDP 优化器的因式分解一致。从算法的角度来看，我们展示了如何构建必要的证书以及如何有效地解决极小极大问题。我们针对具有不精确近似图的强凸极小极大问题的算法可能具有独立的兴趣。我们将理论结果与初步数值实验相结合，表明 CertSDP 在大型稀疏精确 QMP 类 SDP 上显着优于当前最先进的方法。