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.

翻译：本文提出一种新的存储最优一阶方法（FOM）——CertSDP，用于高精度求解特定类别的半定规划（SDP）问题。所考虑的精确类QMP半定规划具有以下特征：低秩解、先验已知半定规划解在某一小子空间上的限制，以及严格互补性等标准正则性假设。关键地，我们展示了如何利用严格互补性证书构造一个低维强凸极小极大问题，其优化器与半定规划优化器的因子分解一致。从算法角度，我们阐明了必要证书的构造方法以及该极小极大问题的高效求解途径。伴随理论结果，初步数值实验表明，在大型稀疏精确类QMP半定规划问题上，CertSDP显著优于当前最先进方法。