This paper introduces HALLaR, a new first-order method for solving large-scale semidefinite programs (SDPs) with bounded domain. HALLaR is an inexact augmented Lagrangian (AL) method where the AL subproblems are solved by a novel hybrid low-rank (HLR) method. The recipe behind HLR is based on two key ingredients: 1) an adaptive inexact proximal point method with inner acceleration; 2) Frank-Wolfe steps to escape from spurious local stationary points. In contrast to the low-rank method of Burer and Monteiro, HALLaR finds a near-optimal solution (with provable complexity bounds) of SDP instances satisfying strong duality. Computational results comparing HALLaR to state-of-the-art solvers on several large SDP instances arising from maximum stable set, phase retrieval, and matrix completion show that the former finds higher accurate solutions in substantially less CPU time than the latter ones. For example, in less than 20 minutes, HALLaR can solve a maximum stable set SDP instance with dimension pair $(n,m)\approx (10^6,10^7)$ within $10^{-5}$ relative precision.

翻译：本文提出HALLaR，一种求解有界域大规模半定规划（SDP）的新型一阶方法。HALLaR是非精确增广拉格朗日（AL）方法，其中AL子问题通过新颖的混合低秩（HLR）方法求解。HLR的构建基于两个关键要素：1）带内部加速的自适应非精确近端点方法；2）用于逃离虚假局部平稳点的Frank-Wolfe步骤。与Burer和Monteiro的低秩方法不同，HALLaR能为满足强对偶性的SDP实例找到近优解（具有可证明的复杂度界）。将HALLaR与最先进的求解器在多个大规模SDP实例（源自最大稳定集、相位恢复和矩阵补全）上进行计算比较，结果表明前者能以显著更少的CPU时间获得更高精度的解。例如，在不到20分钟内，HALLaR可求解维度对$(n,m)\approx (10^6,10^7)$的最大稳定集SDP实例，并达到$10^{-5}$的相对精度。