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：本文介绍HALLaR，一种求解有界域大规模半定规划问题的新一阶方法。HALLaR是一种不精确增广拉格朗日方法，其中增广拉格朗日子问题通过新颖的混合低秩方法求解。HLR方法的核心包含两个关键要素：1) 带有内加速的自适应不精确邻近点方法；2) 用于逃离虚假局部驻点的Frank-Wolfe步。与Burer-Monteiro的低秩方法相比，HALLaR能为满足强对偶性的半定规划实例找到（具有可证明复杂度界的）近优解。将HALLaR与多个先进求解器在最大稳定集、相位恢复和矩阵补全等大规模半定规划实例上进行的计算对比表明，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即可求解维度对$(n,m)\approx (10^6,10^7)$的最大稳定集半定规划问题，达到$10^{-5}$的相对精度。