  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.
