A Dual Riemannian ADMM Algorithm for Low-Rank SDPs with Unit Diagonal

Abstract: This paper proposes a dual Riemannian alternating direction method of multipliers (ADMM) for solving low-rank semidefinite programs with unit diagonal constraints. We recast the ADMM subproblem as a Riemannian optimization problem over the oblique manifold by performing the Burer-Monteiro factorization. Global convergence of the algorithm is established assuming that the subproblem is solved to certain optimality. Numerical experiments demonstrate the excellent performance of the algorithm. It outperforms, by a significant margin, a few advanced SDP solvers (MOSEK, COPT, SDPNAL+, ManiSDP) in terms of accuracy, efficiency, and scalability on second-order SDP relaxations of dense and sparse binary quadratic programs.