An Iterative Eigensolver for Rank-Constrained Semidefinite Programming

Rank-constrained semidefinite programming (SDP) arises naturally in various applications such as max-cut, angular (phase) synchronization, and rigid registration. Based on the alternating direction method of multipliers, we develop an iterative solver for this nonconvex form of SDP, where the dominant cost per iteration is the partial eigendecomposition of a symmetric matrix. We prove that if the iterates converge, then they do so to a KKT point of the SDP. In the context of rigid registration, we perform several numerical experiments to study the convergence behavior of the solver and its registration accuracy. As an application, we use the solver for wireless sensor network localization from range measurements. The resulting algorithm is shown to be competitive with existing optimization methods for sensor localization in terms of speed and accuracy.