  Orthogonality constrained optimization is widely used in applications from
science and engineering. Due to the nonconvex orthogonality constraints, many
numerical algorithms often can hardly achieve the global optimality. We aim at
establishing an efficient scheme for finding global minimizers under one or
more orthogonality constraints. The main concept is based on noisy gradient
flow constructed from stochastic differential equations (SDE) on the Stiefel
manifold, the differential geometric characterization of orthogonality
constraints. We derive an explicit representation of SDE on the Stiefel
manifold endowed with a canonical metric and propose a numerically efficient
scheme to simulate this SDE based on Cayley transformation with theoretical
convergence guarantee. The convergence to global optimizers is proved under
second-order continuity. The effectiveness and efficiency of the proposed
algorithms are demonstrated on a variety of problems including homogeneous
polynomial optimization, computation of stability number, and 3D structure
determination from Common Lines in Cryo-EM.
