  Recently, the proximal Newton-type method and its variants have been
generalized to solve composite optimization problems over the Stiefel manifold
whose objective function is the summation of a smooth function and a nonsmooth
function. In this paper, we propose an adaptive quadratically regularized
proximal quasi-Newton method, named ARPQN, to solve this class of problems.
Under some mild assumptions, the global convergence, the local linear
convergence rate and the iteration complexity of ARPQN are established.
Numerical experiments and comparisons with other state-of-the-art methods
indicate that ARPQN is very promising. We also propose an adaptive
quadratically regularized proximal Newton method, named ARPN. It is shown the
ARPN method has a local superlinear convergence rate under certain reasonable
assumptions, which demonstrates attractive convergence properties of
regularized proximal Newton methods.
