  In this paper, we propose a cubic-regularized Riemannian optimization method
(RDRSOM), which partially exploits the second order information and achieves
the iteration complexity of $\mathcal{O}(1/\epsilon^{3/2})$. In order to reduce
the per-iteration computational cost, we further propose a practical version of
(RDRSOM), which is an extension of the well known Barzilai-Borwein method and
achieves the iteration complexity of $\mathcal{O}(1/\epsilon^{3/2})$. We apply
our method to solve a nonlinear formulation of the wireless sensor network
localization problem whose feasible set is a Riemannian manifold that has not
been considered in the literature before. Numerical experiments are conducted
to verify the high efficiency of our algorithm compared to state-of-the-art
Riemannian optimization methods and other nonlinear solvers.
