  The cubic regularization method (CR) and its adaptive version (ARC) are
popular Newton-type methods in solving unconstrained non-convex optimization
problems, due to its global convergence to local minima under mild conditions.
The main aim of this paper is to develop a momentum-accelerated adaptive cubic
regularization method (ARCm) to improve the convergent performance. With the
proper choice of momentum step size, we show the global convergence of ARCm and
the local convergence can also be guaranteed under the \KL property. Such
global and local convergence can also be established when inexact solvers with
low computational costs are employed in the iteration procedure. Numerical
results for non-convex logistic regression and robust linear regression models
are reported to demonstrate that the proposed ARCm significantly outperforms
state-of-the-art cubic regularization methods (e.g., CR, momentum-based CR,
ARC) and the trust region method. In particular, the number of iterations
required by ARCm is less than 10\% to 50\% required by the most competitive
method (ARC) in the experiments.
