  In this paper, we propose a cubic regularized Newton (CRN) method for solving
convex-concave saddle point problems (SPP). At each iteration, a cubic
regularized saddle point subproblem is constructed and solved, which provides a
search direction for the iterate. With properly chosen stepsizes, the method is
shown to converge to the saddle point with global linear and local superlinear
convergence rates, if the saddle point function is gradient Lipschitz and
strongly-convex-strongly-concave. In the case that the function is merely
convex-concave, we propose a homotopy continuation (or path-following) method.
Under a Lipschitz-type error bound condition, we present an iteration
complexity bound of $\mathcal{O}\left(\ln \left(1/\epsilon\right)\right)$ to
reach an $\epsilon$-solution through a homotopy continuation approach, and the
iteration complexity bound becomes
$\mathcal{O}\left(\left(1/\epsilon\right)^{\frac{1-\theta}{\theta^2}}\right)$
under a H\"{o}lderian-type error bound condition involving a parameter $\theta$
($0<\theta<1$).
