  We study the smooth minimax optimization problem $\min_{\bf x}\max_{\bf y}
f({\bf x},{\bf y})$, where $f$ is $\ell$-smooth, strongly-concave in ${\bf y}$
but possibly nonconvex in ${\bf x}$. Most of existing works focus on finding
the first-order stationary points of the function $f({\bf x},{\bf y})$ or its
primal function $P({\bf x})\triangleq \max_{\bf y} f({\bf x},{\bf y})$, but few
of them focus on achieving second-order stationary points. In this paper, we
propose a novel approach for minimax optimization, called Minimax Cubic Newton
(MCN), which could find an
$\big(\varepsilon,\kappa^{1.5}\sqrt{\rho\varepsilon}\,\big)$-second-order
stationary point of $P({\bf x})$ with calling ${\mathcal
O}\big(\kappa^{1.5}\sqrt{\rho}\varepsilon^{-1.5}\big)$ times of second-order
oracles and $\tilde{\mathcal
O}\big(\kappa^{2}\sqrt{\rho}\varepsilon^{-1.5}\big)$ times of first-order
oracles, where $\kappa$ is the condition number and $\rho$ is the Lipschitz
continuous constant for the Hessian of $f({\bf x},{\bf y})$. In addition, we
propose an inexact variant of MCN for high-dimensional problems to avoid
calling expensive second-order oracles. Instead, our method solves the cubic
sub-problem inexactly via gradient descent and matrix Chebyshev expansion. This
strategy still obtains the desired approximate second-order stationary point
with high probability but only requires $\tilde{\mathcal
O}\big(\kappa^{1.5}\ell\varepsilon^{-2}\big)$ Hessian-vector oracle calls and
$\tilde{\mathcal O}\big(\kappa^{2}\sqrt{\rho}\varepsilon^{-1.5}\big)$
first-order oracle calls. To the best of our knowledge, this is the first work
that considers the non-asymptotic convergence behavior of finding second-order
stationary points for minimax problems without the convex-concave assumptions.
