  We provide a first-order oracle complexity lower bound for finding stationary
points of min-max optimization problems where the objective function is smooth,
nonconvex in the minimization variable, and strongly concave in the
maximization variable. We establish a lower bound of
$\Omega\left(\sqrt{\kappa}\epsilon^{-2}\right)$ for deterministic oracles,
where $\epsilon$ defines the level of approximate stationarity and $\kappa$ is
the condition number. Our analysis shows that the upper bound achieved in (Lin
et al., 2020b) is optimal in the $\epsilon$ and $\kappa$ dependence up to
logarithmic factors. For stochastic oracles, we provide a lower bound of
$\Omega\left(\sqrt{\kappa}\epsilon^{-2} + \kappa^{1/3}\epsilon^{-4}\right)$. It
suggests that there is a significant gap between the upper bound
$\mathcal{O}(\kappa^3 \epsilon^{-4})$ in (Lin et al., 2020a) and our lower
bound in the condition number dependence.
