  This paper resolves a longstanding open question pertaining to the design of
near-optimal first-order algorithms for smooth and
strongly-convex-strongly-concave minimax problems. Current state-of-the-art
first-order algorithms find an approximate Nash equilibrium using
$\tilde{O}(\kappa_{\mathbf x}+\kappa_{\mathbf y})$ or
$\tilde{O}(\min\{\kappa_{\mathbf x}\sqrt{\kappa_{\mathbf y}},
\sqrt{\kappa_{\mathbf x}}\kappa_{\mathbf y}\})$ gradient evaluations, where
$\kappa_{\mathbf x}$ and $\kappa_{\mathbf y}$ are the condition numbers for the
strong-convexity and strong-concavity assumptions. A gap still remains between
these results and the best existing lower bound
$\tilde{\Omega}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$. This paper
presents the first algorithm with $\tilde{O}(\sqrt{\kappa_{\mathbf
x}\kappa_{\mathbf y}})$ gradient complexity, matching the lower bound up to
logarithmic factors. Our algorithm is designed based on an accelerated proximal
point method and an accelerated solver for minimax proximal steps. It can be
easily extended to the settings of strongly-convex-concave, convex-concave,
nonconvex-strongly-concave, and nonconvex-concave functions. This paper also
presents algorithms that match or outperform all existing methods in these
settings in terms of gradient complexity, up to logarithmic factors.
