  In this paper, we revisit the smooth and strongly-convex-strongly-concave
minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020)
established the lower bound $\Omega\left(\sqrt{\kappa_x\kappa_y} \log
\frac{1}{\epsilon}\right)$ on the number of gradient evaluations required to
find an $\epsilon$-accurate solution, where $\kappa_x$ and $\kappa_y$ are
condition numbers for the strong convexity and strong concavity assumptions.
However, the existing state-of-the-art methods do not match this lower bound:
algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation
complexity $\mathcal{O}\left(
\sqrt{\kappa_x\kappa_y}\log^3\frac{1}{\epsilon}\right)$ and $\mathcal{O}\left(
\sqrt{\kappa_x\kappa_y}\log^3 (\kappa_x\kappa_y)\log\frac{1}{\epsilon}\right)$,
respectively. We fix this fundamental issue by providing the first algorithm
with $\mathcal{O}\left(\sqrt{\kappa_x\kappa_y}\log\frac{1}{\epsilon}\right)$
gradient evaluation complexity. We design our algorithm in three steps: (i) we
reformulate the original problem as a minimization problem via the pointwise
conjugate function; (ii) we apply a specific variant of the proximal point
algorithm to the reformulated problem; (iii) we compute the proximal operator
inexactly using the optimal algorithm for operator norm reduction in monotone
inclusions.
