  We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) +
y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions
and admit first-order gradient oracles. Surprisingly, no known first-order
algorithms have hitherto achieved the lower complexity bound of
$\Omega((\sqrt{\frac{L_x}{\mu_x}} + \frac{\|A\|}{\sqrt{\mu_x \mu_y}} +
\sqrt{\frac{L_y}{\mu_y}}) \log(\frac1{\varepsilon}))$ for solving this problem
up to an $\varepsilon$ primal-dual gap in the general parameter regime, where
$L_x, L_y,\mu_x,\mu_y$ are the corresponding smoothness and strongly convexity
constants.
  We close this gap by devising the first optimal algorithm, the Lifted
Primal-Dual (LPD) method. Our method lifts the objective into an extended form
that allows both the smooth terms and the bilinear term to be handled optimally
and seamlessly with the same primal-dual framework. Besides optimality, our
method yields a desirably simple single-loop algorithm that uses only one
gradient oracle call per iteration. Moreover, when $f$ is just convex, the same
algorithm applied to a smoothed objective achieves the nearly optimal iteration
complexity. We also provide a direct single-loop algorithm, using the LPD
method, that achieves the iteration complexity of
$O(\sqrt{\frac{L_x}{\varepsilon}} + \frac{\|A\|}{\sqrt{\mu_y \varepsilon}} +
\sqrt{\frac{L_y}{\varepsilon}})$. Numerical experiments on quadratic minimax
problems and policy evaluation problems further demonstrate the fast
convergence of our algorithm in practice.
