  In this paper we study the convex-concave saddle-point problem $\min_x \max_y
f(x) + y^T \mathbf{A} x - g(y)$, where $f(x)$ and $g(y)$ are smooth and convex
functions. We propose an Accelerated Primal-Dual Gradient Method (APDG) for
solving this problem, achieving (i) an optimal linear convergence rate in the
strongly-convex-strongly-concave regime, matching the lower complexity bound
(Zhang et al., 2021), and (ii) an accelerated linear convergence rate in the
case when only one of the functions $f(x)$ and $g(y)$ is strongly convex or
even none of them are. Finally, we obtain a linearly convergent algorithm for
the general smooth and convex-concave saddle point problem $\min_x \max_y
F(x,y)$ without the requirement of strong convexity or strong concavity.
