  The primal-dual hybrid gradient (PDHG) algorithm is popular in solving
min-max problems which are being widely used in a variety of areas. To improve
the applicability and efficiency of PDHG for different application scenarios,
we focus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework
covering PDHG, alternating direction method of multipliers (ADMM), and other
methods. We give the optimal convergence condition of PrePDHG in the sense that
the key parameters in the condition can not be further improved, which fills
the theoretical gap in the-state-of-art convergence results of PrePDHG, and
obtain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The
theoretical analysis is achieved by establishing the equivalence between
PrePDHG and indefinite proximal ADMM. Besides, we discuss various choices of
the proximal matrices in PrePDHG and derive some interesting results. For
example, the convergence condition of diagonal PrePDHG is improved to be tight,
the dual stepsize of the balanced augmented Lagrangian method can be enlarged
to $4/3$ from $1$, and a balanced augmented Lagrangian method with symmetric
Gauss-Seidel iterations is also explored. Numerical results on the matrix game,
projection onto the Birkhoff polytope, earth mover's distance, and CT
reconstruction verify the effectiveness and superiority of PrePDHG.
