  In this paper, based a novel primal-dual dynamical model with adaptive
scaling parameters and Bregman divergences, we propose new accelerated
primal-dual proximal gradient splitting methods for solving bilinear
saddle-point problems with provable optimal nonergodic convergence rates. For
the first, using the spectral analysis, we show that a naive extension of
acceleration model for unconstrained optimization problems to a quadratic game
is unstable. Motivated by this, we present an accelerated primal-dual hybrid
gradient (APDHG) flow which combines acceleration with careful velocity
correction. To work with non-Euclidean distances, we also equip our APDHG model
with general Bregman divergences and prove the exponential decay of a Lyapunov
function. Then, new primal-dual splitting methods are developed based on proper
semi-implicit Euler schemes of the continuous model, and the theoretical
convergence rates are nonergodic and optimal with respect to the matrix
norms,\, Lipschitz constants and convexity parameters. Thanks to the primal and
dual scaling parameters, both the algorithm designing and convergence analysis
cover automatically the convex and (partially) strongly convex objectives.
Moreover, the use of Bregman divergences not only unifies the standard
Euclidean distances and general cases in an elegant way, but also makes our
methods more flexible and adaptive to problem-dependent metrics.
