  Splitting schemes are a class of powerful algorithms that solve complicated
monotone inclusion and convex optimization problems that are built from many
simpler pieces. They give rise to algorithms in which the simple pieces of the
decomposition are processed individually. This leads to easily implementable
and highly parallelizable algorithms, which often obtain nearly
state-of-the-art performance.
  In this paper, we provide a comprehensive convergence rate analysis of the
Douglas-Rachford splitting (DRS), Peaceman-Rachford splitting (PRS), and
alternating direction method of multipliers (ADMM) algorithms under various
regularity assumptions including strong convexity, Lipschitz differentiability,
and bounded linear regularity. The main consequence of this work is that
relaxed PRS and ADMM automatically adapt to the regularity of the problem and
achieve convergence rates that improve upon the (tight) worst-case rates that
hold in the absence of such regularity. All of the results are obtained using
simple techniques.
