  In this paper, we study the nonexpansive properties of metric resolvent, and
present a convergence rate analysis for the associated fixed-point iterations
(Banach-Picard and Krasnosel'skii-Mann types). Equipped with a variable metric,
we develop the global ergodic and non-ergodic iteration-complexity bounds in
terms of both solution distance and objective value. A byproduct of our
expositions also extends the proximity operator and Moreau's decomposition
identity to arbitrary variable metric. It is further shown that many classes of
the first-order operator splitting algorithms, including alternating direction
methods of multipliers, primal-dual hybrid gradient and Bregman iterations, can
be expressed by the fixed-point iterations of a simple metric resolvent, and
thus, the convergence can be analyzed within this unified framework.
