  Many popular first order algorithms for convex optimization, such as
forward-backward splitting, Douglas-Rachford splitting, and the alternating
direction method of multipliers (ADMM), can be formulated as averaged iteration
of a nonexpansive mapping. In this paper we propose a line search for averaged
iteration that preserves the theoretical convergence guarantee, while often
accelerating practical convergence. We discuss several general cases in which
the additional computational cost of the line search is modest compared to the
savings obtained.
