  We develop a framework for quantitative convergence analysis of Picard
iterations of expansive set-valued fixed point mappings. There are two key
components of the analysis. The first is a natural generalization of
single-valued averaged mappings to expansive, set-valued mappings that
characterizes a type of strong calmness of the fixed point mapping. The second
component to this analysis is an extension of the well-established notion of
metric subregularity -- or inverse calmness -- of the mapping at fixed points.
Convergence of expansive fixed point iterations is proved using these two
properties, and quantitative estimates are a natural byproduct of the
framework. To demonstrate the application of the theory, we prove for the first
time a number of results showing local linear convergence of nonconvex cyclic
projections for inconsistent (and consistent) feasibility problems, local
linear convergence of the forward-backward algorithm for structured
optimization without convexity, strong or otherwise, and local linear
convergence of the Douglas--Rachford algorithm for structured nonconvex
minimization. This theory includes earlier approaches for known results, convex
and nonconvex, as special cases.
