## A Lyapunov function construction for a non-convex Douglas-Rachford iteration. (arXiv:1708.08697v2 [math.OC] UPDATED)

While global convergence of the Douglas-Rachford iteration is often observed
in applications, proving it is still limited to convex and a handful of other
special cases. Lyapunov functions for difference inclusions provide not only
global or local convergence certificates, but also imply robust stability,
which means that the convergence is still guaranteed in the presence of
persistent disturbances. In this work, a global Lyapunov function is
constructed by combining known local Lyapunov functions for simpler, local
sub-problems via an explicit formula that depends on the problem parameters.
Specifically, we consider the scenario where one set consists of the union of
two lines and the other set is a line, so that the two sets intersect in two
distinct points. Locally, near each intersection point, the problem reduces to
the intersection of just two lines, but globally the geometry is non-convex and
the Douglas-Rachford operator multi-valued. Our approach is intended to be
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