  More than 40 years ago, Lions and Mercier introduced in a seminal paper the
Douglas-Rachford algorithm. Today, this method is well recognized as a
classical and highly successful splitting method to find minimizers of the sum
of two (not necessarily smooth) convex functions. While the underlying theory
has matured, one case remains a mystery: the behaviour of the shadow sequence
when the given functions have disjoint domains.
  Building on previous work, we establish for the first time weak and value
convergence of the shadow sequence generated by the Douglas-Rachford algorithm
in a setting of unprecedented generality. The weak limit point is shown to
solve the associated normal problem which is a minimal perturbation of the
original optimization problem. We also present new results on the geometry of
the minimal displacement vector.
