  The Douglas-Rachford algorithm (DRA) is a powerful optimization method for
minimizing the sum of two convex (not necessarily smooth) functions. The vast
majority of previous research dealt with the case when the sum has at least one
minimizer. In the absence of minimizers, it was recently shown that for the
case of two indicator functions, the DRA converges to a best approximation
solution. In this paper, we present a new convergence result on the the DRA
applied to the problem of minimizing a convex function subject to a linear
constraint. Indeed, a normal solution may be found even when the domain of the
objective function and the linear subspace constraint have no point in common.
As an important application, a new parallel splitting result is provided. We
also illustrate our results through various examples.
