  The Douglas-Rachford algorithm is widely used in sparse signal processing for
minimizing a sum of two convex functions. In this paper, we consider the case
where one of the functions is weakly convex but the other is strongly convex so
that the sum is convex. We provide a condition that ensures the convergence of
the same Douglas-Rachford iterations, provided that the strongly convex
function is smooth. We also present a modified Douglas-Rachford algorithm that
does not impose a smoothness condition for the convex function. We then provide
a discussion on the convergence speed of the two types of algorithms and
demonstrate the discussion with numerical experiments.
