  Douglas-Rachford Splitting (DRS) methods based on the proximal point
algorithms for the Poisson and Gaussian log-likelihood functions are proposed
for ptychography and phase retrieval. Fixed point analysis shows that the DRS
iterated sequences are always bounded explicitly in terms of the step size and
that the fixed points are attracting if and only if the fixed points are
regular solutions. This alleviates two major drawbacks of the classical
Douglas-Rachford algorithm: slow convergence when the feasibility problem is
consistent and divergent behavior when the feasibility problem is inconsistent.
Fixed point analysis also leads to a simple, explicit expression for the
optimal step size in terms of the spectral gap of an underlying matrix. When
applied to the challenging problem of blind ptychography, which seeks to
recover both the object and the probe simultaneously, Alternating Minimization
with the DRS inner loops, even with a far from optimal step size, converges
geometrically under the nearly minimum conditions established in the uniqueness
theory.
