  While convergence of the Alternating Direction Method of Multipliers (ADMM)
on convex problems is well studied, convergence on nonconvex problems is only
partially understood. In this paper, we consider the Gaussian phase retrieval
problem, formulated as a linear constrained optimization problem with a
biconvex objective. The particular structure allows for a novel application of
the ADMM. It can be shown that the dual variable is zero at the global
minimizer. This motivates the analysis of a block coordinate descent algorithm,
which is equivalent to the ADMM with the dual variable fixed to be zero. We
show that the block coordinate descent algorithm converges to the global
minimizer at a linear rate, when starting from a deterministically achievable
initialization point.
