  We present a convex relaxation-based algorithm for large-scale general phase
retrieval problems. General phase retrieval problems include i.a. the
estimation of the phase of the optical field in the pupil plane based on
intensity measurements of a point source recorded in the image (focal) plane.
The non-convex problem of finding the complex field that generates the correct
intensity is reformulated into a rank constraint problem. The nuclear norm is
used to obtain the convex relaxation of the phase retrieval problem. A new
iterative method, indicated as Convex Optimization-based Phase Retrieval
(COPR), is presented, with each iteration consisting of solving a convex
problem. In the noise-free case and for a class of phase retrieval problems the
solutions of the minimization problems converge linearly or faster towards a
correct solution. Since the solutions to nuclear norm minimization problems can
be computed using semidefinite programming, and this tends to be an expensive
optimization in terms of scalability, we provide a fast ADMM algorithm that
exploits the problem structure. The performance of the COPR algorithm is
demonstrated in a realistic numerical simulation study, demonstrating its
improvements in reliability and speed with respect to state-of-the-art methods.
