  Phase retrieval has important applications in optical imaging, communications
and sensing. Lifting the dimensionality of the problem allows phase retrieval
to be approximated as a convex optimization problem in a higher-dimensional
space. Convex optimization-based phase retrieval has been shown to yield high
accuracy, yet its low-complexity implementation has not been explored. In this
paper, we study three fundamental approaches for its low-complexity
implementation: the projected gradient method, the Nesterov accelerated
gradient method, and the alternating direction method of multipliers (ADMM)
method. We derive the corresponding estimation algorithms and evaluate their
complexities. We compare their performance in the application area of
direct-detection mode-division multiplexing. We demonstrate that they yield
negligible estimation penalties (less than 0.2 dB for transmitter processing
and less than 0.6 dB for receiver equalization) while yielding low
computational cost, as their implementation complexities all scale
quadratically in the number of unknown parameters. Among the three methods,
ADMM achieves convergence after the smallest number of iterations.
