  We consider the problem of estimating the phases of K mixed complex signals
from a multichannel observation, when the mixing matrix and signal magnitudes
are known. This problem can be cast as a non-convex quadratically constrained
quadratic program which is known to be NP-hard in general. We propose three
approaches to tackle it: a heuristic method, an alternate minimization method,
and a convex relaxation into a semi-definite program. The last two approaches
are showed to outperform the oracle multichannel Wiener filter in
under-determined informed source separation tasks, using simulated and speech
signals. The convex relaxation approach yields best results, including the
potential for exact source separation in under-determined settings.
