Coherent X-ray diffractive imaging (CDI) has emerged as a tool of choice for providing spatially resolved structures of nanoscale objects at DOE's light source facilities. In CDI, the phase of the wave scattered by the object must be recovered algorithmically from the measured diffraction pattern in order to obtain the object's structure. We review various formulations of the phase retrieval problem in coherent X-ray diffraction imaging as a structured nonconvex optimization problems. We review matrix-lifting techniques that provide convex semi-definite programming relaxations of the nonconvex CDI inverse problem. Unfortunately, the resulting SDP formulations square the size of the problem and are computationally intractable for realistic applications. We suggest a new hybrid local-global optimization technique that provides good solution at reasonable computational cost.