Coherent x-ray diffractive imaging is a form of microscopy that does not require optics to image a sample. Instead, it measures diffraction intensity generated by placing a sample in the path of a coherent x-ray beam, and numerically inverts this diffraction using nonlinear optimization techniques. This algorithmic approach solves the "phase problem", which arises due to the inability of x-ray detectors to measure complex valued wave fields. Knowledge of the experimental arrangement is used to enforce constraints in the optimization algorithms to ensure effective numerical recovery. We discuss various schemes of using nonlinear optimization to image samples from their diffraction intensities.