A novel control optimization technique using the adjoint of the perturbed and linearized Navier-Stokes equations is implemented in conjunction with a high-fidelity numerical scheme for aeroacoustic optimization of complex geometry systems. The adjoint equations are formulated to provide sensitivity information used in a gradient-based minimization of an aerodynamic sound cost functional. The forward and adjoint Navier-Stokes equations are discretized on multiple, overlapping meshes in generalized curvilinear coordinates with time-dependent mappings. We apply the adjoint-based optimization to a turbulent Mach 1.3 perfectly-expanded jet and craft it specifically to reduce the far-field sound radiation. The formulation is verified on anti-sound (acoustic cancellation) model flows, in which the adjoint is used to optimize the cancelling sound source. Anti-sound is, of course, impractical for most aeroacoustic flow applications such as jet noise suppression. Given the complexity of the jet turbulence and the subtlety of the sound generation process, there is currently little insight available about how to do this. In our formulation, the adjoint provides a definitive direction in control space in which to modify the near-nozzle control to suppress the jet noise. Our adjoint-based optimization approach appears to be unique in providing controls that affect the noise reduction. The optimization is done within a large-eddy simulation framework using a baseline jet simulation which matches experimental data. Simulations are ongoing, but the optimization algorithm is shown to have found a quieter state.