As wind energy is established as a sustainable alternative source of electricity, very large-scale wind farms with hundreds of turbines are becoming increasingly common. For the optimal design of wind farm layouts, the number of decision variables is at least twice the number of turbines (e.g., the Cartesian coordinates of each turbine). As the number of turbines increases, the computational cost incurred by the optimization solver to converge to a satisfactory solution increases as well. This issue represents a serious limitation in the computer-aided design of large wind farms. Moreover, the wind farm domains are typically highly constrained including land-availability and proximity constraints. These non-linear constraints increase the complexity of the optimization problem and decrease the likelihood of obtaining even a feasible solution. Several approaches have been proposed for micrositing of wind turbines, including random searches, mixed-integer programs, and metaheuristics. Each of these methods has its own trade-off between the quality of optimized layouts and the computational cost of obtaining the solution. In this paper, we demonstrate the capability of non-linear mathematical programming for optimizing very large-scale wind farms by leveraging explicit, analytical derivatives for the objective and constraint functions, thus overcoming the aforementioned limitations while also providing convergence and local optimality guarantees. For that purpose, two large farms with hundreds of turbines and significant land-use constraints are solved on a standard personal computer.