Free flight is an emerging paradigm in air traffic management. Conflict detection and resolution is the heart of any free-flight concept. The problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft is addressed by the rigorous numerical trajectory optimization methods. The conflict problem is posed as an optimal control problem of finding trajectories that minimize a certain objective function while the safe separation between each aircraft pair is maintained. The initial and final positions of the aircraft are known and aircraft models with detailed nonlinear point-mass dynamics are considered. The protection zone around the aircraft is modeled to be cylindrical in shape. A novel formulation of the cylindrical protection zone is proposed by the use of continuous variables. The optimal control problem is converted to a finite dimensional nonlinear program (NLP) by the use of collocation on finite elements. The NLP is solved by the use of an interior point algorithm that incorporates a novel line search method. A reliable initialization strategy that yields a feasible solution on simple models is also proposed and adapted to detailed models. Several resolution scenarios are illustrated. The practical issue of flyability of the generated trajectories is addressed by the ability of our mathematical programming framework to accommodate detailed dynamic models.