A comparison between Mixed Integer Programming and Nonlinear Programming Techniques for 3 D Conflict Resolution of Multiple Aircraft

We consider the problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft using numerical trajectory optimization methods. The conflict problem is posed as an optimal control problem of finding trajectories that minimize a certain objective function while maintaining the safe separation between each aircraft pair. We assume the origin and destination of the aircraft are known and consider aircraft models with simplified linear kinematics. The main objective of this report is to compare two different approaches to the solution of the problem. In the first approach, the optimal control is converted to a finite dimensional Nonlinear Program (NLP) by using collocation on finite elements and by reformulating the disjunctions involved in modeling the protected zones by using continuous variables. We solve the NLP using an Interior Point algorithm that incorporates a novel line search method. In the second approach the optimal control is converted to a finite dimensional Mixed Integer Linear Program (MILP) using Euler discretization and reformulating the disjunctions involved with the protected zones by using binary variables and Big-M techniques. Based on results of extensive random simulations, we compare time complexity and optimality of the solutions obtained with the MILP approach and the NLP approach.