A comparison of formulations for aircraft deconfliction
Renan Spencer Trindade, Claudia D'Ambrosio
TL;DR
This work tackles aircraft deconfliction under heading-angle deviations by comparing a classic MINLP formulation with a separable univariate reformulation that expresses nonlinear trigonometric constraints as sums of univariate functions. By introducing auxiliary variables and a BigM strategy, the authors obtain a reformulated problem (M2) with a linearized objective and univariate constraints, enabling effective use of open-source global solvers such as Couenne, SCIP, and SCMINLP. Computational results on Circle Problem and Random Circle Problem instances show improved primal solvability and competitive dual bounds with the reformulation, with SCMINLP delivering strong feasible-solution quality and SCIP often yielding the best feasible solutions on the reformulated model. The study demonstrates the potential of open-source tools for aircraft deconfliction and highlights complementary solver strengths, suggesting hybrid approaches for larger-scale problems.
Abstract
In this work, we aim to compare different methods and formulations to solve a problem in air traffic management to global optimality. In particular, we focus on the aircraft deconfliction problem, where we are given n aircraft, their position at time 0, and their (straight) trajectories. We wish to identify and solve potential pairwise conflict by temporarily modifying the aircraft's trajectory. A pair of aircraft is in conflict when they do not respect a minimum, predefined safety distance. In general, conflicts could be solved both varying the aircraft's speed or trajectory, but in this paper we only consider the latter, more precisely heading-angle deviations. The problem has been formulated as a mixed integer nonlinear program (MINLP). We compare this formulation, solved by open-source MINLP solvers for global optimization, against a reformulation that shows a larger number of variables and constraints but only separable nonconvexities. We solve such a separable formulation with the same MINLP solvers or the Sequential Convex Mixed Integer Nonlinear Programming method. The separable formulation, despite being larger, facilitates some solvers in finding good-quality solutions.