Two decomposition algorithms for solving a minimum weight maximum clique model for the air conflict resolution problem

Abstract

International audienceIn this article, we tackle the conflict resolution problem using a new variant of the minimum-weight maximum-clique model. The problem involves identifying maneuvers that maintain the required separation distance between all pairs of a set of aircraft while minimizing fuel costs. We design a graph in which the vertices correspond to a finite set of maneuvers and the edges connect conflict-free maneuvers. A maximum clique of minimal weight yields a conflict-free situation that involves all the aircraft and minimizes the costs induced. The innovation of the model is its cost structure: the costs of the vertices cannot be determined a priori, since they depend on the vertices in the clique. We formulate the problem as a mixed integer linear program. Since the modeling of the aircraft dynamics and the computation of trajectories is separated from the solution process, the model is flexible. As a consequence, our mathematical framework is valid for any hypotheses. In particular, the aircraft can perform dynamic velocity, heading, and flight-level changes. To solve instances involving a large number of aircraft spread over several flight levels, we introduce two decomposition algorithms. The first is a sequential mixed integer linear optimization procedure that iteratively refines the discretization of the maneuvers to yield a trade-off between computational time and cost. The second is a large neighborhood search heuristic that uses the first procedure as a subroutine. The best solutions for the available set of maneuvers are obtained in less than 10 seconds for instances with up to 250 aircraft randomly allocated to 20 flight levels

Constant speed optimal reciprocal collision avoidance

“…For each aircraft, a number of alternative trajectories and a matrix of pairwise conflicts were precalculated, taking various uncertainties into account, before the optimization process was performed. This approach separated the problem model from its resolution and was close to the graph model proposed by Lehouillier et al (2017).…”

Complex number formulation and convex relaxations for aircraft conflict resolution

“…State-of-the-art computational results for the horizontal aircraft conflict resolution problem can be found in [5], [8], [11], [13], [24]. Some of these approaches [5], [13] consider heading and speed control separately or maneuver-discretised formulations [24] which can lead to suboptimal solutions. In [11], an exact implementation of the combined speed and heading control problem is tested.…”

Aircraft Deconfliction via Mathematical Programming: Review and Insights

“…A different discrete optimization approach is presented by Lehouillier et al (2017b), who also propose to use a graph as modelling tool. In this case, nodes stand for aircraft maneuvers, while edges connect those that are conflict-free.…”