zbMATH — the first resource for mathematics

Solving air traffic conflict problems via local continuous optimization. (English) Zbl 1339.90356
Summary: This paper first introduces an original trajectory model using B-splines and a new semi-infinite programming formulation of the separation constraint involved in air traffic conflict problems. A new continuous optimization formulation of the tactical conflict-resolution problem is then proposed. It involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory. Encouraging numerical experiments show that this approach is viable on realistic test problems. Not only does one not need to rely on the traditional, discretized, combinatorial optimization approaches to this problem, but, moreover, local continuous optimization methods, which require relatively fewer iterations and thereby fewer costly function evaluations, are shown to improve the performance of the overall global optimization of this non-convex problem.

90C90 Applications of mathematical programming
90B90 Case-oriented studies in operations research
90C59 Approximation methods and heuristics in mathematical programming
65K05 Numerical mathematical programming methods
90C34 Semi-infinite programming
PDF BibTeX Cite
Full Text: DOI
[1] Alliot, J. M.; Bosc, J. F.; Durand, N.; Maugis, L., CATS: A complete air traffic simulator, IEEE Digital Avionics Systems Conference, American Institute of Aeronautics and Astrophysics, 2, 8.2-30-8.2-37, (1997), USA IEEE
[2] Alonso-Ayuso, A.; Escudero, L. F.; Martin-Campo, F. J., Collision avoidance in the air traffic management: A mixed integer linear optimization approach, IEEE Transactions on Intelligent Transportation Systems, 12, 47-57, (2011)
[3] Byrd, R. H.; Gilbert, J. C.; Nocedal, J., A trust region method based on interior point techniques for nonlinear programming, Mathematical Programming, 89, 149-185, (2000) · Zbl 1033.90152
[4] Conn, A. R.; Gould, N. I.M., An exact penalty function for semi-infinite programming, Mathematical Programming, 37, 19-40, (1987) · Zbl 0623.90069
[5] Conn, A. R.; Scheinberg, K.; Vicente, L. N., Introduction to derivative-free optimization. MPS-SIAM series on optimization, 173-205, (2009), USA Society for Industrial and Applied Mathematics Philadelphia, PA
[6] Delahaye, D.; Puechmorel, S., Modeling and optimization of air traffic, (2013), Wiley-ISTE USA · Zbl 1298.90080
[7] Dimarogonas, D. V.; Kyriakopoulos, K. J., A feedback stabilization and collision avoidance scheme for multiple independent nonholonomic non-point agents, Proceedings of the IEEE international symposium on robotics and automation, 820-825, (2005), IEEE USA
[8] do Carmo, M. P., Riemannian geometry, (1992), Birkhäuser Switzerland
[9] Dougui, N.; Delahaye, D.; Puechmorel, S.; Mongeau, M., A light-propagation model for aircraft trajectory planning, Journal of Global Optimization, 56(3), 873-895, (2013) · Zbl 1272.90122
[10] Duncan, M., Applied geometry for computer graphics and computer-aided design, (2005), Springer UK
[11] Durand, N.; Alliot, J. M., Ant colony optimization for air traffic conflict resolution, Proceedings of the 8th USA/Europe air traffic management research and development seminar, (2009), Napa USA
[12] Durand, N., Alliot, J. M., Noailles, J.1996. Automatic aircraft conflict resolution using genetic algorithms In: 11th annual ACM symposium on applied computing (ACM/SAC 96)ACM: USA
[13] Goldberg, D. E., Genetic algorithms in search optimization and machine learning, (1989), Addison-Wesley USA · Zbl 0721.68056
[14] Kuchar, J. K.; Yang, L. C., A review of conflict detection and resolution modeling methods, IEEE Transactions on Intelligent Transportation Systems, 1, 179-189, (2000)
[15] Médioni, F.; Durand, N.; Alliot, J. M., Air traffic conflict resolution by genetic algorithms, Artificial evolution, Lecture Notes in Computer Science, 1063, 370-383, (1995), Springer USA
[16] Milam, M. B.; Mushambi, K.; Murray, R. M., A new computational approach to real-time trajectory generation for constrained mechanical systems, Proceedings of the 39th IEEE conference on decision and control, (2000), IEEE USA
[17] Olive, X., Résolution de conflits par algorithmes stochastiques parallèles (conflict resolution via parallel stochastic algorithms) (ph.D. thesis), (2006), École Nationale Supérieure de l’ Aéronautique et de l’ Espace Toulouse, France
[18] Pallottino, L.; Feron, E.; Bicchi, A., Conflict resolution problems for air traffic management systems solved with mixed integer programming, IEEE Transactions on Intelligent Transportation Systems, 3, 3-11, (2002)
[19] Powell, M. J. D.2009. The Bound Optimization BY Quadratic Approximation (BOBYQA) algorithm for bound constrained optimization without derivatives Technical Report NA2009/06 CambridgeEngland
[20] Roussos, G.; Kyriakopoulos, K. J., Towards constant velocity navigation and collision avoidance for autonomous nonholonomic aircraft-like vehicles, Proceedings of the Joint 48th IEEE conference on decision and control and 28th Chinese control conference, (2009), IEEE USA
[21] SESAR Joint Undertaking (2009). SESAR Joint Undertaking Brochure: Today’s partner for tomorrow’s aviation. Technical Report. Brussels, Belgium.
[22] Stein, O., How to solve a semi-infinite optimization problem, European Journal of Operational Research, 233, 312-320, (2012) · Zbl 1292.90300
[23] Visweswariah, C.; Haring, R.; Conn, A. R., Noise considerations in circuit optimization, IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, 19, 679-690, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.