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Detecting and solving aircraft conflicts using bilevel programming

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Abstract

We present two bilevel programming formulations for the aircraft deconfliction problem: one based on speed regulation in k dimensions, the other on heading angle changes in 2 dimensions. We propose three reformulations of each problem based on KKT conditions and on two different duals of the lower-level subproblems. We also propose a cut generation algorithm to solve the bilevel formulations. Finally, we present computational results on a variety of instances.

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Notes

  1. 1 NM = 1852 m.

References

  1. Villiers, J.: Automatisation du contro\(\hat{\text{l}}\)e de la circulation aèrienne: “ERASMUS”, une voie conviviale pour franchir le mur de la capacitè. Institut du Transport Aèrien 58 (2004)

  2. Drogoul, F., Averty, P., Weber, R.: ERASMUS Strategic Deconfliction to Benefit SESAR. In: Proceedings of the 8th USA/Europe Air Traffic Management Research and Development Seminar, Napa, USA, pp. 1–10 (2009)

  3. SESAR (Single European Sky ATM Research) project. https://ec.europa.eu/transport/modes/air/sesar_en. Accessed 15 Feb 2021

  4. Lao, M., Tang, J.: Cooperative multi-UAV collision avoidance based on distributed dynamic optimization and causal analysis. Appl. Sci. 7(1), 83 (2017)

    Article  Google Scholar 

  5. Claes, D., Tuyls, K.: Multi robot collision avoidance in a shared workspace. Auton. Robots 42(8), 1749–1770 (2018)

    Article  Google Scholar 

  6. Co, C.G., Tanchoco, J.M.A.: A review of research on AGVS vehicle management. Eng. Costs Prod. Econ. 21(1), 35–42 (1991)

    Article  Google Scholar 

  7. Cerulli, M., D’Ambrosio, C., Liberti, L.: Flying safely by bilevel programming. In: Advances in Optimization and Decision Science for Society, Services and Enterprises: ODS, Genoa, Italy, September 4–7, 2019—AIRO Springer Serie, vol. 3, pp. 197–206 (2019)

  8. Rey, D., Rapine, C., Constans, S., Fondacci, R.: A mixed integer linear model for potential conflict minimization by speed modulations. In: ICRAT 2010. Fourth International Conference on Research in Air Transportation (2010)

  9. Rey, D., Rapine, C., Dixit, V.V., Waller, S.T.: Equity-oriented aircraft collision avoidance model. IEEE Trans. Intell. Transp. Syst. 16(1), 172–183 (2015)

    Article  Google Scholar 

  10. Cafieri, S., Durand, N.: Aircraft deconfliction with speed regulation: new models from mixed-integer optimization. J. Glob. Optim. 58(4), 613–629 (2014)

    Article  MathSciNet  Google Scholar 

  11. Cafieri, S., D’Ambrosio, C.: Feasibility pump for aircraft deconfliction with speed regulation. J. Glob. Optim. 71(3), 501–515 (2018)

    Article  MathSciNet  Google Scholar 

  12. Vela, A., Solak, S., Singhose, W., Clarke, J.: A mixed integer program for flight-level assignment and speed control for conflict resolution. In: Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly with 2009 28th Chinese Control Conference, Shanghai, pp. 5219–5226 (2009)

  13. Alonso-Ayuso, A., Escudero, L.F., Martín-Campo, F.J.: Collision avoidance in air traffic management: a mixed-integer linear optimization approach. IEEE Trans. Intell. Transp. Syst. 12(1), 47–57 (2010)

    Article  Google Scholar 

  14. Dias, F.H., Hijazi, H., Rey, D.: Disjunctive linear separation conditions and mixed-integer formulations for aircraft conflict resolution by speed and altitude control (2019). arXiv preprint arXiv:1911.12997

  15. Bilimoria, K.D.: A geometric optimization approach to aircraft conflict resolution. In: 18th Applied Aerodynamics Conference (2000)

  16. Pallottino, L., Feron, E., Bicchi, A.: Conflict resolution problems for air traffic management systems solved with mixed integer programming. IEEE Trans. Intell. Transp. Syst. 3(1), 3–11 (2002)

    Article  Google Scholar 

  17. Cafieri, S., Omheni, R.: Mixed-integer nonlinear programming for aircraft conflict avoidance by sequentially applying velocity and heading angle changes. Eur. J. Oper. Res. 260(1), 283–290 (2017)

    Article  MathSciNet  Google Scholar 

  18. Frazzoli, E., Mao, Z.-H., Oh, J.-H., Feron, E.: Resolution of conflicts involving many aircraft via semidefinite programming. J. Guid. Control Dyn. 24(1), 79–86 (2001)

    Article  Google Scholar 

  19. Rey, D., Hijazi, H.: Complex number formulation and convex relaxations for aircraft conflict resolution. In: IEEE 56th Annual Conference on Decision and Control, pp. 88–93 (2017)

  20. Stein, O.: How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 223, 312–320 (2012)

    Article  MathSciNet  Google Scholar 

  21. Dempe, S., Kalashnikov, V., Prez-Valds, G.A., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, Berlin (2015)

    Book  Google Scholar 

  22. Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0–1 programming problems. J. Optim. Theory Appl. 93(2), 273–300 (1997)

    Article  MathSciNet  Google Scholar 

  23. Kleinert, T., Labbé, M., Plein, F., Schmidt, M.: There’s no free lunch: on the hardness of choosing a correct Big-M in bilevel optimization (2019). Optimization online preprint. http://www.optimization-online.org/DB_HTML/2019/04/7172.html

  24. Dorn, W.S.: Self-dual quadratic programs. J. Soc. Ind. Appl. Math. 9(1), 51–54 (1961)

    Article  MathSciNet  Google Scholar 

  25. Dorn, W.S.: Duality in quadratic programming. Q. Appl. Math. 18(2), 155–162 (1960)

    Article  MathSciNet  Google Scholar 

  26. Wolfe, P.: A duality theorem for non-linear programming. Q. Appl. Math. 19, 239–244 (1961)

    Article  MathSciNet  Google Scholar 

  27. Kleinert, T., Labbé, M., Plein, F., Schmidt, M.: Closing the gap in linear bilevel optimization: a new valid primal–dual inequality (2020). preprint

  28. Zare, M.H., Borrero, J.S., Zeng, B.: A note on linearized reformulations for a class of bilevel linear integer problems. Ann. Oper. Res. 272, 99–117 (2019)

    Article  MathSciNet  Google Scholar 

  29. Fang, S., Lin, C., Wu, S.: Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme. J. Comput. Appl. Math. 129, 89–104 (2001)

    Article  MathSciNet  Google Scholar 

  30. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Cengage Learning, Boston (2002)

  31. Sahinidis, N.V., Tawarmalani, M.: BARON 7.2.5: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2005)

  32. Gill, P.E.: User’s guide for SNOPT version 7, 2 (2006)

  33. Belotti, P., Lee, J., Liberti, L., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)

    Article  MathSciNet  Google Scholar 

  34. Wächter, A., Biegler, L.: On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming Math. Program. 106, 25–57 (2006)

  35. Alonso-Ayuso, A., Escudero, L.F., Martín-Campo, F.J.: Exact and approximate solving of the aircraft collision resolution problem via turn changes. Transp. Sci. 50, 263–274 (2016)

    Article  Google Scholar 

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Correspondence to Martina Cerulli.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was partly funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 764759 ETN “MINOA”.

This publication was supported by the Chair “Integrated Urban Mobility”, backed by L’X—École Polytechnique and La Fondation de l’École polytechnique and sponsored by Uber. The Partners of the Chair shall not under any circumstances accept any liability for the content of this publication, for which the authors shall be solely liable.

Appendices

Appendix

Returning aircraft to original trajectories

The optimal heading angle change \(\theta _i^\star \) for each aircraft i is obtained by solving Eqs. (14a)–(14c). The trajectory deviation is followed until necessary to guarantee the safety distance, then the aircraft must return to their initial trajectories. Following what is done in [17, 35], for each pair of aircraft the convex unconstrained QP (22) is solved as a post-processing step to return each aircraft to its original flight plan as soon as possible after conflict resolution:

$$\begin{aligned} \min \limits _{t_{ij}} \left\| \begin{pmatrix} (x_{i}^0 - x_{j}^0) + t_{ij}(\cos (\phi _i+\theta _i^\star )v_i-\cos (\phi _j+\theta _j^\star )v_j) \\ (y_{i}^0 - y_{j}^0) + t_{ij}(\sin (\phi _i+\theta _i^\star )v_i-\sin (\phi _j+\theta _j^\star )v_j) \end{pmatrix} \right\| ^2. \end{aligned}$$
(22)

The objective function of the problem (22) is the relative squared Euclidean distance between aircraft, which is computed using the optimal heading angles of the proposed bilevel problem (14a)–(14c).

Once the optimal solution \(\tau _{ij}^\star \) for problem (22) is found, we compute

$$\begin{aligned} T_i^\star := \max \limits _{j:i\ne j} \tau _{ij}^\star \end{aligned}$$
(23)

as the optimal time for which aircraft i can return to its initial trajectory after the deconfliction (there will be, for each i, a different \(\tau _{ij}\) for every pair of aircraft (ij)).

Knowing \((x_i(T_i^\star ),y_i(T_i^\star ))\) and the exit point from the air sector, it is easy to determine the new trajectory each aircraft has to follow in order to go back to its initial trajectory, as shown in Fig. 7.

As clarified in [35], when the aircraft are returning to the initial trajectories, new conflicts may occur. In order to ensure a conflict-free situation, the HACADP must be solved again. Sometimes, the maneuver to return to the initial trajectory must start when the aircraft is already close to the boundary of the air sector. This could lead to an angle variation exceeding the bounds. In this case, Alonso-Ayuso et al. [35] proposes turning at the maximum bound and sending a warning message to the air traffic controllers of the following sector notifying that the aircraft is arriving in that sector at a different entry point w.r.t. the scheduled one.

Fig. 7
figure 7

New trajectories of two aircraft that, after conflict resolution, return to their initial trajectories (dashed lines)

Cut dominance

The time per iteration taken by Algorithm 1 and Algorithm 2 increases with the number of cuts added to the formulation. In fact, while solutions of the lower-level subproblems are easily computed in closed form in Step 4 of both algorithms respectively, increasing the number of quadratic constraints (20) and (21) yields a time increase when solving \(R_h\). It would therefore be desirable to remove as many unnecessary cuts as possible. Consequently, we study the existence of dominance relationships between cuts in the proposed cut generation algorithms.

Proposition 5

Let (ij) be a pair of aircraft. There is no dominance relationship between any pair of the cuts added for (ij) by either Algorithms 1 or 2.

Proof

Let \(h, h'\) be two different iterations of the cut generation algorithm in which cuts were added for the pair (ij). We consider the time instants at which i and j are closest in each iteration, \(\tau _{ij}^h\) and \(\tau _{ij}^{h'}\), with \(\tau _{ij}^h\ne \tau _{ij}^{h'}\). Taking the case of Algorithm 1 (the proof for Algorithm 2 is analogous), we proceed by contradiction and suppose that the cut added in iteration h dominates that of iteration \(h'\). That is, for all feasible \(q_i, q_j\):

$$\begin{aligned} \sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h}(q_i v_i u_{ik} - q_j v_j u_{jk}))^2 \ge d^2 \end{aligned}$$

implies

$$\begin{aligned} \sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h'}(q_i v_i u_{ik} - q_j v_j u_{jk}))^2 \ge d^2, \end{aligned}$$

or, equivalently,

$$\begin{aligned}&\sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h}(q_i v_i u_{ik} - q_j v_j u_{jk}))^2 \\&\quad \le \sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h'}(q_i v_i u_{ik} - q_j v_j u_{jk}))^2. \end{aligned}$$

In particular, for \(q_i, q_j\) equal to the solution obtained at iteration \(h'\), \(q_i^{h'}, q_j^{h'}\), the inequality

$$\begin{aligned}&\sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h}(q_i^{h'} v_i u_{ik} - q_j^{h'} v_j u_{jk}))^2 \\&\quad \le \sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h'}(q_i^{h'} v_i u_{ik} - q_j^{h'} v_j u_{jk}))^2 \end{aligned}$$

must hold. Being \(\tau _{ij}^{h'}\) a minimizer of the function

$$\begin{aligned} f(t):=\sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + t(q_i^{h'} v_i u_{ik} - q_j^{h'} v_j u_{jk}))^2, \end{aligned}$$

it has to be:

$$\begin{aligned}&\sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h}(q_i^{h'} v_i u_{ik} - q_j^{h'} v_j u_{jk}))^2 \\&\quad = \sum _{k\in K} ((x_{ik}^0 - x_{jk}^0) + \tau _{ij}^{h'}(q_i^{h'} v_i u_{ik} - q_j^{h'} v_j u_{jk}))^2. \end{aligned}$$

Since \(\tau _{ij}^h\ne \tau _{ij}^{h'}\), it yields that f(t) attains its minimum at two different points. This is only possible if f(t) is constant for all t, i.e., if \(q_i^{h'} v_i u_{ik} = q_j^{h'} v_j u_{jk}\). But, if a cut is added for (ij) at iteration \(h'\) is because the separation distance was violated also at \(t=0\) (since f is constant), something that we discarded by assumption. \(\square \)

Proposition 5 ensures that a cut added in a certain iteration for a pair of aircraft will not be dominated by any cut added in future iterations for the same pair. However, whether several cuts together dominate a single cut is a question that remains open. Similarly, the study of potential dominance between cuts involving conflicting triplets of aircraft in special cases would be worth-studying for future work.

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Cerulli, M., D’Ambrosio, C., Liberti, L. et al. Detecting and solving aircraft conflicts using bilevel programming. J Glob Optim 81, 529–557 (2021). https://doi.org/10.1007/s10898-021-00997-1

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