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A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization

Authors Ralf Borndörfer , Fabian Danecker , Martin Weiser

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Author Details

Ralf Borndörfer
  • Zuse Institute, Berlin, Germany
  • Freie Universität Berlin, Germany
Fabian Danecker
  • Zuse Institute, Berlin, Germany
Martin Weiser
  • Zuse Institute, Berlin, Germany

Funding

This research was funded by the DFG Research Center of Excellence MATH^+ - Berlin Mathematics Research Center, Project TrU 4.

Acknowledgements

We thank three anonymous referees for helpful comments that improved the presentation of this paper.

Cite AsGet BibTex

Ralf Borndörfer, Fabian Danecker, and Martin Weiser. A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/OASIcs.ATMOS.2022.2

Abstract

We present an efficient algorithm that finds a globally optimal solution to the 2D Free Flight Trajectory Optimization Problem (aka Zermelo Navigation Problem) up to arbitrary precision in finite time. The algorithm combines a discrete and a continuous optimization phase. In the discrete phase, a set of candidate paths that densely covers the trajectory space is created on a directed auxiliary graph. Then Yen’s algorithm provides a promising set of discrete candidate paths which subsequently undergo a locally convergent refinement stage. Provided that the auxiliary graph is sufficiently dense, the method finds a path that lies within the convex domain around the global minimizer. From this starting point, the second stage will converge rapidly to the optimum. The density of the auxiliary graph depends solely on the wind field, and not on the accuracy of the solution, such that the method inherits the superior asymptotic convergence properties of the optimal control stage.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Continuous functions
  • Mathematics of computing → Discretization
  • Mathematics of computing → Discrete optimization
  • Mathematics of computing → Continuous optimization
  • Mathematics of computing → Nonconvex optimization
  • Mathematics of computing → Graph algorithms
Keywords
  • shortest path
  • flight planning
  • free flight
  • discretization error bounds
  • optimal control
  • discrete optimization
  • global optimization

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