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    <title language="eng">Convergence Properties of Newton's Method for Globally Optimal Free Flight Trajectory Optimization</title>
    <abstract language="eng">The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.</abstract>
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    <author>Ralf Borndörfer</author>
    <submitter>Fabian Danecker</submitter>
    <author>Fabian Danecker</author>
    <author>Martin Weiser</author>
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      <title>ZIB-Report</title>
      <number>23-19</number>
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      <language>eng</language>
      <type>uncontrolled</type>
      <value>shortest path</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>flight planning</value>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>free flight</value>
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      <value>optimal control</value>
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    <title language="eng">A Discrete-Continuous Algorithm for Free Flight Planning</title>
    <abstract language="eng">We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.</abstract>
    <parentTitle language="eng">Algorithms</parentTitle>
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    <author>Ralf Borndörfer</author>
    <submitter>Fabian Danecker</submitter>
    <author>Fabian Danecker</author>
    <author>Martin Weiser</author>
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      <language>eng</language>
      <type>uncontrolled</type>
      <value>shortest path</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>flight planning</value>
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    <subject>
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      <type>uncontrolled</type>
      <value>free flight</value>
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      <value>discrete-continuous algorithm</value>
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      <value>discrete optimization</value>
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    <title language="eng">Newton's Method for Global Free Flight Trajectory Optimization</title>
    <abstract language="eng">Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton's method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.</abstract>
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    <id>9204</id>
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    <title language="eng">Convergence Properties of Newton’s Method for Globally Optimal Free Flight Trajectory Optimization</title>
    <abstract language="eng">The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.</abstract>
    <parentTitle language="eng">23rd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2023)</parentTitle>
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    <title language="eng">Newton's Method for Global Free Flight Trajectory Optimization</title>
    <abstract language="eng">Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton's method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.</abstract>
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    <title language="eng">Error Bounds for Discrete-Continuous Free Flight Trajectory Optimization</title>
    <abstract language="eng">Flight planning, the computation of optimal routes in view of flight time and  fuel consumption under given weather conditions, is traditionally done by finding globally shortest paths in a predefined airway network. Free flight trajectories, not restricted to a network, have the potential to reduce the costs significantly, and can be computed using locally convergent continuous optimal control methods.&#13;
&#13;
Hybrid methods that start with a discrete global search and refine with a fast continuous local optimization combine the best properties of both approaches, but rely on a good switchover, which requires error estimates for discrete paths relative to continuous trajectories.&#13;
    &#13;
Based on vertex density and local complete connectivity, we derive localized and a priori bounds for the flight time of discrete paths relative to the optimal continuous trajectory, and illustrate their properties on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter bounds using a posteriori error estimators.</abstract>
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    <title language="eng">A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization</title>
    <abstract language="eng">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&#13;
solution, such that the method inherits the superior asymptotic convergence properties of the optimal control stage.</abstract>
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    <submitter>Martin Weiser</submitter>
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    <author>Martin Weiser</author>
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    <title language="eng">A Discrete-Continuous Algorithm for Free Flight Planning</title>
    <abstract language="eng">We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.</abstract>
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    <identifier type="doi">10.3390/a14010004</identifier>
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    <enrichment key="SourceTitle">Algorithms</enrichment>
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    <author>Ralf Borndörfer</author>
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    <author>Fabian Danecker</author>
    <author>Martin Weiser</author>
    <series>
      <title>ZIB-Report</title>
      <number>20-33</number>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>shortest path</value>
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    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>flight planning</value>
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      <language>eng</language>
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      <value>free flight</value>
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      <value>discrete-continuous algorithm</value>
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      <language>eng</language>
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      <language>eng</language>
      <type>uncontrolled</type>
      <value>discrete optimization</value>
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    <collection role="institutes" number="traffic">Mathematics of Transportation and Logistics</collection>
    <collection role="persons" number="borndoerfer">Borndörfer, Ralf</collection>
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