Error Bounds for Discrete-Continuous Shortest Path Problems with Application to Free Flight Trajectory Optimization

TL;DR

This paper derives error bounds for discrete approximations of continuous shortest path problems, specifically applied to free flight trajectory optimization, enabling better graph design for more accurate route planning.

Contribution

It introduces a priori and localized error bounds for flight time in discrete paths, improving understanding of discretization effects in hybrid optimization methods.

Findings

Localization significantly reduces error bounds by four orders of magnitude.

Graph density directly influences the accuracy of discrete path approximations.

Bounds suggest potential for further improvement with a posteriori estimators.

Abstract

Two-stage methods addressing continuous shortest path problems start local minimization from discrete shortest paths in a spatial graph. The convergence of such hybrid methods to global minimizers hinges on the discretization error induced by restricting the discrete global optimization to the graph, with corresponding implications on choosing an appropriate graph density. A prime example is flight planning, i.e., the computation of optimal routes in view of flight time and fuel consumption under given weather conditions. Highly efficient discrete shortest path algorithms exist and can be used directly for computing starting points for locally convergent optimal control methods. We derive a priori and localized error bounds for the flight time of discrete paths relative to the optimal continuous trajectory, in terms of the graph density and the given wind field. These bounds allow designing graphs with an optimal local connectivity structure. The properties of the bounds are illustrated 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 error bounds by a posteriori estimators.