Convergent Network Approximation for the Continuous Euclidean Length Constrained Minimum Cost Path Problem

In many path planning situations we would like to find a path of constrained Euclidean length in R2 that minimises a line integral. We call this the Continuous Length-Constrained Minimum Cost Path Problem (C-LCMCPP). Generally, this will be a non-convex optimization problem, for which continuous approaches only ensure locally optimal solutions. However, network discretisations yield weight constrained network shortest path problems (WCSPPs), which can in practice be solved to global optimality, even for large networks; we can readily find a globally optimal solution to an approximation of the C-LCMCPP. Solutions to these WCSPPs yield feasible solutions and hence upper bounds. We show how networks can be constructed, and a WCSPP in these networks formulated, so that the solutions provide lower bounds on the global optimum of the continuous problem. We give a general convergence scheme for our network discretisations and use it to prove that both the upper and lower bounds so generated converge to the global optimum of the C-LCMCPP, as the network discretisation is refined. Our approach provides a computable lower bound formula (of course the upper bounds are readily computable). We give computational results showing the lower bound formula in practice, and compare the effectiveness of our network construction technique with that of standard grid-based approaches in generating good quality solutions. We find that for the same computational effort, we are able to find better quality solutions, particularly when the length-constraint is tighter.