Convergent network approximation for the continuous euclidean length constrined minimum cost path problem

In many path-planning situations we would like to find a path of constrained Euclideanlength in R2 that minimizes a line integral. We call this the Continuous Length-ConstrainedMinimum Cost Path Problem (C-LCMCPP). Generally, this will be a nonconvex optimizationproblem, for which continuous approaches ensure only locally optimal solutions. However, networkdiscretizations yield weight constrained network shortest path problems (WCSPPs), which can inpractice be solved to global optimality, even for large networks; we can readily find a globally optimalsolution to an approximation of the C-LCMCPP. Solutions to these WCSPPs yield feasiblesolutions and hence upper bounds. We show how networks can be constructed, and a WCSPP inthese networks formulated, so that the solutions provide lower bounds on the global optimum of thecontinuous problem. We give a general convergence scheme for our network discretizations and useit to prove that both the upper and lower bounds so generated converge to the global optimum ofthe C-LCMCPP, as the network discretization is refined. Our approach provides a computable lowerbound formula (of course, the upper bounds are readily computable). We give computational resultsshowing the lower bound formula in practice, and compare the effectiveness of our network constructiontechnique with that of standard grid-based approaches in generating good quality solutions. Wefind that for the same computational effort, we are able to find better quality solutions, particularlywhen the length constraint is tighter.