For a flying military vehicle, avoiding detection can be a key objective. To achieve this, flying the leastprobability-of-detection path from A to B through a field of detectors is a fundamental strategy. While most of the previous optimization models aim to minimize the cumulative radar exposure, this paper derives a model that can directly minimize the probability of being detected. Furthermore, a variational dynamic programming method is applied to this model which allows one to find a precise locally optimal path with low computational complexity, even when Doppler effects increase the dimension of this problem. In addition, a homotopy method with exceptionally low computational complexity is derived to adjust the optimal path when the detection rate function changes due to the removal of detectors, the addition of detectors or other changes of detectors. Finally, the paper also shows how to apply a convex optimization method to find optimal positions of detectors when vehicles can do path planning. (C) 2015 Elsevier Ltd. All rights reserved.