Heuristic Strategies of Modified Levenberg–Marquardt Algorithm for Fitting Transition Curves

Abstract

Horizontal curve identification is important both for road safety management and railway maintenance. Parameters of a transition curve are introduced to perform an orthogonal least-squares fitting. During such a fitting process, the Gauss-Newton (GN) method may fail to converge because of an ill-conditioned Hessian matrix. A biobjective fitting model is introduced, and the Levenberg–Marquardt (LM) algorithm is specified to perform the fitting of transition curves. The LM parameter is updated heuristically during iterations according to the specific information explored instead of the standard preset way. Further, another heuristic strategy is proposed to search a path to the optimum instead of the traditional greedy strategy. The heuristic strategies were compared with traditional ones by fitting a transition curve of a railway to the measured points. Monte Carlo simulations were employed to test the robustness and efficiency of the modified LM algorithm, with different initial values, all converging to the same optimum. Results showed that the heuristic strategy for updating the LM parameter has a better robustness than the preset way, and the heuristic strategy for searching a path converges much faster than the traditional one, for which visual interpretations are provided.