缓和曲线正交拟合的Levenberg-Marquardt算法

为了由测量点识别既有线路中的缓和曲线参数，研究了基于参数方程的缓和曲线正交拟合迭代优化方法. 首先，通过特征值分析，阐明了由于病态性的存在，在迭代过程中，常规的Gauss-Newton （GN）算法会发散. 其次，提出了双目标优化模型，将GN算法与最速下降法结合，确定了正交拟合缓和曲线的Levenberg-Marquardt （LM）算法. 同时提出了在寻优过程中，评估当前迭代位置距离最优位置的远近来动态设置LM参数. 最后以一段缓和曲线的实测点为例，随机取样了5 000例初值，采用蒙特卡罗方法对比了GN算法和LM算法拟合缓合曲线的性能. 试验结果表明:GN算法拟合缓合曲线不收敛；对于不同的初始值，LM算法都收敛到相同的最优值，体现了LM算法具有良好的稳健性；LM算法的迭代次数最少为5次，最大为50次，平均为16.8次，迭代次数和初值与最优值位置的远近相关. 

Levenberg-Marquardt Algorithm for Orthogonal Fitting of Transition Curves

To identify the parameters of transition curves in as-built alignments by measured points, orthogonal least-squares fitting is studied on the basis of the parameter equation of transition curves. First, eigenvalue analysis has clarified that the Gauss-Newton (GN) algorithm usually fails to converge because of the existence of the ill-condition. Next, a bi-objective optimization model is proposed and the Levenberg-Marquardt (LM) algorithm combining the GN algorithm with the steepest descent method is constructed to fit a transition curve to points orthogonally. The LM parameter is updated dynamically during iterations according to the evaluation of the distance between the current and the optimum locations. Finally, Monte Carlo simulations are employed to test the performances of the GN and LM algorithms with measured points and the same 5 000 initial values. Experimental results show that the GN algorithm diverges while the LM algorithm converges to the same optimum under different initial values. The number of iterations, with an average of 16.8 times and the minimum of 5 times and the maximum of 50 times, is related to the distance between the initial and the optimum locations. The LM algorithm shows a better robustness than the GN algorithm.