利用不等式约束求解病态问题的新算法

A New Algorithm for Ill-Posed Problems with Inequality Constraints

病态问题是大地测量数据处理中常见的问题,充分利用平差过程所给的先验信息可以确保参数的可靠性和有效性。提出了一种利用不等式约束求解病态问题的新算法,该算法将先验信息表示为不等式形式,并与病态模型构成不等式约束平差模型。结合Karush-Kuhn-Tucker条件可将该模型转化为线性互补问题,然后利用Lemke算法求解。该法避免了对病态矩阵求逆,保证了参数解的唯一性和稳定性。最后,本文模拟了未知参数附先验信息的Hilbert矩阵及全球定位系统(Global Positioning System,GPS)快速定位实验,并结合多种经典的病态平差方法,验证了Lemke算法在处理病态问题上的有效性。

The ill-posed problem is a common problem in geodetic data processing.Making full use of the prior information given by the adjustment process could ensure the reliability and effectiveness of the parameters.In this paper,a new algorithm was proposed to solve ill-posed problems by using inequality constraints.The algorithm represents the prior information in the form of inequality and forms an inequality constrained adjustment model with ill-posed model.Combined with Karush-Kuhn-Tucker condition,the model could be transformed into a linear complementarity problem,and then solved by Lemke algorithm.The method avoids matrix inversion and ensures the uniqueness and stability of parameter solution.Finally,this paper simulated the Hilbert matrix with prior information of unknown parameters and GPS fast positioning experiment,and combined with a variety of classical ill-posed adjustment methods,verified the effectiveness of Lemke algorithm in dealing with the ill-posed problem.