引用本文: 谢建, 龙四春, 李黎, 李博超. 不等式约束加权整体最小二乘的凝聚函数法[J]. 武汉大学学报 ( 信息科学版), 2018, 43(10): 1526-1530.
XIE Jian, LONG Sichun, LI Li, LI Bochao. An Aggregate Function Method for Weighted Total Least Squares with Inequality Constraints[J]. Geomatics and Information Science of Wuhan University, 2018, 43(10): 1526-1530.
 Citation: XIE Jian, LONG Sichun, LI Li, LI Bochao. An Aggregate Function Method for Weighted Total Least Squares with Inequality Constraints[J]. Geomatics and Information Science of Wuhan University, 2018, 43(10): 1526-1530.

## An Aggregate Function Method for Weighted Total Least Squares with Inequality Constraints

• 摘要: 误差向量的方差-协方差阵是一般对称正定矩阵下的附不等式约束加权整体最小二乘平差模型，研究了其参数估计和精度评定问题。首先，将残差平方和极小化函数在整体最小二乘准则下转化为只包含模型参数的目标函数，同时将所有的不等式约束表示成一个等价的凝聚约束函数，并运用乘子罚函数策略将不等式约束加权整体最小二乘平差问题转化为相应的无约束最优化问题，并用BFGS方法求解。然后，将误差方程和约束函数线性展开，推导了最优解和观测量间的近似线性函数关系，运用方差-协方差传播律得到了最优解的近似方差。最后，用数值实例验证了方法的有效性和可行性。

Abstract: When the variance-covariance matrix of the error is fairly symmetric and positively definite, parameter estimation and accuracy assessment of weighted total least squares adjustment with inequality constraints(ICWTLS) are investigated in this paper. First, the problem of minimizing the residual sum of squares are converted to an optimization problem with only the model parameters under the total least squares, and all the inequality constraints are transformed into an equivalent aggregate constraint. Accordingly, the ICWTLS problem is converted to unconstrained optimization problem by a penalty function approach and is then solved by BFGS optimization method. Then, the observation equation and constraints function are expanded to the first order Taylor series and the linear approximation between the solution and observations is derived as well as the approximate dispersion matrix of the solution under the variance propagation law. Finally, a numerical simulation is given to indicate the validity and feasibility of this method.

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