﻿ 非线性不等式约束平差的一种新岭估计算法
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 大地测量与地球动力学  2022, Vol. 42 Issue (5): 494-498  DOI: 10.14075/j.jgg.2022.05.010

### 引用本文

LI Wenna, SONG Yingchun, DENG Wei, et al. A New Ridge Estimation Algorithm for Nonlinear Inequality Constrained Adjustment[J]. Journal of Geodesy and Geodynamics, 2022, 42(5): 494-498.

### Foundation support

National Natural Science Foundation of China, No.41674009, 41574005.

### 第一作者简介

LI Wenna, postgraduate, majors in surveying adjustment and data processing, E-mail: 2312150159@qq.com.

### 文章历史

1. 中南大学地球科学与信息物理学院, 长沙市麓山南路932号，410083;
2. 中南林业科技大学土木工程学院, 长沙市韶山南路498号，410004

1 非线性不等式约束平差模型及其算法

 $\boldsymbol{L}=\boldsymbol{A} \boldsymbol{X}+\boldsymbol{e}$ (1)

 $\left\{\begin{array}{l} \boldsymbol{L}=\boldsymbol{A X}+\boldsymbol{e} \\ \text {s. t. }\|\boldsymbol{X}\|^{2} \leqslant c \end{array}\right.$ (2)

 \begin{aligned} &\min (\boldsymbol{L}-\boldsymbol{A} \boldsymbol{X})^{\mathrm{T}}(\boldsymbol{L}-\boldsymbol{A} \boldsymbol{X}) \\ &\text {s. t. }\|\boldsymbol{X}\|^{2}=c \end{aligned} (3)

 $\boldsymbol{X}(\lambda)=\left(\boldsymbol{A}^{\mathrm{T}} \boldsymbol{A}+\lambda \boldsymbol{I}\right)^{-1} \boldsymbol{A}^{\mathrm{T}} \boldsymbol{L}$ (4)

 $\boldsymbol{A}=\boldsymbol{P}\left[\begin{array}{l} \boldsymbol{B} \\ 0 \end{array}\right] \boldsymbol{Q}^{\mathrm{T}}$ (5)

 $\boldsymbol{B}=\boldsymbol{C S D}^{\mathrm{T}}$ (6)

 $\boldsymbol{A}=\boldsymbol{P}\left[\begin{array}{c} \boldsymbol{C S D}^{\mathrm{T}} \\ 0 \end{array}\right] \boldsymbol{Q}^{\mathrm{T}}=\boldsymbol{U} \boldsymbol{\varSigma} \boldsymbol{V}^{\mathrm{T}}$

 $\boldsymbol{X}_{1}(\lambda)=\left(\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}+\lambda \boldsymbol{I}\right)^{-1} \boldsymbol{B}^{\mathrm{T}} \boldsymbol{L}_{1}$ (7)

 $\min \left\|\left(\begin{array}{c} \boldsymbol{B} \\ \sqrt{\lambda} \boldsymbol{I} \end{array}\right) \boldsymbol{X}_{1}(\lambda)-\left(\begin{array}{c} \boldsymbol{L}_{1} \\ 0 \end{array}\right) \right\|^{2}$ (8)

 $\boldsymbol{W}^{\mathrm{T}}\left(\begin{array}{c|c} \boldsymbol{B} & \boldsymbol{L}_{1} \\ \sqrt{\lambda} \boldsymbol{I} & 0 \end{array}\right)=\left(\begin{array}{c|c} \boldsymbol{B}_{\lambda} & \boldsymbol{Z}_{1} \\ 0 & \boldsymbol{Z}_{2} \end{array}\right)$ (9)

 $\boldsymbol{B}_{\lambda} \boldsymbol{X}_{1}(\lambda)=\boldsymbol{Z}_{1}$ (10)

X1(λ)=QTX(λ)得‖X(λ)‖2=‖X1(λ)‖2

 $\begin{gathered} \omega(\lambda)=\|\boldsymbol{X}(\lambda)\|^{2}=\left\|\boldsymbol{X}_{1}(\lambda)\right\|^{2}=\\ \boldsymbol{L}_{1}^{\mathrm{T}} \boldsymbol{B}\left(\boldsymbol{B}^{\mathrm{T}} \boldsymbol{B}+\lambda \boldsymbol{I}\right)^{-2} \boldsymbol{B}^{\mathrm{T}} \boldsymbol{L}_{1} \end{gathered}$ (11)

 $\omega(\lambda)=\boldsymbol{L}_{1}^{\mathrm{T}} \boldsymbol{C} \boldsymbol{S}\left(\boldsymbol{S}^{2}+\lambda \boldsymbol{I}\right)^{-2} \boldsymbol{S C}^{\mathrm{T}} \boldsymbol{L}_{1}=\sum\limits_{i=1}^{n}\left(\frac{\sigma_{i} \widetilde{L}_{i}}{\sigma_{i}^{2}+\lambda}\right)^{2}$ (12)

 $\lambda_{i+1}=\lambda_{i}-\frac{\omega\left(\lambda_{i}\right)}{\omega^{\prime}\left(\lambda_{i}\right)-\left(\omega^{\prime \prime}\left(\lambda_{i}\right) \omega\left(\lambda_{i}\right) / 2 \omega^{\prime}\left(\lambda_{i}\right)\right)}$ (13)

2 参数的确定方法

 $\sigma_{1} \geqslant \cdots \geqslant \sigma_{n-1} \geqslant \sigma_{n}$

 $c_{\mathrm{LS}} \approx\left(\frac{\tilde{L}_{n}}{\sigma_{n}}\right)^{2} \gg\left(\frac{\tilde{L}_{i}}{\sigma_{i}}\right)^{2}, 1 \leqslant i \leqslant n-1$

 $\frac{\sigma_{i}^{2} \widetilde{L}_{i}^{2}}{\left(\sigma_{i}^{2}+\lambda\right)^{2}} \ll \frac{\sigma_{n}^{2} \widetilde{L}_{n}^{2}}{\left(\sigma_{n}^{2}+\lambda\right)^{2}}, 1 \leqslant i \leqslant n-1$

 $c \approx \frac{\sigma_{n}^{2} \tilde{L}_{n}^{2}}{\left(\sigma_{n}^{2}+\lambda\right)^{2}} \approx \frac{\sigma_{n}^{4} c_{\mathrm{LS}}}{\left(\sigma_{n}^{2}+\lambda\right)^{2}}$

 $\hat{\lambda}=\sigma_{n}^{2}\left(\sqrt{\frac{c_{\mathrm{LS}}}{c}}-1\right) \approx \lambda^{*}$ (14)

ω($\hat{\lambda}$)的值不会远大于c，否则令$\hat{\lambda}$作为迭代初始值相比于0作为迭代初始值的优势不明显。

 $\varphi(\sigma)=(\sigma+\hat{\lambda} / \sigma)^{2}$

 $\varphi^{\prime}\left(\sigma^{*}\right)=2\left(\sigma^{*}+\hat{\lambda} / \sigma^{*}\right)\left(1-\hat{\lambda} /\left(\sigma^{*}\right)^{2}\right)=0$

 $\sigma^{*}=\sqrt{\hat{\lambda}}, \varphi\left(\sigma^{*}\right)=4 \hat{\lambda}$

σ=σn时，φ(σn)=σn2·cLS/c。如果$\varphi(\hat{\sigma})$=φ(σn)，且$\hat{\sigma} \neq \sigma_{n}$，有：

 $\hat{\sigma}=\frac{\hat{\lambda}}{\sigma_{n}}=\sigma_{n}\left(\sqrt{\frac{c_{\mathrm{LS}}}{c}}-1\right)$

$\theta=\sqrt{\frac{c_{\mathrm{LS}}}{c}}-1$，则$\sigma^{*}=\sigma_{n} \sqrt{\theta}, \hat{\sigma}=\sigma_{n} \theta$。为便于分析，根据以上变量，绘制φ(σ)在θ>1和θ≤1情况下的大致图形(图 12)。

 图 1 θ>1时φ(σ)的图像 Fig. 1 Image of φ(σ) when θ>1

 图 2 θ≤1时φ(σ)的图像 Fig. 2 Image of φ(σ) when θ≤1

1) 假设φ(σi)≥φ(σn), i < n，其等同于：

 $\left(\sigma_{i}-\sigma_{n}\right)\left(\sigma_{i}-\sigma_{n} \theta\right) \geqslant 0$ (15)

 $\omega(\hat{\lambda})=\sum\limits_{i=1}^{n} \tilde{L}_{i}^{2} / \varphi\left(\sigma_{i}\right) \leqslant\|\tilde{\boldsymbol{L}}\|^{2} / \varphi\left(\sigma_{n}\right)$ (16)

$\eta=\|\tilde{\boldsymbol{L}}\|^{2} / \tilde{L}_{n}^{2} \geqslant 1$，可以将式(16)简化为：

 $\begin{gathered} \omega(\hat{\lambda}) \leqslant \frac{\|\tilde{\boldsymbol{L}}\|^{2}}{\sigma_{n}^{2}} \cdot \frac{c}{c_{\mathrm{LS}}}=\frac{\|\widetilde{\boldsymbol{L}}\|^{2}}{\widetilde{L}_{n}^{2}} \cdot \frac{\widetilde{L}_{n}^{2}}{\sigma_{n}^{2}} \cdot \frac{c}{c_{\mathrm{LS}}} \leqslant\\ \frac{\left\|\tilde{\boldsymbol{L}}\right\|^{2}}{{\widetilde{L}}_{n}^{2}} c=\eta c \end{gathered}$

 $\left\|\boldsymbol{X}_{1}(\hat{\lambda})\right\|^{2} / c \leqslant \eta$

2) 假设θ>1，且 B的奇异值有一些分布在区间Y内，则存在：

 $\varphi\left(\sigma_{i}\right) \geqslant \varphi\left(\sigma^{*}\right)=4 \sigma_{n}^{2} \theta, i \leqslant n$

 $\begin{gathered} \frac{\left\|\boldsymbol{X}_{1}(\hat{\lambda})\right\|^{2}}{c} \leqslant \frac{\|\widetilde{\boldsymbol{L}}\|^{2}}{4 \sigma_{n}^{2} \theta c} \leqslant \\ (\eta / 4) \sqrt{\frac{c_{\mathrm{LS}}}{c}}\left(1-\sqrt{\frac{c}{c_{\mathrm{LS}}}}\right) \leqslant(\eta / 2) \sqrt{c_{\mathrm{LS}} / c} \end{gathered}$

3 数值实验 3.1 数值模拟实验

 图 3 本文算法与岭估计法的比较 Fig. 3 Comparison of our algorithm and ridge estimation algorithm

3.2 病态测边网实验

 图 4 空间测边网[11] Fig. 4 Spatial geodesic netwo

4 结语

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A New Ridge Estimation Algorithm for Nonlinear Inequality Constrained Adjustment
LI Wenna1     SONG Yingchun1     DENG Wei1     XIE Xuemei2
1. School of Geosciences and Info-Physics, Central South University, 932 South-Lushan Road, Changsha 410083, China;
2. School of Civil Engineering, Central South University of Forestry and Technology, 498 South-Shaoshan Road, Changsha 410004, China
Abstract: This paper uses the a priori information to constrain the parameters, establishes a nonlinear inequality constrained adjustment model, and proposes a new ridge estimation algorithm for solving this adjustment model. The computational results of the two ill-posed examples in this paper show that the new ridge estimation algorithm is feasible, and the accuracy of the calculation results is higher than that of ordinary ridge estimation.
Key words: ill-posed problem; prior information; nonlinear; ridge estimation; double diagonalization