On non-combinatorial weighted total least squares with inequality constraints

Observation systems known as errors-in-variables (EIV) models with model parameters estimated by total least squares (TLS) have been discussed for more than a century, though the terms EIV and TLS were coined much more recently. So far, it has only been shown that the inequality-constrained TLS (ICT...

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Vydáno v:Journal of geodesy Ročník 88; číslo 8; s. 805 - 816
Hlavní autor: Fang, Xing
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014
Springer Nature B.V
Témata:
Earth and Environmental Science
Earth Sciences
Geodetics
Geophysics/Geodesy
Optimization
Original Article
Quadratic programming
Variables
Sequential quadratic programming (SQP)
Non-convex optimization
Second-order sufficient conditions
Linear inequality constraints
Weighted total least squares (WTLS)
Errors-in-variables (EIV) model
Karush–Kuhn–Tucker (KKT) necessary conditions
Active set method
WTLS contours
ISSN:0949-7714, 1432-1394
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Shrnutí:Observation systems known as errors-in-variables (EIV) models with model parameters estimated by total least squares (TLS) have been discussed for more than a century, though the terms EIV and TLS were coined much more recently. So far, it has only been shown that the inequality-constrained TLS (ICTLS) solution can be obtained by the combinatorial methods, assuming that the weight matrices of observations involved in the data vector and the data matrix are identity matrices. Although the previous works test all combinations of active sets or solution schemes in a clear way, some aspects have received little or no attention such as admissible weights, solution characteristics and numerical efficiency. Therefore, the aim of this study was to adjust the EIV model, subject to linear inequality constraints. In particular, (1) This work deals with a symmetrical positive-definite cofactor matrix that could otherwise be quite arbitrary. It also considers cross-correlations between cofactor matrices for the random coefficient matrix and the random observation vector. (2) From a theoretical perspective, we present first-order Karush–Kuhn–Tucker (KKT) necessary conditions and the second-order sufficient conditions of the inequality-constrained weighted TLS (ICWTLS) solution by analytical formulation. (3) From a numerical perspective, an active set method without combinatorial tests as well as a method based on sequential quadratic programming (SQP) is established. By way of applications, computational costs of the proposed algorithms are shown to be significantly lower than the currently existing ICTLS methods. It is also shown that the proposed methods can treat the ICWTLS problem in the case of more general weight matrices. Finally, we study the ICWTLS solution in terms of non-convex weighted TLS contours from a geometrical perspective.
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ISSN:0949-7714
1432-1394
DOI:10.1007/s00190-014-0723-y