TY - JOUR

T1 - An algorithmic approach to the total least-squares problem with linear and quadratic constraints

AU - Schaffrin, Burkhard

AU - Felus, Yaron A.

N1 - Funding Information:
Acknowledgements: The authors would like to thank D.M. Sima for providing the code for the regularized total least-squares solver. The first author would also like to acknowledge the support of the German Science Foundation (DFG Mercator Professorship); he is also most grateful to Prof. G. Schmitt, his host at the University of Karlsruhe (TH), Germany. The second author was partially supported by the National Geospatial-Intelligence Agency under contract No.HM1582-04-1-2026.

PY - 2009

Y1 - 2009

N2 - Proper incorporation of linear and quadratic constraints is critical in estimating parameters from a system of equations. These constraints may be used to avoid a trivial solution, to mitigate biases, to guarantee the stability of the estimation, to impose a certain "natural" structure on the system involved, and to incorporate prior knowledge about the system. The Total Least-Squares (TLS) approach as applied to the Errors-In-Variables (EIV) model is the proper method to treat problems where all the data are affected by random errors. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear constraints and TLS problems with a quadratic constraint. In this contribution, a new algorithm is presented to solve TLS problems with both linear and quadratic constraints. The new algorithm is developed using the Euler-Lagrange theorem while following an optimization process that minimizes a target function. Two numerical examples are employed to demonstrate the use of the new approach in a geodetic setting.

AB - Proper incorporation of linear and quadratic constraints is critical in estimating parameters from a system of equations. These constraints may be used to avoid a trivial solution, to mitigate biases, to guarantee the stability of the estimation, to impose a certain "natural" structure on the system involved, and to incorporate prior knowledge about the system. The Total Least-Squares (TLS) approach as applied to the Errors-In-Variables (EIV) model is the proper method to treat problems where all the data are affected by random errors. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear constraints and TLS problems with a quadratic constraint. In this contribution, a new algorithm is presented to solve TLS problems with both linear and quadratic constraints. The new algorithm is developed using the Euler-Lagrange theorem while following an optimization process that minimizes a target function. Two numerical examples are employed to demonstrate the use of the new approach in a geodetic setting.

KW - Adjustment with constraints

KW - Non-convex optimization

KW - Total least-squares

UR - http://www.scopus.com/inward/record.url?scp=62549146224&partnerID=8YFLogxK

U2 - 10.1007/s11200-009-0001-2

DO - 10.1007/s11200-009-0001-2

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AN - SCOPUS:62549146224

SN - 0039-3169

VL - 53

SP - 1

EP - 16

JO - Studia Geophysica et Geodaetica

JF - Studia Geophysica et Geodaetica

IS - 1

ER -