Weighted Total Least Squares with Constraints: Alternative Derivation without Using Lagrange Multipliers

Abstract

This technical note presents alternative derivations for the weighted total least-squares (WTLS) problem subject to weighted and hard constraints. The derivations do not take into account the Lagrange multipliers, and the final results are shown to be identical to those presented by a recently published article in the same journal. The final WTLS estimates are formulated by the standard least-squares theory. The method, which is formulated using the Gauss–Newton method, provides an alternative to the Newton method. The formulation is generally presented for the weighted linear(ized) constraints. As a by-product, hard constraints turn out to be a special case of the general formulation of the weighted constraints. An alternative derivation is presented for the WTLS problem subjected only to hard constraints. For this purpose, the parametric representation of the hard constraints is used, the general solution of which is obtained as the sum of a particular solution and the homogeneous solution. As a special case, the WTLS formulation (without the constraints) can also directly be followed.