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WLS general formulation is

(2.31a)

subject to

(2.31b)

(2.31c)

where m is the number of measurements.

Figure 2.2 depicts the objective function for the WLS estimator in the case of one single error. The violet and green curves represent a high and low weighting factor ω_i, respectively. Note that the weighted measurement error y_i(x) = w_i ⋅ (h_i(x) − z_i) appears as y_i(x)² in the objective function J(x).

Figure 2.2 Objective function for the WLS estimator as a function of the error (one measurement).

WLS estimation is typically obtained by solving the first‐order optimality conditions of problem (2.31) via the Newton–Raphson method [18]. Note that nonlinear problem (2.31) is already a mathematical programming problem that is ready to be solved by an appropriate solver, e.g. MINOS [11], within an optimization environment such as AMPL [13] or GAMS [12].