Nonlinear least squares regression of multiple weighted simultaneous equations using a general purpose optimization routine

General purpose optimization routines, such as the Simplex, the conjugate gradient, a quasi Newton, etc., can be used for the nonlinear least squares regression of parameters for systems of simultaneous equations. 

The nonlinear regression is done by using the total weighted square error as the objective function and then minimization of it using by varying the regression parameters.

When implementing this algorithm, most of the development work is usually devoted to creating the optimization routine.  After that is created, the implementation of the nonlinear least squares regression of the model/regression parameters to the data is fairly straight forward.  The sum of the weighted square errors is used as the objective function and the model/regression parameters are used as the optimization parameters.