GPS-X Technical Reference

Optimizer 394
In Equation 14.15, the value of 10 -7 is referred to as the finite-difference relative
perturbation size in GPS-X and can be changed in the Derivative Information subsection
of the Optimizer form.
The gradient provided in the solution report is calculated at the solution. If the solution is
a local minimum, the elements of the gradient vector should be close to zero. Because
poor scaling of the parameters and the objective function can make the gradient appear
large, the relative gradient is also reported. The elements of the relative gradient are
calculated by scaling the gradient elements using the following equation:
Equation 14.16
The infinity norm of the relative gradient vector is also reported. If the solution is a local
minimum, this norm should be close to zero.
The gradient is only reported when the report objective function gradient and Hessian
switch in the Derivative Information sub-section in the Optimizer form is set to ON.
Model Sensitivity Coefficients
The model sensitivity coefficients express the local sensitivity of the process model to
infinitesimal changes in the optimization variables (subset of the model parameters).
They are the partial derivatives of the model with respect to the optimization variables. A
sensitivity coefficient can be calculated for each target (i.e. response) variable with
respect to each optimization variable at each data point. The sensitivity coefficients are
calculated using the forward-difference approximation shown below:
Equation 14.17
In this equation f i,j is the jth target variable at the ith data point. The step size is calculated
in the same way as for the gradient.
The sensitivity coefficients are only reported when the report model sensitivity
coefficients switch in the Derivative Information sub-section in the Optimizer form is
set to ON.
GPS-X Technical Reference