4.3.4 Constraints

As the optimizer requests new steps with a local knowledge of the value vector and the Jacobian matrix, it is possible that without additional features the used models are not defined or stable^4.4 for the requested parameter vector. The use of constraints for the parameter values allows to confine the requested parameters in a stable area.

The transformation makes it necessary to distinguish between internal and external parameter values. The external values are those set on input and are requested from the optimizer executable from the model. For these values the constraints are active. The internal parameters are those used by the optimizer kernel and are not bounded by constraints.

The function to transform constrained, external parameters x_external to unconstrained, internal values x_internal is

$$x_{internal} = \tan \left ( \frac{ x_{external} - \frac... ...{\frac{x_{max} + x_{min}}{2}} \cdot \frac{\pi \ f}{2} \right )$$ (4.42)

and the reverse function is

$$x_{external} = \frac{\arctan (x_{internal}) }{ \frac{\pi ... ... \frac{x_{max} + x_{min}}{2} + \frac{x_{max} - x_{min}}{2}$$ (4.43)

where all values are found on the main branch of the $\arctan$ function.

For numerical stability the factor f is included in (4.42). For a value f < 1 the minimum and maximum values (x_min and x_max) are not mapped to $- \infty$ and $+ \infty$. Thus the start values can be set equal to an upper or lower boundary and that the finite differences in these points can be calculated at all. On the other hand this factor implies a possible overshoot o of

$$o^{[\%]} = \frac{100}{f} - 100$$ (4.44)

for the factor f = 0.99999 this value is about $0.001\ \%$. This has to be considered when the values of the constraints are assigned. For a parameter with a range from 0 - $1 \cdot 10^{8}$ the possible range is $- 1 \cdot 10^{4}$ - $1.00001 \cdot 10^{8}$. If this is ignored stability problems in the optimization cycle can arise, for example if simulations with negative parameter values are requested.

In Figure 4.2 the mapping of the internal and external parameter is shown. The real, external parameters from the x axis are transformed using the solid curve to the internal values on the y axis. The dotted lines are the minimum and maximum values of the external parameter. The dashed lines are the tangents of the solid tangents function. These are the absolute minimum and maximum values that can occur during optimization resulting from the overshoot calculated in (4.44).

Figure 4.2: Transformation for the bound constraints.

Footnotes

... stable^4.4

In some cases simulators do not converge for certain parameter sets. This makes it also necessary to reduce the input parameter space.