Both scaling—substituting scalar multiples of optimization parameters for those in the original model—and reparameterization—substituting invertible functions of optimization parameters for the parameters themselves—are tools that sometimes assist us to get solutions or else to reduce the computational effort to get solutions. Of course, scaling is but a simple form of reparameterization.
In this chapter, we will look at this topic. Unfortunately, while scaling and reparameterization are often recommended and are mathematically relatively simple, they are not trivial to use because they need to be applied carefully and, as with many topics in nonlinear optimization, the results may both help and impede the process of obtaining good results.
16.1 Why scale or reparameterize?
The essence of nonlinearity of a function is that the value of cannot be expressed in terms where the parameters appear to power 1. This can be important when we want to understand the functional surface of . In some regions, a standard change—either absolute or percentage—in one of the parameters x will result in a small change in the value of while in others the change ...