In Chapter 3, we defined a possible typology of every function (or landscape). Given that a test case comprises a restricted number of functions, its structure cannot be completely in accordance with this typology. Choices will have to be made as well as compromises that bias the test case.
5.1. Structure of a representative test case
As we have seen, unimodal functions are negligible and, therefore, the test case should not include any. In practice, however, it is useful to have at least one, to ensure that the optimizer to be tested is capable of finding the minimum in a simple case. It should be noted that, from a certain point of view, all unimodal functions are equivalent (see section A.6).
In addition, except in dimension 1, the proportion of functions with m global minima should be zero for any m, which is obviously impossible with a limited test case. A possible compromise could be a case including a proportion 1 of functions with a single global minimum, a proportion 2 of functions with two global minima and so on up to a given value of m, using formula [3.9].
The constraint is that . In fact, this tradeoff is also valid for the first dimension, except that, ...