Iterative Optimizers

6Difficulty vs Dimension

Expressions such as “Let us study the same problem in different dimensions” can sometimes be seen. This is very often misleading, because even if landscapes are defined by formulae that are mathematically identical with the dimension as the only variable parameter, their structures are usually different. We shall examine two examples for which it is clearly the case and, on the contrary, two examples in which the structure is preserved when the dimension increases.

6.1. Rosenbrock function

The landscape is defined by the function f:

[6.1]

with in general k = 100. The problem is not separable. It is unimodal for D ≤ 3 and the minimum is at point (1, … , 1), therefore on a diagonal of the search space. However, it becomes bimodal for 4 ≤ D ≤ 7, where the second (local) minimum is variable according to the value of D. Thereby, it can not be said that this is the “same problem” in different dimensions. A detailed study of this behavior is given in Shang and Qiu (2006).

6.2. Griewank function

It is defined by

Its minimum is at the origin of coordinates and is equal to zero.

Figure 6.1. The Griewank function in dimensions 1 and 2. For a color version of this ...