Iterative Optimizers

3Landscape Typology

In dimension D = 1, on a digital computer, it can be considered that the definition space contains N equidistant values, namely , similarly to the space of values. A landscape is then a mapping from I_N onto I_N, more generally, in dimension D, from onto I_N. In general, we have N = 2^K−1 + 1, where K is an integer such as 32, 64, 128 and so on.

To establish a typology of the set of landscapes, I shall consider only the following three criteria, relatively easy to take into account in practice:

• – the distance–value correlation;
• – plateaus (their size, their value);
• – minima (basins of attraction and values).

A priori, the relative positions of basins of attraction could play a role, but experimentally, it proves to be of little significance (see section A.8). In this quite formal chapter, we shall talk about functions rather than landscapes, and triplets will be largely used. Remember a few definitions, seen in section 2.7:

A triplet is a set {x₁, x₂, x₃}, , for which x₁ ≠ x₂, x₁ ≠ x₃ and x₂ ≠ x₃.

For every function f from onto I_N, three subsets of triplets are ...