Iterative Optimizers

8The Explo2 Algorithm

Coding exploitation is relatively simple, even when using a D-sphere S (i, αr). In fact, it suffices to randomly sample in its interior, in a uniform or non-uniform manner (see source code A.12.1).

With a D-cube, it is even easier to sample randomly in its interior and if this must be done according to a probabilistic distribution denser toward the center, then we can, for example, apply a similar approach to that given for the greedy algorithm (see section A.11): each coordinate of the new point is an instance of the distribution

[8.1]

up to a linear transformation (translation to center in i and homothety to adjust the support to the side of the D-cube). The larger K is, the more closely the distribution approaches the normal distribution.

However, for exploration, it is more complicated. How, indeed, can we sample inside a complexly shaped domain, which is the complementary (with respect to the search space) of a union of D-spheres or D-cubes? In the implementation covered in Chapter 7, it is well understood that when the number of sampled positions (and thereby of potential exploitation domains) increases, this procedure is likely to take longer, before a satisfactory point is found.

Nonetheless, the experiment shows that the number of attempts remains reasonable in general, except in dimension 1, where it is better to use the specific method. It should ...