On the impact of objective function transformations on evolutionary and black-box algorithms

Abstract

Different objective functions characterize different problems. However, certain fitness transformations can lead to easier problems although they are still a model of the considered problem. In this article, the class of not worsening transformations for a simple population-based evolutionary algorithm (EA) is described completely. That is the class of functions that transfers easy problems in easy ones and difficult problems in difficult ones. Surprisingly, this class $\mathcal{T}_{{\rm rank}}$ for the rank-based EA equals that for all black-box algorithms. The importance of the black-box algorithms' knowledge of the transformation is also pointed out. Hence, a comparison with the class $\mathcal{T}_{{\rm prop}}$ of not worsening transformations for a similar EA which applies fitness-proportional selection, shows that $\mathcal{T}_{{\rm rank}}$ is a proper superset of $\mathcal{T}_{{\rm prop}}$. Moreover, $\mathcal{T}_{{\rm rank}}$ is a proper subset of the corresponding class for random search. Finally, the minimal and maximal classes of not worsening transformations are described completely, too.