Analyzing the search space transformation invariance properties of metaheuristics for avoiding over-customization

In the last three decades, a large number of metaheuristics inspired by biological evolution, swarm behaviors, and natural phenomenons have been proposed for solving black-box optimization problems. While metaheuristics successively showed superior performance over their predecessors on benchmark problems, it has been criticized that some of them tend to search towards the origin and can only perform well on problems with optimal variables of zero. While experimental studies on such over-customization issues have been given in the literature, there is a lack of theoretical analysis methods. In this paper, we suggest a comprehensive method to theoretically study the potential over-customization issues of metaheuristics, by means of deriving the conditions of search space transformation invariance properties of variation operators. The proposed method is used to theoretically analyze the translation, scale, and rotation invariance properties of several representative metaheuristics, and the conclusions are further verified by a variety of experiments. Our conclusions reveal that some metaheuristics are sensitive to the transformations of search spaces, which means that their performance superiority on specific problems may not demonstrate their effectiveness in more generic scenarios. At last, we give some advices on the avoidance of the over-customization in developing metaheuristics. We hope the conclusions of this paper are beneficial for the development of new metaheuristics.