Inbreeding properties of geometric crossover and non-geometric recombinations

Geometric crossover is a representation-independent generalization of traditional crossover for binary strings. It is defined in a simple geometric way by using the distance associated with the search space. Many interesting recombination operators for the most frequently used representations are geometric crossovers under some suitable distance. Showing that a given recombination operator is a geometric crossover requires finding a distance for which offspring are in the metric segment between parents. However, proving that a recombination operator is not a geometric crossover requires excluding that one such distance exists. It is, therefore, very difficult to draw a clear-cut line between geometric crossovers and non-geometric crossovers. In this paper we develop some theoretical tools to solve this problem and we prove that some well-known operators are not geometric. Finally, we discuss the implications of these results.

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