# A Mathematical Unification of Geometric Crossovers Defined on Phenotype Space

### Authors

### Abstract

Geometric crossover is a representation-independent defini-tion of crossover based on the distance of the search space

interpreted as a metric space. It generalizes the traditional

crossover for binary strings and other important recombina-

tion operators for the most frequently used representations.

Using a distance tailored to the problem at hand, the ab-

stract definition of crossover can be used to design new prob-

lem specific crossovers that embed problem knowledge in the

search. This paper is motivated by the fact that genotype-

phenotype mapping can be theoretically interpreted using

the concept of quotient space in mathematics. In this paper,

we study a metric transformation, the quotient metric space,

that gives rise to the notion of quotient geometric crossover.

This turns out to be a very versatile notion. We give many

example applications of the quotient geometric crossover.