Geometry of evolutionary algorithms

• Authors: Alberto Moraglio
• Venue: Proceedings of the 14th annual conference companion on Genetic and evolutionary computation
• Year: 2012

The aim of the tutorial is to introduce a formal, unified point of view on evolutionary algorithms across representations based on geometric ideas, and to present the benefits for both theory and practice brought by this novel perspective. The key idea behind the geometric framework is that search operators are not defined directly on solution representations, but are defined on the structure of the search space by means of simple geometric shapes, such as balls and segments, that delimit the region of space that includes all possible offspring with respect to the location of their parents. These geometric definitions can be then rewritten as equivalent but operational definitions involving the underlying representation. For example, the operator termed "uniform geometric crossover" is defined as to produce offspring that are uniformely distributed in the segment between parents. When the uniform geometric crossover is instantiated to the space of real vectors endowed with the Euclidean distance, and to the space of binary strings with the Hamming distance it comes to coincide to familiar operators, the blend crossover for real vectors and the uniform crossover for binary strings, respectively. This natural dualility of geometric search operators allows us to define exactly the same search operator across representations in a strong mathematical sense. This possibility forms the basis of a geometric framework for the unification of evolutionary algorithms across representations.