1. Product Geometric Crossover Alberto Moraglio and Riccardo Poli Abstract Geometric crossover is a representation-independent definition of crossover based on the distance of the search space. It generalizes the traditional crossover for binary strings and other important recombination operators for the most frequently used representations. Using a distance tailored for the problem at hand, the abstract definition of crossover can be used to design new problem specific crossovers that embed problem knowledge in the search. We introduce the important notion of product geometric crossover that allows to build new geometric crossovers combining pre-existing geometric crossovers in a simple way. Geometric Crossover It is based on the notion of metric line segment A binary operator GX is a geometric crossover if all offspring are in a segment between its parents Geometric crossover is dependent on the metric xy Traditional Crossover The traditional crossover is geometric under the Hamming distance 10110 11011 A B A B 11010 X X 2 1 3 H(A,X) + H(X,B) = H(A,B) Problem Knowledge To embed problem knowledge in the search, pick a geometric crossover associated with a good distance for the problem at hand A good distance for the problem at hand is a distance that makes the fitness landscape smooth Geometricity How can we know that a recombination is a geometric crossover? By proof: find a metric such that all offspring are in the segment between parents under that metric By construction: apply the formal definition of crossover to an edit distance and derive the associated crossover By geometricity-preserving transformations: new method that allows to build new geometric crossovers from existing ones Geometricity-preserving Syntax Manipulation The commutative diagram is very interesting when the metric transformation mt determines an induced geometricity preserving crossover transformation gt that has a simple interpretation in terms of syntactic manipulation. This allows us to get new geometric crossovers by simple geometricity- preserving syntax manipulation without explicit use of the underlying metric spaces. Metric Transformations Metric transformation: an operation that produces a metric space from other metric spaces Many existing transformations: sub-spaces, product spaces, quotient spaces, linear combination, convex transformation & many more Commutative Diagram Linking Metric and Crossover Transformations M: metric space M’: new metric space mt: metric transformation gx: application of the formal definition of geometric crossover X: geometric crossover associated with M X’: geometric crossover associated with M’ gt: induced geometricity-preserving crossover transformation associated with the metric transformation mt Product Geometric Crossover Definition: Let and be geometric crossovers. A product geometric crossover is a recombination that applies the crossover to the first position and to the second position. Non-independence: the application of and do not need to be independent Theorem: any product geometric crossover is a geometric crossover under the distance given by the sum of the distances of the compounding crossovers Traditional Crossover as Product Geometric Crossover Discrete metric of A: Corollary: the traditional crossover is geometric under Hamming distance Proof: traditional crossover is product geometric crossover of n crossovers associated to the discrete metric Sudoku and Product Crossover We have used the product geometric crossover to derive (i) a new geometric crossover for the entire sudoku grid by using simple geometric crossover for each row seen as permutations (ii) the distance associated with it. This has allowed us to analyse the geometric fitness landscape associated with this new operator and predict its good performance. Future Investigations Beside vectors considered here, other basic representations such as sets, permutations, sequences, trees and graphs can be seen as structures containing generic objects. We will study the metric transformations associated with them to derive more complex crossovers. Types of Product Crossover - Multi-crossover: same representation, same crossover n times - Hybrid-crossover: same representation, different crossover for each projection - Mixed-representation crossover: different representation for each projection - Dependent mixed-representation crossover: ex. Topology and Weights of Neural Network