Inbreeding Properties of Geometric Crossover and Non- geometric Recombinations Alberto Moraglio & Riccardo Poli ECAI 2006

Contents I.Geometric Crossover II.Inbreeding Properties of Geometric Crossover III.Notable Non-geometric Recombinations IV.Conclusions

I. Geometric Crossover

Geometric Crossover 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 x y

Geometric Crossover The traditional n-point crossover is geometric under the Hamming distance A B A B X X H(A,X) + H(X,B) = H(A,B)

Many Recombinations are Geometric Traditional Crossover extended to multary strings Recombinations for real vectors PMX, Cycle Crossovers for permutations Homologous Crossover for GP trees Ask me for more examples over a coffee!

Being geometric crossover is important because…. We know how the search space is searched by geometric crossover for any representation: convex search We know a rule-of-thumb on what type of landscapes geometric crossover will perform well: “smooth” landscape This is just a beginning of general theory, in the future we will know more!

Non-geometricity questions Existence of non-geometric crossovers: is any recombination a geometric crossover given a suitable distance? Proving non-geometricity: given a recombination, how can we prove that it is non-geometric crossover? Geometricity: to prove that an operator is geometric we need to find a metric for which offspring are in the metric segment between parents under this metric. Non-geometricity: to prove that a recombination is non- geometric requires to show that it is not geometric under any distance. This is difficult because there are infinitely many distances to rule out!!

II. Inbreeding Properties

Inbreeding Properties Properties of geometric crossover arising only from its axiomatic definition (metric axioms) Valid for any distance, probability distribution and solution representation. So all geometric crossovers have them. Based on inbreeding: breeding between close relatives

Purity Theorem: When both parents are the same P1, their child must be P1.

Convergence Theorem: C is the child of P1 and P2 and C is not P1. Then the recombination of C and P2 cannot produce P1.

Partition Theorem: C is the child of P1 and P2. Then the recombination of P1 and C and the recombination of C and P2 cannot produce the same offspring unless the offspring is C.

III. Non-geometric Recombinations

Non-geometricity and Inbreeding properties It is possible to prove non-geometricity of a recombination operator under any distance, any probability distribution and any represenation producing a single counter- example to any inbreeding property because they must hold for all geometric crossovers. Then if they do not hold, the operator is non- geometric.

Extended line crossover P1 P2 C Theorem: Extended line crossover is non-geometric. Proof: the converge property does not hold.

Koza’s subtree swap crossover P1P2=P1 C1C2 Theorem: Koza’s crossover is non-geometric. Proof: the property of purity does not hold.

Davis’s order crossover Theorem: Davis’s order crossover is non-geometric. Proof: the converge property does not hold.

Summary Geometric crossover: offspring are in the segment between parents under a suitable distance Proving non-geometricity is difficult: need to prove non- geometricity under all distances! Inbreeding properties of crossover (purity, convergence, partition): hold for all geometric crossovers, follow logically from axiomatic definition of crossover only Imbreeding properties allows to prove non-geometricity in a very simple way: producing a simple counter-example Non-geometric recombinations: Extended-line recombination, Koza’s subtree swap crossover, Davis’s order crossover These are notable exceptions: many other well-known recombination operators are geometric

Thank you for your attention! Questions?