Abstract
Previous work proposed to unify an algebraic theory of fitness landscapes and a geometric framework of evolutionary algorithms (EAs). One of the main goals behind this unification is to develop an analytical method that verifies if a problem’s landscape belongs to certain abstract convex landscape classes, where certain recombination-based EAs (without mutation) have polynomial runtime performance. This paper advances such unification by showing that: (a) crossovers can be formally classified according to geometric or algebraic axiomatic properties; and (b) the population behaviour induced by certain crossovers in recombination-based EAs can be formalised in the geometric and algebraic theories. These results make a significant contribution to the basis of an integrated geometric-algebraic framework with which analyse recombination spaces and recombination based EAs.
The full version of this article, including some omitted proofs, can be found online at https://www.researchgate.net/profile/Marcos_Diez_Garcia.
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Notes
- 1.
Geometric mutation is defined by requiring that the mutated offspring belongs to the d-metric ball of its single parent.
- 2.
Certain crossover families (e.g. one-point and masked crossovers) have logarithmic upper-bounds for the closure iteration number, with increasing dimension of the search space [16]. In short, idempotency does not require excessively many iterations.
- 3.
Notice that a crossover section with all parent’s genes is also a valid contiguous section.
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García, M.D., Moraglio, A. (2019). A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search. In: Liefooghe, A., Paquete, L. (eds) Evolutionary Computation in Combinatorial Optimization. EvoCOP 2019. Lecture Notes in Computer Science(), vol 11452. Springer, Cham. https://doi.org/10.1007/978-3-030-16711-0_12
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