Abstract
Based on a geometric theory of evolutionary algorithms, it was shown that all evolutionary algorithms equipped with a geometric crossover and no mutation operator do the same kind of convex search across representations, and that they are well-matched with generalised forms of concave fitness landscapes for which they provably find the optimum in polynomial time [13]. Analysing the landscape structure is essential to understand the relationship between problems and evolutionary algorithms. This paper continues such investigations by considering the following challenge: develop an analytical method to recognise that the fitness landscape for a given problem provably belongs to a class of concave fitness landscapes. Elementary landscapes theory provides analytic algebraic means to study the landscape structure [15]. This work begins linking both theories to better understand how such method could be devised using elementary landscapes. Examples on the well known One Max, Leading Ones, Not-All-Equal Satisfiability and Weight Partition problems illustrate the fundamental concepts supporting this approach.
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Notes
- 1.
Extensions for non-regular and non-symmetric neighbourhoods are possible, but regular and symmetric ones will be assumed here.
- 2.
The reader is referred to the Mathematica notebook publicly available online at: https://github.com/marcosdg/ppsn-2018, for the examples details.
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García, M.D., Moraglio, A. (2018). Bridging Elementary Landscapes and a Geometric Theory of Evolutionary Algorithms: First Steps. In: Auger, A., Fonseca, C., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds) Parallel Problem Solving from Nature – PPSN XV. PPSN 2018. Lecture Notes in Computer Science(), vol 11102. Springer, Cham. https://doi.org/10.1007/978-3-319-99259-4_16
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