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f (C1 xi ) =
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1 C1 nl
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(151)
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where C2 x1 , , C2 xC2 ns form the population C2 sample, and C1 nl is the total number of individuals in population C1 that defeat individual C2 xl The competitive tness sharing method rewards those population C1 individuals that beat population C2 individuals, which few other population C1 individuals could beat It is therefore not necessarily the case that the best population C1 individual beats the most population C2 individuals Tournament fitness [27] holds a number of single elimination, binary tournaments to determine a relative tness ranking Tournament tness results in a tournament tree, with the root element as the best individual For each level in the tree, two opponents are randomly selected from that level, and the best of the two advances to the next level In the case of an odd number of competitors, a single individual from the current level moves to the next level After tournament ranking, any of the standard selection operators (refer to Section 85) can be used to select parents
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Hall of Fame Elitism is a mechanism used in standard EAs to ensure that the best parents of a current generation survive to the next generation To be able to survive for more generations, an individual has to be highly t in almost every population For coevolution, Rosin and Belew [740] introduced the hall of fame to extend elitism in time At
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152 Competitive Coevolution
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each generation the best individual of a population is stored in that population s hall of fame The hall of fame may have a limited size, in which case a new individual to be inserted in the hall of fame will replace the worst individual (or the oldest one) Individuals from one population now compete against a sample of the current opponent population and its hall of fame The hall of fame prevents overspecialization
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Generic Competitive Coevolutionary Algorithm
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The generic CCE algorithm in Algorithm 151 assumes that two competing populations are used For a single population CCE algorithm, refer to Algorithm 152 For both these algorithms, any standard EA can be used to evolve each population for one iteration (ie perform reproduction and select the new population) Table 151 provides a summary of some publications for EAs used, including PSO Algorithm 151 Competitive Coevolutionary Algorithm with Two Populations Initialize two populations, C1 and C2 ; while stopping condition(s) not true do for each C1 xi , i = 1, , C1 ns do Select a sample of opponents from C2 ; Evaluate the relative tness of C1 xi with respect to this sample; end for each C2 xi , i = 1, , C2 ns do Select a sample of opponents from C1 ; Evaluate the relative tness of C2 xi with respect to this sample; end Evolve population C1 for one generation; Evolve population C2 for one generation; end Select the best individual from the solution population, S1 ;
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Algorithm 152 Single Population Competitive Coevolutionary Algorithm Initialize one population, C; while stopping condition(s) not true do for each Cxi , i = 1, , Cns do Select a sample of opponents from C to exclude Cxi ; Evaluate the relative tness of Cxi with respect to the sample; end Evolve population C for one generation; end Select the best individual from C as the solution;
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Algorithm 151 assumes that C1 is the solution population and C2 is the test population
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15 Coevolution
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Table 151 Algorithms Used to Achieve Adaptation in CCE Algorithm GA EP ES GP PSO Reference [37, 132, 657, 750, 855] [119, 120, 268, 532] [60, 914] [343, 357, 358, 480] [286, 580, 654]
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In CCE it is also possible to use two competing solution populations, in which case the individuals in one population serve as the test cases of individuals in the other populations Here the solution returned by the algorithm is the best individual over both populations The performance of a CCE can be greatly improved if the two competing populations are as diverse as possible Diversity can be maintained by incorporating mechanisms to facilitate niche formation Shared sampling and the tness sharing methods achieve this goal Alternatively, any niching (speciation) algorithm can be implemented
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