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124 Evolution Strategy Operators
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Kursawe [492] used a self-adaptation scheme where 1 n nx , and each individual uses a di erent number of deviation parameters, n i (t) At each generation, t, the number of deviation parameters can be increased or decreased at a probability of 005 If the number of deviation parameters increases, ie n i (t) = n i (t 1), then the new deviation parameter is initialized as in ,i (t) (t) = 1 n ,i (t 1)
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Evolution Strategy Operators
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Evolution strategies use the three main operators of EC, namely selection, crossover, and mutation These operators are discussed in Sections 1241, 1242, and 1243 respectively
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Selection Operators
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Selection is used for two tasks in an ES: (1) to select parents that will take part in the recombination process and (2) to select the new population For selecting the parents for the crossover operator, any of the selection methods reviewed in Section 85 can be used Usually, parents are randomly selected For each generation, o spring are generated from parents and mutated After crossover and mutation, the individuals for the next generation are selected Two main strategies have been developed: ( + ) ES: In this case (also referred to as the plus strategies) the ES generates o spring from parents, with 1 < The next generation consists of the best individuals selected from parents and o spring The ( + ) ES strategy implements elitism to ensure that the ttest parents survive to the next generation ( , ) ES: In this case (also referred to as the comma strategies), the next generation consists of the best individuals selected from the o spring Elitism is not used, and therefore this approach exhibits a lower selective pressure than the plus strategies Diversity is therefore larger than for the plus strategies, which results in better exploration The ( , ) ES requires that 1 < < , Using the above notation, ES are collectively referred to as ( + )-ES The ( + ) notation has been extended to ( , , ), where denotes the maximum lifespan of an individual If an individual exceeds its lifespan, it is not selected for the next population Note that ( , )-ES is equivalent to ( , 1, )-ES The best selection strategy to use depends on the problem being solved Highly convoluted search spaces need more exploration, for which the ( , )-ES are more applicable
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12 Evolution Strategies
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Because information about the characteristics of the search space is usually not available, it is not possible to say which selection scheme will be more appropriate for an arbitrary function For this reason, Huang and Chen [392] developed a fuzzy controller to decide on the number of parents that may survive to the next generation The fuzzy controller receives population diversity measures as input, and attempts to balance exploration against exploitation Runarsson and Yao [746] developed a continuous selection method for ES, which is essentially a continuous version of ( , )-ES The basis of this selection method is that the population changes continuously, and not discretely after each generation There is no selection of a new population at discrete generational intervals Selection is only used to select parents for recombination, based on a tness ranking of individuals As soon as a new o spring is created, it is inserted in the population and the ranking is immediately updated The consequence is that, at each creation of an o spring, the worst individual among the parents and o spring is eliminated
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Crossover Operators
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In order to introduce recombination in ES, Rechenberg [709] proposed that the (1+1)ES be extended to a ( + 1)-ES (refer to Section 121) The ( + 1)-ES is therefore the rst ES that utilized a crossover operator In ES, crossover is applied to both the genotype (vector of decision variables) and the strategy parameters Crossover is implemented somewhat di erently from other EAs Crossover operators di er in the number of parents used to produce a single o spring and in the way that the genetic material and strategy parameters of the parents are , combined to form the o spring In general, the notation ( / , + ) is used to indicate that parents are used per application of the crossover operator Based on the value of , the following two approaches can be found: Local crossover ( = 2), where one o spring is generated from two randomly selected parents Global crossover (2 < ), where more than two randomly selected parents are used to produce one o spring The larger the value of , the more diverse the generated o spring is compared to smaller values Global crossover with large improves the exploration ability of the ES In both local and global crossover, recombination is done in one of two ways: Discrete recombination, where the actual allele of parents are used to construct the o spring For each component of the genotype or strategy parameter vectors, the corresponding component of a randomly selected parent is used , The notation ( / D + ) is used to denote discrete recombination Intermediate recombination, where allele for the o spring is a weighted average of the allele of the parents (remember that oating-point representations , are assumed for the genotype) The notation ( / I + ) is used to denote intermediate recombination
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