131 Basic Differential Evolution

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Equation (131) expresses the total number of directions that can be explored per generation To increase the exploration power of DE, the number of directions can be increased by increasing the population size and/or the number of di erentials used At this point it is important to emphasize that the original DE was developed for searching through continuous-valued landscapes The sections that follow will show that exploration of the search space is achieved using vector algebra, applied to the individuals of the current population

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Mutation

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The DE mutation operator produces a trial vector for each individual of the current population by mutating a target vector with a weighted di erential This trial vector will then be used by the crossover operator to produce o spring For each parent, xi (t), generate the trial vector, ui (t), as follows: Select a target vector, xi1 (t), from the population, such that i = i1 Then, randomly select two individuals, xi2 and xi3 , from the population such that i = i1 = i2 = i3 and i2 , i3 U (1, ns ) Using these individuals, the trial vector is calculated by perturbing the target vector as follows: ui (t) = xi1 (t) + (xi2 (t) xi3 (t)) (132)

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where (0, ) is the scale factor, controlling the amplication of the di erential variation Di erent approaches can be used to select the target vector and to calculate di erentials as discussed in Section 132

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Crossover

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The DE crossover operator implements a discrete recombination of the trial vector, ui (t), and the parent vector, xi (t), to produce o spring, xi (t) Crossover is implemented as follows: uij (t) if j J (133) xij (t) = xij (t) otherwise where xij (t) refers to the j-th element of the vector xi (t), and J is the set of element indices that will undergo perturbation (or in other words, the set of crossover points) Di erent methods can be used to determine the set, J , of which the following two approaches are the most frequently used [811, 813]: Binomial crossover: The crossover points are randomly selected from the set of possible crossover points, {1, 2, , nx }, where nx is the problem dimension Algorithm 131 summarizes this process In this algorithm, pr is the probability that the considered crossover point will be included The larger the value of pr , the more crossover points will be selected compared to a smaller value This means that more elements of the trial vector will be used to produce the o spring, and less of the parent vector Because a probabilistic decision is made as to the

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13 Differential Evolution inclusion of a crossover point, it may happen that no points may be selected, in which case the o spring will simply be the original parent, xi (t) This problem becomes more evident for low dimensional search spaces To enforce that at least one element of the o spring di ers from the parent, the set of crossover points, J , is initialized to include a randomly selected point, j

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Exponential crossover: From a randomly selected index, the exponential crossover operator selects a sequence of adjacent crossover points, treating the list of potential crossover points as a circular array The pseudocode in Algorithm 132 shows that at least one crossover point is selected, and from this index, selects the next until U (0, 1) pr or |J | = nx Algorithm 131 Di erential Evolution Binomial Crossover for Selecting Crossover Points j U (1, nx ); J J {j }; for each j {1, , nx } do if U (0, 1) < pr and j = j then J J {j}; end end Algorithm 132 Di erential Evolution Exponential Crossover for Selecting Crossover Points J {}; j U (0, nx 1); repeat J J {j + 1}; j = (j + 1) mod nx ; until U (0, 1) pr or |J | = nx ;

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