Selection in Visual Studio .NET

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Selection
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Selection is applied to determine which individuals will take part in the mutation operation to produce a trial vector, and to determine which of the parent or the o spring will survive to the next generation With reference to the mutation operator, a number of selection methods have been used Random selection is usually used to select the individuals from which di erence vectors are calculated For most DE implementations the target vector is either randomly selected or the best individual is selected (refer to Section 132) To construct the population for the next generation, deterministic selection is used: the o spring replaces the parent if the tness of the o spring is better than its parent; otherwise the parent survives to the next generation This ensures that the average tness of the population does not deteriorate
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131 Basic Differential Evolution
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General Di erential Evolution Algorithm
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Algorithm 133 provides a generic implementation of the basic DE strategies Initialization of the population is done by selecting random values for the elements of each individual from the bounds de ned for the problem being solved That is, for each individual, xi (t), xij (t) U (xmin,j , xmax,j ), where xmin and xmax de ne the search boundaries Any of the stopping conditions given in Section 87 can be used to terminate the algorithm Algorithm 133 General Di erential Evolution Algorithm Set the generation counter, t = 0; Initialize the control parameters, and pr ; Create and initialize the population, C(0), of ns individuals; while stopping condition(s) not true do for each individual, xi (t) C(t) do Evaluate the tness, f (xi (t)); Create the trial vector, ui (t) by applying the mutation operator; Create an o spring, xi (t), by applying the crossover operator; if f (xi (t)) is better than f (xi (t)) then Add xi (t) to C(t + 1); end else Add xi (t) to C(t + 1); end end end Return the individual with the best tness as the solution;
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Control Parameters
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In addition to the population size, ns , the performance of DE is in uenced by two control parameters, the scale factor, , and the probability of recombination, pr The e ects of these parameters are discussed below: Population size: As indicated in equation (131), the size of the population has a direct in uence on the exploration ability of DE algorithms The more individuals there are in the population, the more di erential vectors are available, and the more directions can be explored However, it should be kept in mind that the computational complexity per generation increases with the size of the population Empirical studies provide the guideline that ns 10nx The nature of the mutation process does, however, provide a lower bound on the number of individuals as ns > 2nv + 1, where nv is the number of di erentials used For nv di erentials, 2nv di erent individuals are required, 2 for each di erential The
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242 additional individual represents the target vector
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13 Differential Evolution
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Scaling factor: The scaling factor, (0, ), controls the ampli cation of the di erential variations, (xi2 xi3 ) The smaller the value of , the smaller the mutation step sizes, and the longer it will be for the algorithm to converge Larger values for facilitate exploration, but may cause the algorithm to overshoot good optima The value of should be small enough to allow di erentials to explore tight valleys, and large enough to maintain diversity As the population size increases, the scaling factor should decrease As explained in Section 1311, the more individuals in the population, the smaller the magnitude of the difference vectors, and the closer individuals will be to one another Therefore, smaller step sizes can be used to explore local areas More individuals reduce the need for large mutation step sizes Empirical results suggest that large values for both ns and often result in premature convergence [429, 124], and that = 05 generally provides good performance [813, 164, 19] Recombination probability: The probability of recombination, pr , has a direct in uence on the diversity of DE This parameter controls the number of elements of the parent, xi (t), that will change The higher the probability of recombination, the more variation is introduced in the new population, thereby increasing diversity and increasing exploration Increasing pr often results in faster convergence, while decreasing pr increases search robustness [429, 164] Most implementations of DE strategies keep the control parameters constant Although empirical results have shown that DE convergence is relatively insensitive to di erent values of these parameters, performance (in terms of accuracy, robustnes, and speed) can be improved by nding the best values for control parameters for each new problem Finding optimal parameter values can be a time consuming exercise, and for this reason, self-adaptive DE strategies have been developed These methods are discussed in Section 1333
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