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124 Evolution Strategy Operators
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Figure 122 Directed Mutation Operator for ES results in an asymmetrical mutation probability distribution Here the step size is larger for the x2 axis than for the x1 axis, and positive directions are preferred As each component of the genotype is mutated independently, it is su cient to de ne a 1-dimensional asymmetrical probability density function Hildebrand et al proposed the function, x2 2 if x < 0 (1+ 1+c) e (1234) fD (x) = x2 2 if x 0 (1+ 1+c) e (1+c) where c > 0 is the positive directional value The directional mutation method uses only deviations as strategy parameters, but associates a directional value, cj , with each deviation, j Both and c are self-adapted, giving a total of 2nx strategy parameters This is computationally more e cient than using a nx (nx 1)/2-sized rotational vector, and provides more information about preferred search directions and step sizes than deviations alone If D(c, ) denotes the asymmetric distribution, then xij (t) = Dj (cij (t), ij (t)) Ostermeier and Hansen [645] developed a coordinate system invariant mutation operator, with self-adaptation as discussed in Section 1233 Genotypes are mutated using both deviations and correlations, as follows: xl (t) = 1 xi (t) + l N (0, Cl (t))
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226 where Cl (t) =
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1 with lk (t) N (0, i l (t) Ci (t)) and nm is the mutation strength For large values of nm , the mutation strength is small, because a large sample, 1 , 2 , , nm , provides a closer approximation to the original distribution than a smaller sample Ostermeier and Hansen suggested that nm = nx
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Evolution Strategy Variants
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Previous sections have already discussed a number of di erent self-adaptation and mutation strategies for ES This section describes a few ES implementations that di er somewhat from the generic ES algorithm summarized in Algorithm 121
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Polar Evolution Strategies
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Bian et al [66], and Sierra and Echeverr [788] independently proposed that the ia components of genotype be transformed to polar coordinates Instead of the original genotype, the polar genotypes are evolved For an nx -dimensional Cartesian coordinate, the corresponding polar coordinate is given as (1237) (r, nx 2 , , 1 , ) where 0 < 2 , 0 q for q = 1, , nx 2, and r > 0 Each individual is therefore represented as (1238) i (t) = (xp (t), i (t)) i where , xp = (r, nx 2, , 1 , ) Polar coordinates are transformed back to Cartei sian coordinates as follows: x1 x2 x3 xi xn = = = = = = = r cos sin 1 sin 2 sin nx 2 r sin sin 1 sin 2 sin nx 2 r cos 1 sin 2 sin nx 2 (1239) r cos i 2 sin i 1 sin nx 2
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The mutation operator uses deviations to adjust the and q angles: = ( l (t) + ,l (t)N (0, 1)) mod 2 lq (t) + (t)Nq (0, 1)) mod lq (t) = ( lq l (1240) (1241)
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125 Evolution Strategy Variants Algorithm 122 Polar Evolution Strategy Set the generation counter, t = 0; Initialize the strategy parameters, , ,q , q = 1, , nx 2; Create and initialize the population, C(0), as follows:; for i = 1, , do r = 1; i (0) U (0, 2 ); iq (0) U (0, ), q = 1, , nx 2; xp (0) = (r, i (0), i (0)); i i (0) = (xp (0), i (0)); i end for each individual, l (0) C(0) do Transform polar coordinate xp (0) to Cartesian coordinate nxi (0); i Evaluate the tness, f (xi (0)); end while stopping condition(s) not true do for l = 1, , , generate offspring do Randomly choose two parents; Create o spring, l (t), using local, discrete recombination; Mutate l (t) to produce l (t); Transform xp (t) back to Cartesian xl (t); l Evaluate the tness, f (xi (t)); end Select individuals from the o spring to form C(t + 1); t = t + 1; end
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where l (t) and lq (t), q = 1, , nx 2 refer to the components of the o spring, l (t), l = 1, , produced by the crossover operator, and l (t) = ( ,l (t), ,l1 (t), ,l2 (t), , ,l(nx 2) (t)), is its strategy parameter vector Note that r = 1 is not mutated The polar ES as used in [788] is summarized in Algorithm 122
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