113 Strategy Parameters

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with fmax (t) the maximum tness of the current population Take note that the objective here is to maximize f , and that f (xi (t)) returns a positive value If f (xi (t)) is a small value, then ij (t) will be large (bounded above by 1), which results in large mutations Deviations are scaled by a uniform number in the range [0, 1] to ensure a mix of small and large step sizes Ma and Lai [542] proposed that deviations be proportional to normalized tness values: (1140) xij (t) = xij (t) + ij i (t)Nij (0, 1) where ij is the proportionality constant, and deviations are calculated as i (t) =

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ns l=1

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f (xi (t)) f (xl (t))

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(1141)

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with ns the size of the population This approach assumes f is minimized Yuryevich and Wong [943] proposed that ij (t) = (xmax,j xmin,j ) fmax (t) f (xi (t)) + fmax (t) (1142)

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to combine both boundary information and tness information In the above xmin and xmax specify the bounds in decision space, and > 0 is an o set parameter to ensure non-zero deviations Usually, is a small value This approach assumes that f is maximized The inclusion of boundary constraints forces large mutations for components with a large domain, and small mutations if the domain is small Swain and Morris [827] set deviations proportional to the distance from the best individual, ie y (1143) ij (t) = ij | j (t) xij (t)| + where > 0, and the proportionality constant is calculated as ij = E(xmin , xmax ) (1144)

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with [0, 2], and E(xmin , xmax ) gives the width of the search space as the Euclidean distance between the vectors xmin and xmax The parameter, , de nes a search neighborhood Larger values of promote exploration, while smaller values promote exploitation A good idea is to adapt over time, starting with large values that are decreased over time O spring are generated using xij (t) = xij (t) dir(xij ) ij (t)Nij (0, 1) where the direction of the update is dir(xij ) = sign( j xij ) y (1146) (1145)

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198 Gao [308] suggested that ij (t) = 1 j f (xi (t)) + j

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11 Evolutionary Programming

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fmax (t) fmin (t)

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(1147)

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where it is proposed that = 25; fmax and fmin refer to the largest and smallest tness values of the current population

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Self-Adaptation

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The emphasis of EP is on developing behavioral models EP is derived from simulations of adaptive behavior Previous sections have already indicated the strong in uence that strategy parameters have on the behavior of individuals, as quanti ed via the tness function Two of the major problems concerning strategy parameters are the amount of mutational noise that should be added, and the severity (ie step sizes) of such noise To address these problems, and to produce truly self-organizing behavior, strategy parameters can be evolved (or learned ) in parallel with decision variables An EP that utilizes such mechanisms is referred to as a self-adaptive EP Self-adaptation is not unique to EP According to Fogel et al [277], the idea of selfadaptation stretches back as far as 1967 with proposals by Rechenberg However, Schwefel [769] provided the rst detailed account of self-adaptation in the context of evolution strategies (ES) (also refer to 12) With reference to EP, Fogel et al [271] provided the rst suggestions for self-adaptive EP Since then, a number of self-adaptation methods have been proposed These methods can be divided into three broad categories [40]: Additive methods: The rst self-adaptive EP as proposed by Fogel et al [265] is an additive method where ij (t + 1) = ij (t) + ij (t)Nij (0, 1) (1148)

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with referred to as the learning rate In the rst application of this approach, = 1/6 If ij (t) 0, then ij (t) = , where is a small positive constant (typically, = 0001) to ensure positive, non-zero deviations As an alternative, Fogel [266] proposed ij (t + 1) = ij (t) + where f (a) = a f ( ij (t))Nij (0, 1) if a > 0 if a 0 (1149)

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(1150)

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ensures that the square root is applied to a positive, non-zero value Multiplicative methods: Jiang and Wang [418] proposed a multiplicative adjustment, where t ij (t + 1) = (0)( 1 e 2 nt + 3 ) (1151) where 1 , 2 and 3 are control parameters, and nt is the maximum number of iterations

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113 Strategy Parameters Lognormal methods: Borrowed from the ES literature [277], ij (t + 1) = ij (t)e( Ni (0,1)+ with O spring are produced using xij (t) = xij (t) + ij (t)Nij (0, 1) = = 1 2 nx 1 2nx

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Nij (0,1))

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(1152)

(1153) (1154)

(1155)

Self-adaptive EP showed the undesirable behavior of stagnation due to the tendency that strategy parameters converge too fast The consequence is that deviations become small too fast, thereby limiting exploration The search stagnates for some time until strategy parameters grow su ciently large due to random variation One solution to this problem is to impose a lower bound on the values of ij However, this triggers another problem of deciding when ij values are to be considered as too small Liang et al [524] provided a solution by considering dynamic lower bounds: min (t + 1) = min (t) nm (t) (1156)

where min (t) is the lower bound at time step (generation) t, [025, 045] is the reference rate, and nm (t) is the number of successful consecutive mutations (ie the number of mutations that results in improved tness values) This approach is based on the 1/5 success rule of Rechenberg [709] (refer to 12) Matsumura et al [565] developed the robust EP (REP) where the representation of each individual is expanded to allow for n strategy parameter vectors to be associated with each individual, as follows (xi (t), i0 , , ik , in ) (1157)

where i0 is referred to as the active strategy parameter vector, obtained through application of three mutation operators on the other strategy parameter vectors Component values of the strategy parameter vectors are mutated as follows: Duplication: i0j (t) = ilj (t) = i0j (t) i(l 1)j (t) (1158) (1159)

for l {1, 2, , n } Then ikj (t) is self-adapted by application of the lognormal method of equation (1152) on the ikj (t) for k = 0, 1, , n

200 Deletion: i(l 1)j (t) in j (t) = ilj (t)