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In an attempt to improve the convergence speed of EP, Kim et al [462] proposed the accelerated EP (AEP), which uses two variation operators: A directional operator to determine the direction of the search based tness scores, and
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the Gaussian mutation operator given in equation (117) Individuals are represented as i (t) = (xi (t), i (t), ai (t)) (1167)
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where ij { 1, 1}, j = 1, , nx gives the search direction for each component of the i-th individual, and ai represents the age of the individual Age is used to force wider exploration if o spring are worse than their parents O spring generation consists of two steps The rst step updates age parameters for each individual, and determines search directions (assuming minimization): ai (t) = and ij (t) = sign(xij (t) xij (t 1)) ij (t 1) if f (xi (t)) < f (xi (t 1)) otherwise (1169) 1 if f (xi (t)) < f (xi (t 1)) ai (t 1) + 1 otherwise (1168)
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If the tness of an individual improved, the search will continue in the direction of the improvement If the tness does not improve, the age is incremented, which will result in larger step sizes as follows: If ai (t) = 1, then i (t) = 1 f (xi (t)) xij (t) = xij (t) + ij (t)|N (0, i (t))| Otherwise, if ai (t) > 1, i (t) = 2 f (xi (t))ai (t) xij (t) = xij (t) + N (0, i (t)) where 1 and 2 are positive constants Selection occurs by having an o spring compete directly with its parent using absolute tness Wen et al [896] used a similar approach, but using the dynamic strategy parameter approach given in equation (1141) (1172) (1173) (1170) (1171)
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Choi and Oh [126] proposed an EP algorithm based on backpropagation learning of feedforward neural networks (refer to Section 322) The best individual, y(t), of the current population, C(t), calculated as y(t) = xi (t) : f (xi (t)) = min {f (xi (t))}
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is taken as the target The temporal error between this target, y(t), and the individual, xi (t), is then used by the mutation operator to improve exploration For each parent, xi (t), an o spring is generated as follows (assuming minimization): x xij (t) = xij (t) + xij (t) + ij (t) where y xij (t) = ( j (t) xij (t))|Nij (0, 1)| xij (t) = i (t) xij (t 1) + ij (t 1) x with > 0 the learning rate, > 0 the momentum rate, and i (t) = 1 if f (xi (t 1)) < f (xi (t 1)) 0 otherwise (1178) (1176) (1177) (1175)
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A very simple approach to improve the exploitation ability of EP, is to add a hillclimbing facility to generated o spring While a better tness can be obtained, hillclimbing is applied to each o spring [235] Alternatively, gradient descent has been used to regenerate o spring [920, 779] For each o spring, xi (t), recalculate the o spring using f (1179) xij (t) = xij (t) i (t) xij (t) where the learning rate is calculated as i (t) =
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nx f j=1 xij (t) nx f f 2f j=1 xih (t) xij (t) xih (t) xij (t)
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As an alternative to gradient descent, Birru et al [70] used conjugate gradient search (refer to Section 323), where line searches are performed for each component of the o spring The initial search direction is the downhill gradient, with subsequent search directions chosen along subsequent gradient components that are orthogonal to all previous search directions Birru et al [70] also proposed a derivitive-free local search method to re ne o spring The stochastic search developed by Solis and Wets [802] is applied to each o spring at a speci ed probability Based on this probability, if the local search is performed, a limited number of steps is done as summarized in Algorithm 112
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