**CHAPTER 3: ADAPTIVE DIFFERENTIAL EVOLUTION: TAXONOMY**

**3.4 Adaptive Differential Evolution: Procedural Analysis and Comparison**

**3.4.1 DE with Adaptive Parameters and Single Strategy**

**3.4.1.2 Adaptive DE with Single Advanced Strategy**

ο· **DESAP Algorithm **

o *Advanced DESAP Mutation and Crossover Schemes *

In DESAP the base strategy used is a bit different from the standard DE/rand/1/bin and of some sort similar to the strategy introduced in (Abbass, 2002).

*Crossover Scheme: The crossover operator is performed first with some *
probability, ππππ(0,1) < πΏ^{π1} or π = π, where π is a randomly selected variable within
individual π. The updating strategy is as follows,

π^{πβπππ} = π^{π1}+ πΉ β (π^{π2}β π^{π3}) (3.8)

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The ordinary amplification factor πΉ is set to 1, thereby at least one variable in π must be
changed. Otherwise the value of π^{πβπππ} and its control parameters will be set to the same
values associated with π^{π1}.

*Mutation Scheme: The mutation stage is implemented with some mutation *
probability, ππππ(0,1) < π^{π1}, otherwise all the values will remain fixed.

π^{πβπππ}= π^{πβπππ}+ πππππ(0, π^{π1} ) (3.9)

As can be seen from the equation above, that DESAP mutation is not derived from one of the DE standard mutation schemes.

o *DESAP Parameter Control Schemes *

DESAP is proposed not only to update the values of the mutation and crossover control parameters, π and πΏ, but, rather, it adjusts the population size parameter,π as well in a self-adaptive manner. All parameters undergo the evolution and pressure (i.e.

crossover and mutation) in a way analogue to their corresponding individuals. The terms
πΏ and π have the same meaning as πΆπ
and ππ, respectively, π refers to the probability
of applying the mutation scheme whereas the ordinary πΉ is kept fixed during the
evolution process. Mainly, two versions of DESAP have been applied. The population
size of both DESAP versions (Rel and Abs) are initialized by generating, randomly, a
population of (10 Γ π) initial vectors π, where π denotes the number of design
variables which are already recommended by the authors of the original DE method
(Storn & Price, 1995). The mutation probability π_{π} and crossover rate πΏ_{π} are both
initialized to random values generated uniformly between [0,1]. The population size
parameter π_{π} is initialized in DESAP-Abs version to,

π_{π} = πππ’ππ(ππππ‘πππ ππππ’πππ‘πππ π ππ§π + πππππ(0,1)) (3.10)

whereas in DESAP-Rel to,

π_{π} = ππππ(β0.5,0. 5) (3.11)

the updating process is then applied on the parameters πΏ, Ξ· and π , at the same level with their corresponding individuals using the same crossover and mutation schemes (see Equation 3.8-3.9).

** Updating the crossover rate πΏ **

πΏ^{πβπππ}= πΏ^{π1}+ πΉ β (πΏ^{π2}β πΏ^{π3}) (3.12)

πΏ^{πβπππ}= πππππ(0,1) (3.13)

* Updating the mutation probability Ξ· *

π^{πβπππ}= π^{π1}+ πΉ β (π^{π2}β π^{π3}) (3.14)

π^{πβπππ}= πππππ(0,1) (3.15)

* *

*Updating the population size π *

DESAP-Abs: π^{πβπππ} = π^{π1}+ πππ‘(πΉ β (π^{π2}β π^{π3})) (3.16)
DESAP-Rel: π^{πβπππ}= π^{π1}+ πππ‘(πΉ β (π^{π2}β π^{π3})) (3.17)
DESAP-Abs: π^{πβπππ} = π^{πβπππ}+ πππ‘(πππππ(0.5,1)) (3.18)
DESAP-Rel: π^{πβπππ}= π^{πβπππ}+ πππππ(0, π^{π1} ) (3.19)

The ordinary amplification factor πΉ is set to 1. The evolution process of DESAP continues until it achieves a pre-specified population size π, then the new population size is calculated for the next generation as,

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DESAP-Abs: π_{πππ€} = πππ’ππ(β π/π)^{π}_{1} (3.20)
DESAP-Rel: π_{πππ€} = πππ’ππ(π + (π Γ π)) (3.21)

For the next generation and in an attempt to carry forward all the individuals with the
remaining (π_{πππ€}β π) individuals, the condition (π_{πππ€} > π) should be satisfied;

otherwise, carry forward only the first π_{πππ€} individuals of the current generation.

ο· **JADE Algorithm **

o *Advanced JADE Mutation Schemes *

There are different mutation versions of JADE have been proposed in (Zhang &

Sanderson, 2009a) and (Zhang & Sanderson, 2009b), which we refer to in our study.

The first new mutation scheme is called DE/current-to-pbest/1/bin (see Equation 3.22), which it has less greedy property than its previous specification scheme, DE/current-to-best/1/bin, since it utilizes not only the information of the best individual, but the information of the π% good solutions in the current population indeed.

π£_{π,π}^{π‘} = π₯_{π,π}^{π‘} + πΉ_{π}. (π₯_{πππ π‘,π}^{π,π‘} β π₯_{π,π}^{π‘} ) + πΉ_{π}. (π₯_{π1,π}^{π‘} β π₯_{π2,π}^{π‘} ), (3.22)

where π β (0, 1] and π₯_{πππ π‘,π}^{π,π‘} is a random uniform chosen vector as one of the superior
100π% vectors in the current population. The second mutation scheme with an external
*archive, denoted as *π΄, that has been introduced to store the recent explored *inferior *
*individuals that have been excluded from the search process and their differences from *
the individuals in the running population, π. The archive vector π΄ is first initialized to
be empty. Thereafter, solutions that are failed in the selection operation of each
generation are added to this archive. The new mutation operation is then reformulated as
follows,

π£_{π}^{π‘} = π₯_{π}^{π‘}+ πΉ_{π}. (π₯_{πππ π‘}^{π,π‘} β π₯_{π}^{π‘}) + πΉ_{π}. (π₯_{π1}^{π‘} β π₯Μ_{π2}^{π‘} ), (3.23)

where π₯_{π}^{π‘} and π₯_{π1}^{π‘} are generated from π in the same way as in the original JADE,
whereas π₯Μ_{π2}^{π‘} is randomly generated from the union, π΄ βͺ π. Eventually, randomly
selected solutions are going to be removed from the archive if its size exceeds a certain
threshold, say population size ππ, just to keep the archive within a specified dimension.

It is clear that if the archive size has been set to be zero then Equation 3.22 is a special case of Equation 3.23.

Another variant has been proposed to further increase the population diversity, named archive-assisted DE/rand-to-pbest/1 as follows,

π£_{π}^{π‘} = π₯_{π1}^{π‘} + πΉ_{π}. (π₯_{πππ π‘}^{π,π‘} β π₯_{π1}^{π‘} ) + πΉ_{π}. (π₯_{π2}^{π‘} β π₯Μ_{π3}^{π‘} ) (3.24)

o *JADE Parameter Control Schemes *

JADE updates four control parameters (πΉ, πΆπ
, π_{πΉ} and π_{πΆπ
}) during the evolution
process.

*Mutation factor (F) and location parameter of mutation probability distribution (π*_{πΉ}*): *

The mutation probability πΉ_{π} is independently generated at each generation for each
individual π according to the following formula,

πΉ_{π} = πππππ_{π}(π_{πΉ}, 0.1) (3.25)

where πππππ_{π} is a Cauchy distribution with location parameter π_{πΉ} and scale parameter
0.1. If πΉ_{π} β₯ 1 then the value is truncated to be 1 or regenerated if πΉ_{π} β€ 0. The location
parameter π_{πΉ} is first initiated to be 0.5. In this step, JADE shows some similarity in
updating the mean of the distribution, π_{πΆπ
} , to the learning style used in Population

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Based Incremental Learning (PBIL) algorithm (Baluja, 1994; Baluja & Caruana, 1995).

The standard version of the PBIL uses learning rate πΏπ
β (0,1] that must be fixed a
priori. Then, by utilizing Hebbian-inspired rule the difference rate (1 β πΏπ
) is
multiplied by the probability vector (ππ) that represents the combined experience of the
PBIL throughout the evolution process, whereas πΏπ
is multiplied by each bit (i.e. geneβs
value) of the current individual(s) used in the updating process. Likewise, JADE
updates the mutation distribution mean location, π_{πΉ} is updated at the end of each
generation after accumulating the set of all the successful mutation probabilities πΉ_{π}βs at
generation π‘, denoted by π_{πΉ},. The new π_{πΆπ
} is updated as,

π_{πΉ} = (1 β π) β π_{πΉ} + π β ππππ_{πΏ}(π_{πΉ}), (3.26)
where ππππ_{πΏ}(. ) is Lehmer mean,

ππππ_{πΏ}(π_{πΉ}) =^{β}_{β}^{πΉβππΉ}^{πΉ}_{πΉ}^{2}

πΉβππΉ

(3.27)

*Crossover probability (CR) and mean of crossover probability distribution (π*_{πΆπ
}): The
crossover probability πΆπ
_{π} is updated, independently, for each individual according to a
normal distribution,

πΆπ
_{π} = πππππ_{π}(π_{πΆπ
}, 0.1), (3.28)

with mean π_{πΆπ
} and standard deviation 0.1 and truncated to the interval (0, 1]. The mean
π_{πΆπ
} is first initiated to be 0.5. Then, similar to the updating scheme of the mutation
probability mean, the distribution of the crossover mean, π_{πΆπ
} , is updated at each
generation after accumulating the set of all the successful crossover probabilities πΆπ
_{π}βs
at generation π‘, denoted by π_{πΆπ
}, hence calculate its ππππ_{π΄}(π_{πΆπ
}). The new π_{πΆπ
} is
updated by the equation,

π_{πΆπ
} = (1 β π) β π_{πΆπ
}+ π β ππππ_{π΄}(π_{πΆπ
}), (3.29)

where π is a positive constant β (0,1] and ππππ_{π΄}(β) is the usual arithmetic mean.

ο· **MDE_pBX Algorithm **

o *Advanced MDE_pBX Mutation and Crossover Schemes *

*Mutation Scheme: The new proposed mutation scheme DE/current-to-gr*best/1/bin,
utilizes the best individual π₯_{ππ}^{π‘} _{πππ π‘} chosen from the π% group of individuals *randomly *
selected from the current population for each target vector. The group size π of the
MDE_pBX is varying from 5% to 65% of the ππ. The new scheme can be described
as,

π£_{π}^{π‘} = π₯_{π}^{π‘}+ πΉ_{π¦}β (π₯_{ππ}^{π‘} _{πππ π‘}β π₯_{π}^{π‘}+ π₯_{π1}^{π‘} β π₯_{π2}^{π‘} ), (3.30)

where π₯_{π1}^{π‘} πππ π₯_{π2}^{π‘} are two different individuals randomly selected from the current
population and they are also mutually different from the running individual π₯_{π}^{π‘} and
π₯_{ππ}^{π‘} _{πππ π‘}.

*Crossover Scheme: The new proposed recombination scheme π-Best, has been defined *
as a greedy strategy; it is based on the incorporation between a randomly selected
mutant vector perturbed by one of the π top-ranked individual selected from the current
population to yield the trial vector at the same index. Throughout evolution the value of
parameter π is reduced linearly in an adaptive manner (see Equation 3.37).

o *MDE_pBX Parameters Control Schemes *

*Modifications applied to the adaptive schemes in MDE_pBX: The scalar factor *πΉ_{π} and
the crossover rate πΆπ
_{π} of each individual are both altered independently at each
generation using JADE schemes (see Equation 3.25 and Equation 3.28). The new

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modifications have been applied only to πΉ_{π} and πΆπ
_{π} adapting schemes. In MDE_pBX,
both πΉ_{π} and πΆπ
_{π} are subscribed to the same rule of adjusting. Firstly, the values of πΉ_{π}
and πΆπ
_{π} are initialized to 0.5 and 0.6 respectively, then are updated at each generation
in the following way,

πΉ_{π} = π€_{πΉ} β πΉ_{π}+ (1 β π€_{πΉ}) β ππππ_{πππ€}(πΉ_{π π’ππππ π }) (3.31)
πΆπ
_{π}= π€_{πΆπ
}β πΆπ
_{π}+ (1 β π€_{πΆπ
}) β ππππ_{πππ€}(πΆπ
_{π π’ππππ π }) (3.32)

where a set of successful scale factors πΉ_{π π’ππππ π } and a set of successful crossover
probability πΆπ
_{π π’ππππ π } are generated from the current population. And | | stands for the
cardinality of each successful set. The variable π is set to 1.5 as it proves to give better
results on a wide range of test problems. Then the mean power ππππ_{πππ€ } of each set is
calculated as follows,

ππππ_{πππ€}(πΉ_{π π’ππππ π }) = β (π₯^{π} /|πΉ_{π π’ππππ π }|)^{1}^{π}

π₯βπΉ_{π π’ππππ π }

(3.33)

ππππ_{πππ€}(πΆπ
_{π π’ππππ π }) = β (π₯^{π} /|πΆπ
_{π π’ππππ π }|)^{π}^{1}

π₯βπΆπ π π’ππππ π

(3.34)

Together with calculating the weight factors π€_{πΉ} and π€_{πΆπ
} as,

π€_{πΉ} = 0.8 + 0.2 Γ ππππ(0, 1) (3.35)
π€_{πΆπ
} = 0.9 + 0.1 Γ ππππ (0, 1) (3.36)

the πΉ_{π} and πΆπ
_{π} are formulized. As can be seen from Equations 3.35-3.36, the value of
π€_{πΉ} uniformly randomly varies within the range [0.8, 1], while the value of π€_{πΆπ
}

uniformly randomly varies within the range[0.9, 1]. The small random values used to
perturb the parameters πΉ_{π} and ππππ_{πππ€} will reveal an improvement in the performance
of MDE_πBX as it emphasizes slight varies on these two parameters each time πΉ is
generated.

*Crossover amplification factor ( π): Throughout evolution the value of parameter *π is
reduced linearly in the following manner,

π = ππππ [ππ

2 β (1 βπΊ β 1

πΊ_{πππ₯})] (3.37)

where ππππ(π¦) is the βπππππππβ function that outputs the smallest integer β₯ π¦. πΊ =
[1,2,3, β¦ πΊ_{πππ₯}] is the running generation index, πΊ_{πππ₯} is the maximum number of
generations, and ππ is the population size. The reduction monotony of the parameter π
creates the required balance between exploration and exploitation.

ο· **p-ADE Algorithm **

o *Advanced p-ADE Mutation scheme *

A new mutation strategy called DE/rand-to-best/pbest/bin is used; which is, essentially, based on utilizing the best global solution and the best previous solution of each individual that are involved in the differential process, thus bringing in more effective guidance information to generate new individuals for the next generation.

The detailed operation is as follows,

* π£*_{π}^{π‘}= π_{π}^{π‘}β π₯_{π1}^{π‘} + πΎ_{π}^{π‘} β (π₯_{πππ π‘}^{π‘} β π₯_{π}^{π‘}) + πΉ_{π}^{π‘} β (π₯_{ππππ π‘π}^{π‘} β π₯_{π}^{π‘}) (3.38)

where π₯_{πππ π‘}^{π‘} denotes the best individual in the current generation π‘. π₯_{π1}^{π‘} is a random

63

generated individual where π1 β [1, ππ] and π1 β π. π₯_{ππππ π‘π}^{π‘} denotes the best π^{π‘β}βs
previous individual picked up from the previous generation. The mutationβs control
parameters π_{π}^{π‘},πΎ_{π}^{π‘}, and πΉ_{π}^{π‘} of the π^{π‘β} individual are updated using a dynamic adaptive
manner. The most remarkable merit of this mutation technique is the inclusion of three
different working parts at the same time:

* Inertial Part (Inheriting part) represented by * π_{π}^{π‘}β π₯_{π1}^{π‘} where the current
individual,π£_{π}^{π‘}, inherits traits from another individual at generation π‘.

* Social Part (Learning Part) * represented by πΎ_{π}^{π‘}β (π₯_{πππ π‘}^{π‘} β π₯_{π}^{π‘}) where the current
individual,π£_{π}^{π‘}, gains information from the superior individual in the current generation
π‘.

* Cognitive Part (Private Thinking) *represented by πΉ_{π}^{π‘} β (π₯_{ππππ π‘π}^{π‘} β π₯_{π}^{π‘}) where the
current individual,π£_{π}^{π‘}, reinforces its own perception through the evolution process.

The high values of both the inertial and the cognitive part play a key role in intensifying
the exploration searching space, thus improving its ability for finding the global
solution. While the large values of the social part promotes connections among
individuals, thus resulting to speed up the convergence rate. From the previous
description of the main mechanism of π-ADE mutation scheme and the PSO standard
perturbation scheme (Kennedy & Eberhart, 1995; Xin, Chen, Zhang, Fang, & Peng,
2012), we can observe that they are closely related in origin, in particular, for the case
where the mutation (see Equation 3.38) is divided into three learning parts in the same
manner applied by PSO algorithm. In π-ADE there is an additional mechanism which is
called *classification mechanism. This classification mechanism is coupled with the *
mutation scheme to be implemented on the whole population at each generation.

Accordingly, the new mechanism divides the populationβs individuals into three classes:

*Superior individuals: The first individualsβ category where the fitness values of these *
individuals fall in the range π_{π} β π_{ππππ} < βπΈ(π^{2}), where π_{ππππ} is the mean fitness
values and πΈ(π^{2}) is the second moment of the fitness values of all individuals in the
current generation. In this case, the exploration ability of the search process is
enhanced by further intensifying the inertial and cognitive parts in order to increase
the likelihood of the excellent individual to find the global solution in its
neighborhood area. So, the corresponding individual is generated as follows,

π£_{π}^{π‘} = π_{π}^{π‘}β π₯_{π1}^{π‘} + πΉ_{π}^{π‘} β (π₯_{ππππ π‘π}^{π‘} β π₯_{π}^{π‘}) (3.39)

*Inferior individuals: The second individualsβ category where the fitness values of *
these individuals fall in the range π_{π} β π_{ππππ} > πΈ(π^{2}). The individual in this case has
poor traits since its place in the search space is far away from the global optimum.

Therefore, the exploration search ability is also intensified for rapid convergence rate. So, the corresponding individual is generated as follows,

π£_{π}^{π‘}= π_{π}^{π‘}β π₯_{π1}^{π‘} + πΎ_{π}^{π‘} β (π₯_{πππ π‘}^{π‘} β π₯_{π}^{π‘}) (3.40)

*Medium Individuals: The third individualsβ category where the fitness values of these *
individuals fall in the range βπΈ(π^{2}) < π_{π} β π_{ππππ} < πΈ(π^{2}). The individuals in this
category are not superior nor are they inferior; therefore, the complete perturbation
scheme (see Equation 3.38) should be implemented entirely for further enhancing
both the exploitation and exploration abilities.

o *p-ADE Parameter Control Schemes *

* p-ADE comprises four control parameters involved in the search process, including *
three mutation scheme parameters ( π, πΉ and πΎ) and crossover rate πΆπ
. A dynamic

65

adaptive scheme has been proposed to commonly update the four parameters through the run as follows,

π_{π}^{π‘} = π_{πππ}+ (π_{πππ₯}β π_{πππ}) Γ ((2 β ππ₯ π ( π‘

πΊππΓ π π(2))) Γ1
2
+ π_{π}^{π‘}β π_{πππ}^{π‘}

π_{πππ₯}^{π‘} β π_{πππ}^{π‘} Γ1
2 )

(3.41)

πΎ_{π}^{π‘} = πΎ_{πππ}+ (πΎ_{πππ₯}β πΎ_{πππ}) Γ ((ππ₯ π ( π‘

πΊππΓ π π(2)) β 1) Γ1
2
+ π_{π}^{π‘}β π_{πππ}^{π‘}

π_{πππ₯}^{π‘} β π_{πππ}^{π‘} Γ1
2 )

(3.42)

πΉ_{π}^{π‘} = πΉ_{πππ}+ (πΉ_{πππ₯}β πΉ_{πππ}) Γ ((2 β ππ₯ π ( π‘

πΊππΓ π π(2))) Γ1
2
+ π_{πππ₯}^{π‘} β π_{π}^{π‘}

π_{πππ₯}^{π‘} β π_{πππ}^{π‘} Γ1
2 )

(3.43)

πΆπ
_{π}^{π‘} = πΆπ
_{πππ}+ (πΆπ
_{πππ₯}β πΆπ
_{πππ}) Γ ((2 β ππ₯ π ( π‘

πΊππΓ π π(2))) Γ1
2
+ π_{π}^{π‘}β π_{πππ}^{π‘}

π_{πππ₯}^{π‘} β π_{πππ}^{π‘} Γ1
2 )

(3.44)

As can be seen from the above equations, the main adaptive scheme is equally captive to the influence of the number of generations achieved, as well as the fitness values.

Technically, the value of each control parameter varies within its specified range as, π β [0.1, 0.9], πΎ β [0.3, 0.9], πΉ β [0.3, 0.9] and πΆπ β [0.1, 0.9] during the run of the algorithm. Throughout the evolution process, the values of these parameters will gradually decreases; thereby transits the search from exploration to exploitation.