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10.4.4 MIES
Evolution strategies (ES) are metaheuristics that follow the concept of natural
evolution. An individual in an ES contains the information about one solution
candidate. This individual is subject to recombination, mutation and selection
operations. By evolving sets, or populations of individuals, the ES try to find
improved solutions.
In MIES, an individual contains information about real-valued variables, ordinal
integer variables and categorical variables. Parameters of the probability distribution
used in the mutation operator (such as standard deviations or step sizes) are also
stored in the individual for the purpose of self-adaptive parameter control. The latter
parameters are referred to as strategy parameters. As a consequence the domain of
an individual I can be expressed as follows:
I = R 1 × . . . × R n r × Z 1 × . . . × Z n z × D 1 × . . . × D n d × A s
(10.4)
with A s = R
n σ +n ζ
+
× [0, 1] n p being the domain of the strategy variables.
Correspondingly, an individual of a population can be represented as
a = (r 1 , . . . , r n r , z 1 , . . . , z n z , d 1 , . . . , d n d , σ 1 , . . . , σ n σ , ζ 1 , . . . , ζ n ζ , p 1 , . . . , p n p )
(10.5)
The so-called design variables r 1 , . . . , r nr , z 1 , . . . , z nz , d 1 , . . . , d nd determine the
objective function value and thus the fitness of the individual. The strategy variables
σ 1 , . . . , σ nr are standard deviations used in the mutation of the real-valued variables,
and ζ 1 , . . . , ζ nz denote mean step sizes in the mutation of the integer parameters.
Finally, p 1 , . . . , p np denote mutation probabilities (or rates) for the nominal discrete
object parameters. MIES is considered a self-adaptive process because the strength
of the mutation parameters continuously evolves during the optimization. Hence,
the mutation strength itself is also governed by an evolutionary process. The
philosophy behind self-adaptation is that the evolutionary process can solve two
problems simultaneously: the determination of the best strategy variables and the
determination of the best object variables. More details on self-adaptation can be
found in Chap. 11.
The first population P(0) of μ individuals is generated by uniform random
sampling from I. Then, the main loop of the MIES algorithm starts. In a first step,
the algorithm generates the set of λ new offspring individuals with the following
procedure. Two parents are randomly selected from the population, and an offspring
is generated by recombination and mutation. The recombination operator can be
subdivided into two steps, selection of the parents and recombining the selected
parents. The two parents c 1 , c 2 ∈ I are selected randomly, from the parental
generation for each of the offspring individuals. In MIES, two different types of
recombination are used: dominant and intermediate [45]. The first one is adopted
for solution variables and consists of a random selection of one of the corresponding
parental parameters for each offspring vector position. The latter is used for
L. Gentile et al.
10.4.4 MIES
Evolution strategies (ES) are metaheuristics that follow the concept of natural
evolution. An individual in an ES contains the information about one solution
candidate. This individual is subject to recombination, mutation and selection
operations. By evolving sets, or populations of individuals, the ES try to find
improved solutions.
In MIES, an individual contains information about real-valued variables, ordinal
integer variables and categorical variables. Parameters of the probability distribution
used in the mutation operator (such as standard deviations or step sizes) are also
stored in the individual for the purpose of self-adaptive parameter control. The latter
parameters are referred to as strategy parameters. As a consequence the domain of
an individual I can be expressed as follows:
I = R 1 × . . . × R n r × Z 1 × . . . × Z n z × D 1 × . . . × D n d × A s
(10.4)
with A s = R
n σ +n ζ
+
× [0, 1] n p being the domain of the strategy variables.
Correspondingly, an individual of a population can be represented as
a = (r 1 , . . . , r n r , z 1 , . . . , z n z , d 1 , . . . , d n d , σ 1 , . . . , σ n σ , ζ 1 , . . . , ζ n ζ , p 1 , . . . , p n p )
(10.5)
The so-called design variables r 1 , . . . , r nr , z 1 , . . . , z nz , d 1 , . . . , d nd determine the
objective function value and thus the fitness of the individual. The strategy variables
σ 1 , . . . , σ nr are standard deviations used in the mutation of the real-valued variables,
and ζ 1 , . . . , ζ nz denote mean step sizes in the mutation of the integer parameters.
Finally, p 1 , . . . , p np denote mutation probabilities (or rates) for the nominal discrete
object parameters. MIES is considered a self-adaptive process because the strength
of the mutation parameters continuously evolves during the optimization. Hence,
the mutation strength itself is also governed by an evolutionary process. The
philosophy behind self-adaptation is that the evolutionary process can solve two
problems simultaneously: the determination of the best strategy variables and the
determination of the best object variables. More details on self-adaptation can be
found in Chap. 11.
The first population P(0) of μ individuals is generated by uniform random
sampling from I. Then, the main loop of the MIES algorithm starts. In a first step,
the algorithm generates the set of λ new offspring individuals with the following
procedure. Two parents are randomly selected from the population, and an offspring
is generated by recombination and mutation. The recombination operator can be
subdivided into two steps, selection of the parents and recombining the selected
parents. The two parents c 1 , c 2 ∈ I are selected randomly, from the parental
generation for each of the offspring individuals. In MIES, two different types of
recombination are used: dominant and intermediate [45]. The first one is adopted
for solution variables and consists of a random selection of one of the corresponding
parental parameters for each offspring vector position. The latter is used for