IEEE Computational Intelligence Magazine - February 2023 - 27

FIGURE 4 Example of one-point crossover (top) and mutation (bottom) operators applied to a tree for a 2-dimensional optimization problem. The
nodes selected for crossover and mutation are highlighted in red. As a result of the crossover operator, the subtrees having the root in the red node
are swapped; in the case of mutation, the yellow subtree is modified after the application of this operator.
TABLE II Settings of GP4DFs and GA-FSTPSO, number of points sampled to compute the meta-fitness values, and maximum number of
fitness evaluations (budget) used to execute the tests on 2, 30, 50, and 100 dimensions.
DIMENSIONS
2
30
50
100
50
GP4DFs
40
5
GA-FSTPSO
INDIVIDUALS GENERATIONS INDIVIDUALS GENERATIONS PARTICLES ITERATIONS
10
10
10
10
5
100
130
169
100
100
110
10
13
13
selected parent among the m possible parents, and the mutated
individual is added to the offspring population (lines 19 22).
GP4DFs exploits a one-point crossover in which, for each parent,
a node of the tree is randomly selected and the two subtrees,
whose root is the selected node, are swapped as shown in the top
ofFigure 4. Notably, the crossover might swap subtrees belonging
to DFs ofdifferent dimensions. Such characteristic ofthe crossover
is useful when dealing with problems whose global optimum is
located in the same position among different dimensions. The
mutation randomly samples a node ofthe tree, according to a uniform
distribution, and randomly mutates the subtree identified by
the sampled node as shown in the bottom ofFigure 4. It is worth
noting that to avoid unexpected behaviors, both crossover and
mutation can neither sample the root node nor select or insert an
additional root node in a subtree ofthe individuals.
Once all the offspring are generated, their fitness values are
calculated (line 23) and m individuals are selected among the
ðm þÞ individuals through m distinct tournaments, where
each tournament includes five randomly selected individuals
(lines 24 32). In each tournament, only the individual with
the best meta-fitness, as defined in [27], survives and goes to the
next generation. To calculate the meta-fitness, n points are
10
10
13
8
10
10
11
SAMPLED POINTS BUDGET
20
60
76
105
40 000
600000
988000
1951 950
randomly sampled on the dilated landscape and their fitness is
evaluated; then, the best 20% ofpoints are considered and their
average fitness value is computed and used as the fitness value of
the individual. The choice of considering the best 20% of the
sampled points was driven by the results of the analyses performed
in a previous work [27]. As a matter offact, it is empirically
observed that the higher the number ofpoints considered
to calculate the fitness value of a candidate solution, the higher
the probability to include points that are " geometrically distant "
from the dilated region, a situation that might mislead the whole
optimization process of GP4DFs. Different from the uniform
sampling strategy used in GA-FSTPSO, a sampling strategy that
leverages the Sobol sequences [39] is employed here. Sobol
sequences are quasi-random low discrepancy sequences that
allow for efficiently sampling a multidimensional space of
numerical values. Indeed, quasi-random series aim to uniformly
cover the search space with a reduced number of samples, i.e.,
with a lower number ofpoints compared to classic uniform distributions.
In particular, a single Sobol sequence is generated at
the beginning of the GP process and used to calculate all the
meta-fitness values during the entire evolution process. At the
end ofthe generations, the best individual is returned (line 33).
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 27

IEEE Computational Intelligence Magazine - February 2023

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