IEEE Computational Intelligence Magazine - February 2023 - 24

FIGURE 1 The original landscape (left) and a dilated landscape (right)
of the Ackley benchmark function. The less promising regions of the
landscape (yellow areas) are compressed, while the basin of attraction
of the global optimum (blue area) is expanded in the dilated
landscape.
identification a difficult task, as the optimization algorithms
might stagnate in regions like basins containing local optima.
In this context, DFs are introduced to manipulate the fitness
landscape to ease the optimization process [26]. In particular, DFs
can be defined starting from the so-called Basis Functions (see
Section II-A). The defined DFs re-map the original search space
onto a dilated search space by " expanding " promising regions,
possibly around the global optimum, and by " compressing "
other regions characterized by poor fitness values or local optima.
Figure 1 shows an example ofthe application ofDFs. The original
two-dimensional search space of the Ackley benchmark
function (left plot) is dilated (right plot) by compressing the nonpromising
regions, denoted by the yellow areas, while expanding
the basin ofattraction ofthe global optimum, the blue area.
A. Basis Functions
Arbitrarily complex DFs can be obtained by combining different
Basis Functions (BFs), which are monotonically increasing
functions characterized by a parameter regulating the dilation
intensity: the higher the value of the parameter, the stronger
the dilation. In this article, the family of BFs used in [27],
which comprises the linear transformationfaðxÞ and its inverse
f1
a ðxÞ, is extended by introducing two additional functions
gbðxÞ and hgðxÞ, together with their inverse, to transform the
search space by means oflogarithm and exponentiation operations,
respectively. The upper plots of Figure 2 show some
examples of BFs with different values for the parameters a, b,
and g, which lead to dilations with different intensities.
TheDFs can also include the folding operators KrðqrÞ, which
are a family ofBFs whose definition includes other invertible BFs.
In particular, the folding operators require the use ofa BF q,its
inverse q1, a parameter r that determines the dilation effect of
the employed BFs q and q1, and a " folding point " r 2½0; 1.
The folding point r indicates the threshold, beyond which the
folding operator stops using the BF q and starts using its inverse
q1. The bottom panels ofFigure 2 show some examples offolding
operators with different BFs q and folding points r. All BFs
and the folding operator used in this work are defined in Table I.
To obtain an effective dilation, the DFs must be defined by
exploiting information either about the region where the global
optimum is located or about non-promising landscape regions.
24 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
Since in many situations, e.g., real-world problems, such information
is not available, an automatic method (discussed in the next
section) is then proposed to evolve tailored DFs [27].
B. Genetic Algorithms Coupled With FST-PSO
The two-layer evolutionary method proposed in [27], named
GA-FSTPSO, combines GAs [29] and Fuzzy Self-Tuning
PSO (FST-PSO) [30] to automatically identify optimal DFs
for any arbitrary optimization problem. GA-FSTPSO exploits
a ðm þÞ GA, with ¼ m, where m () indicates the number
of parents (offspring) in each generation. Specifically, in each
generation, descendants are created from the selected pairs
of parents through the crossover and mutation operators.
These operators are thus used to blend the characteristics of
two parents and to introduce new genetic material by randomly
altering one or more values of an individual. Finally,
the m best out of all m þ individuals are kept for the next
generation. In addition, GA-FSTPSO exploits FST-PSO [2],
FIGURE 2 Examples of BFs (left) and their corresponding inverse
functions (right), with different parameter values (top) and different
folding points (bottom).
TABLE I Definition of the basis functions used in this work.
BF
DEFINITION
faðxÞ
f1
ða1Þxþ1 a 2ð0; 1Þ
ax
a ðxÞ
gbðxÞ
g1
hgðxÞ
h1
g
KrðqrÞ
K1
(
r ðqrÞ
f1=aðxÞ a 2ð0; 1Þ
lnðbxþ1Þ
lnðbþ1Þ b 2ð0; 1Þ
b ðxÞ
1
b ððb þ 1Þx 1Þ b 2ð0; 1Þ
xg g 2ð0; 1Þ
h1=gðxÞ g 2ð0; 1Þ
1
r qrðx=rÞ
ð1 rÞq1
r ð x
1r r
Krðq1
r Þ
x r
1rÞþrx > r
:
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