IEEE Computational Intelligence Magazine - February 2023 - 28

IV. Results
The performance ofGP4DFs is assessed and compared against GAFSTPSOon
several benchmark functions, characterized by different
features (e.g., multi-modality, separability), specifically defined to
analyze global optimization algorithms [2]. These benchmark functions
have been designed tomimic themain characteristics ofvarious
real-world problems, which are often endowed with complex features
that are barely grasped by basic optimization algorithms [40].
Benchmark function suites for real-parameter numerical optimization
are employed every year in research competitions, like those
hosted by the IEEE Congress on Evolutionary Computation
(CEC) [28], [41] and the Genetic and Evolutionary Computation
Conference (GECCO) [42], [43], [44]. The tests presented here
include a subset of the benchmark functions used in [27], such as
Ackley, Alpine, Michalewicz, and Sphere functions, along with
their shrinked and shifted variants. Moreover, the whole suite of
CEC'17 [28], [45], except for function F2 that was removed from
the competition, is considered in the tests. The complete list ofthe
benchmark functions used in this work is reported in Tables S1-S2.
All tests shown hereafter have been run with 2, 30, 50, and
100 dimensions, for all the benchmark functions. To achieve a
fair comparison between GA-FSTPSO and GP4DFs, both
algorithms employ all the BFs listed in Table I to automatically
build the DFs, with a range for the parameter impacting the
dilation intensity in ½0; 10; note that GA-FSTPSO uses the
real-valued interval, while GP4DFs employs the discrete interval.
For the folding operators, the " folding point " can assume
the values of0.25, 0.5, and 0.75.
Table II reports the settings used for both GP4DFs and GAFSTPSO
on all the problems' dimensions considered in this work.
The settings have been obtained by arbitrarily fixing the budget of
amaximum number offitness evaluations to	20 000 D,as typically
happens for the competitions, where the performance ofthe
meta-heuristics is compared in solving real-parameter numerical
optimization benchmark functions. The setting values for each
number of dimensions of the problems are selected to properly
balance the exploration and exploitation capabilities ofGP4DFs.
Typically, larger populations consider a greater variety ofsolutions,
especially during the first generations, while such diversity
decreases in the later stages of the evolution. On the other hand,
the exploitation capability is enhanced with higher numbers of
generations, as the population undergoes multiple evolution steps,
possibly producing better candidate DFs. For what concerns GAFSTPSO,
the settings were selected to balance the search between
new compositions ofBFs (the GA outer layer) and good parameterization
of the BFs (the FST-PSO inner layer). Finally, GAFSTPSO
applies the crossover and mutation operators with the
same probability (i.e., 0.5), while GP4DFs uses a mutually exclusive
probability, with pc ¼ 0:6and pm ¼ 0:4. In order to collect
statistically sound results, 30 distinct runs were executed for each
benchmark function and each algorithm.
In the analyses, both the convergence speed of the two
approaches and the distribution ofthe best solutions found over
30 runs are taken into account. The convergence plots report,
for each generation ofGP4DFs and of the outer layer of GA28
IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2023
FSTPSO, the median meta-fitness of the best individuals. The
boxplots represent the distributions of the meta-fitness values of
the best solutions found. In addition, the results ofthe statistical
tests, i.e., the p-values obtained by the Mann-Whitney U test
[46], [47], which specify whether the difference between the distributions
is statistically significant or not, are shown in the plots.
Figures 5 and 6, as well as Figures S5- S9 ofthe supplementary
material, show the performance of GP4DFs and GAFSTPSO
on the benchmark functions with 30 dimensions. The
results achieved with 2 dimensions are shown in Figures S1- S4,
the results with 50 dimensions are shown in Figures S10- S14,
while for 100 dimensions, the results are presented in Figures
S15- S19. Due to space limitations, a subset of the considered
benchmark functions defined with D ¼ 30 dimensions is
brought into discussion. These benchmark functions are characterized
by different distinctive features: unimodal functions
(Nobile1, F1, and F3), multimodal (Deceptive, Michalewicz,
and Shubert), shifted and shrinked functions (shifted and
shrinked Rosenbrock), hybrid functions (F14 and F16), and
composed functions (F21 and F29). The boxplots reporting the
distributions of the meta-fitness values indicate that GP4DFs is
better than GA-FSTPSO on the majority of benchmark functions
on 30 dimensions, with a strong statistical difference.
Regarding convergence, GP4DFs was able to find better individuals
than GA-FSTPSO by using a third ofthe budget ofthe
fitness evaluations. Stated otherwise, after only a third ofthe fitness
evaluations, the median meta-fitness value ofthe best individuals
ofGP4DFs (blue lines) is below the median meta-fitness
ofthe best individuals found by GA-FSTPSO at the end ofthe
optimization process (orange lines). In addition, in several cases,
the " warm " initialization strategy allowed GP4DFs to generate
individuals with better meta-fitness values than most ofthe individuals
found by GA-FSTPSO.
By comparing the results ofGP4DFswith those obtained from
GA-FSTPSO, it is also observed that both algorithms found similar
solutions for the benchmark functions with the global optimum
located in the center of the search space (e.g., Ackley), as well as
for the functions where the global optimum has the same coordinate
in all the dimensions (e.g., Rosenbrock). In other cases, the
obtained results highlight that evolving a specialized DF for each
dimension of the search space allows for obtaining better results
than using the same DF for all the dimensions. Moreover,GP4DFs
achieved better results on unimodal functions (F1, F3), multimodal
functions (Deceptive, shifted Rosenbrock), hybrid functions
(F14, F16) and composed functions (F21, F29).
On the contrary, GA-FSTPSO outperformed GP4DFs on a
fewfunctions regardless ofthe numberofdimensions (e.g., Deceptive,
Rosenbrock, and Shubert). In these situations, the solutions
evolved by GP4DFs got stuck in trying to dilate regions optima,
instead of focusing on more promising regions. In fact, both the
Deceptive and Rosenbrock functions are characterized by an easily
reachable local optimum that can trap the whole population; Shubert
is instead characterized by multiple local optima located near
the global optima, a circumstance that can misdirect the identification
of optimal DFs. It can be hypothesized that the evolution
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