IEEE Computational Intelligence Magazine - February 2023 - 29
FIGURE 5 Convergence plots and boxplots representing the distribution of the best solutions found over 30 runs by GP4DFs (blue) and GA-FSTPSO
(orange) on the benchmark functions defined in 30 dimensions. *** p-value 0.001.
operators were not able to help the population escape the local
optima, hindering the exploration ofmore promising search space
regions. Nonetheless, considering the No Free Lunch Theorem
[48], it can be argued that GP4DFs is generally a better choice than
GA-FSTPSO, especially when no information about the location
ofthe global optimum is available.
In the case ofthe benchmark functions defined in 2, 50, and
100 dimensions, the obtained results confirm the observations
related to D ΒΌ 30 dimensions, with the only exception of F3,
where the performance GP4DFs decreases as the number of
dimensions increases. Overall, the obtained results corroborate
the validity ofGP4DFs, especially in the case ofbenchmark functions
specifically defined to resemble the features of real-world
optimization problems (i.e., those ofthe CEC'17 suite).
The diversity of the solutions found by GP4DFs over the 30
executions was examined either comparing the optimal solutions
found during different runs, and the different DFs obtained for the
different dimensions ofthe problem,within the same solution. For
several benchmark functions, GP4DFs found similar solutions
over the 30 runs. Take Figure S20 in the supplementary material
as an example, it shows the dilation obtained on all the dimensions
of shifted Rosenbrock benchmark function, induced by the DFs
encoded in the optimal solutions found over the 30 runs. It can be
observed that, besides converging towards the same solution during
each run, GP4DFs used the sameDF in almost all dimensions.
It is worth noting that GP4DFs found different solutions
over 30 repetitions when the benchmark function is non-separable
or the global optimum is not located in a point having
the same coordinate in all the dimensions, an example of such
case is F1 with 30 dimensions (see Figure S21 in the supplementary
material, available online). In addition, considering
the features of this benchmark function, the application of a
specific DF to a given dimension can influence the choice of
the DFs for the other dimensions; the optimal solutions found
by GP4DFs effectively transform the search space by employing
different DFs to dilate the different dimensions.
V. Conclusion
In this work, a GP-based approach, called GP4DFs, was proposed
to automatically evolve DFs that are used to transform the search
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 29
IEEE Computational Intelligence Magazine - February 2023
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