IEEE Computational Intelligence Magazine - February 2023 - 25

addition, mutation can be applied to modify the structure of an
individual. Mutation replaces a selected part of the individual
with a new randomly generated part, aiming at further increasing
the diversity ofthe offspring population. Finally, the parental population
is replaced by the offspring population. The whole procedure
is repeated until a termination criterion is met (e.g., the
maximumnumber ofgenerations is reached).
III. GP4DFs
A novel approach based on GP, named GP4DFs, is defined to
automatically find a specific DF for each search space dimension,
using a reduced number of fitness evaluations with
respect to GA-FSTPSO [27].
GP4DFs is a novel method employing a strongly typed GP
FIGURE 3 Example of a tree encoding two different DFs for a
2-dimensional optimization problem. In particular, the blue (orange)
subtree encodes the DF applied to the first (second) dimension of the
search space.
[30], an effective and settings-free variant of the well-known
PSO algorithm [31]. FST-PSO uses a fuzzy reasoner for each
particle ofthe swarm to calculate, at run-time, the optimal balance
among all the forces governing the exploration and
exploitation capabilities ofthe particle.
GA-FSTPSO combines GAs and FST-PSO as follows: an
outer layer exploits GAs to evolve the structure, i.e., the composition
of BFs of the DFs, and an inner layer uses FST-PSO
to tune the parameters of the BFs. GA-FSTPSO has some
issues that might hinder its applicability when used to tackle
high-dimensional optimization problems. First, it requires a
high number of fitness evaluations to identify an optimal DF
that can be then effectively used. Second, the same DF is
applied to all the search space dimensions, which might be
unfruitful in the case ofasymmetrical fitness landscapes.
C. Genetic Programming
GP is a family ofEC approaches that evolve a population ofcomputer
programs [32], [33]. GP was initially proposed to find the
optimal computer program for different AI problems, including
Machine Learning problems [32]. While GAs are mostly characterized
by individuals with a fixed length, GP allows for evolving
a set ofindividuals ofdynamic size and shape, such as hierarchical
compositions ofprimitive functions and terminals [32]. The individuals
are built by composing predefined primitive functions,
which may include arithmetic and conditional logical operations,
as well as mathematical and domain-specific functions. On the
other hand, the terminals can be numeric constants or inputs
appropriate to the problem domain.
Similar to GAs, GP is initialized with a population of randomly
generated individuals. During each generation, crossover is
applied to pairs ofparental individuals, selected based on their fitness,
to create new offspring solutions, possibly ofbetter quality.
Considering that parents may have different sizes and shapes, the
generated offspring solutions are combinations of their parents'
subexpressions (e.g., subtrees, subprograms, and subroutines). In
approach [34] to evolve trees encoding D different DFs, where
D is the number of the search space dimensions related to the
optimization problem. Each individual of GP4DFs represents
a candidate solution composed ofa combination ofDFs. Candidate
solutions apply a possibly different transformation to
each search space dimension. More precisely, the d-th DF
encoded by an individual is applied to the d-th dimension of
the search space (where d ¼ 1; ... ; D). To effectively encode
multiple DFs using a unique tree, GP4DFs employs a
" dummy " root node that does not represent any kind ofoperation.
The root node is characterized by D outgoing edges
that are used to separate the D DFs, where each one is tailored
for the corresponding dimension. For instance, the tree represented
in Figure 3 encodes a putative solution for a 2-dimensional
problem. The node denoted by ROOT represents the
" dummy " node, while the blue subtree is applied to the first
dimension of the search space and the orange subtree to the
second dimension. In particular, the DF for the first dimension
encoded by the blue subtree is f6ð0:5 þ xÞ, while the DF for
the second dimension encoded by the orange subtree is
ðlnðg8ðxÞÞÞ=ðh5ðxÞÞ. It is worth noting that a similar implementation
could be realized by considering a multi-tree GP
approach [35], where a single individual can be considered as a
forest oftrees. In such a context, each tree ofthe forest (encoding
a specific DF) would be enumerated and applied to each
dimension ofthe search space. The option ofusing a " dummy "
node as root, allowing for managing individuals as simple trees,
has been chosen because it simplifies the implementation without
modifying the dynamics ofthe algorithm, and because it is a
very common approach that has been employed in several existing
studies (see for instance [36], [37]).
The individuals of GP4DFs are structured as follows: the
internal nodes ofthe subtrees are functional nodes with arity 1
or 2, whose edges point towards a node that encodes a real
value. We employ the arithmetical operators (i.e., a þ b, a b,
a b, a=b, ab, with a; b 2 R) having arity equal to 2. Moreover,
we use the inverse (a1 ¼ 1=a), the exponential (ea, where e is
the Euler's number), the natural logarithm (lnðaÞ), and the sine
and cosine (sinðaÞ, cosðaÞ) operators with arity equal to 1. It is
worth pointing out that the operations a=b and 1=a are protected,
so they are applied if b 6¼ 0 and a 6¼ 0, respectively;
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 25
IEEE Computational Intelligence Magazine - February 2023

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