IEEE Computational Intelligence Magazine - February 2023 - 23

issue, different methods have been proposed to manipulate
search spaces and improve the optimization process. This paper
focuses on Dilation Functions (DFs), which are one ofthe most
promising techniques to manipulate the fitness landscape, by
" expanding " or " compressing " specific regions. The definition
of appropriate DFs is problem dependent and requires a-priori
knowledge ofthe optimization problem. Therefore, it is essential
to introduce an automatic and efficientstrategy toidentify
optimal DFs. With this aim, we propose a novel method based
on Genetic Programming, named GP4DFs, which is capable of
evolving effective DFs. GP4DFs identifies optimal dilations,
where a specific DF is applied to each dimension of the search
space. Moreover, thanks to a knowledge-driven initialization
strategy, GP4DFs converges to better solutions with a reduced
number of fitness evaluations, compared to the state-of-the-art
approaches. The performance ofGP4DFs is assessed on a set of
43 benchmark functions mimicking several features of realworld
optimization problems. The obtained results indicate the
suitability ofthe generated DFs.
features or using surrogate models of the fitness function [17],
[18], [19], [20]. Even though surrogate models generally lead to
evaluating simpler fitness landscapes, their construction is most of
thetimea difficult task. Alternatively, it is possible to directly
apply transformation or modification strategies to the search
space to improve the quality ofthe sampling ofthe candidate solutions
and to try to avoid non-promising regions. Among others,
the Shrinking Space Technique [21] and the Space Transformation
Search [22] apply shrinking and transformation operations,
allowing the optimization methods to focus on promising areas
[21], [23], [24]. In addition, approaches based on " decoders "
have also been introduced to map feasible regions of the search
space into areas that should ideally improve the sampling performed
by the meta-heuristics [25].
Recently, Dilation Functions (DFs) have been proposed to
I. INTRODUCTION
T
he number of studies focusing on the optimization
of real-world problems has drastically increased in
the last few years [1], [2]. Among various real-world
applications, it is worth mentioning engineering
problems, which range from civil to manufacturing and intelligent
system design [3], [4], network engineering [5], systems
biology and bioinformatics [6], biochemical model calibration
[7] and haplotype phasing [8], co-registration ofmedical images
[9], and hydrology [10]. All the problems related to the aforementioned
applications can be formulated as real-valued optimization
problems sharing characteristics such as nonconvexity,
non-separability, non-differentiability, and multimodality
[11]. These characteristics generally hinder the performance
oflocal search algorithms and require the application of
global search algorithms, such as bio-inspired meta-heuristics
[12]. These meta-heuristics include Evolutionary Computation
(EC) [13] and Swarm Intelligence (SI) [14] approaches, where
the former is inspired by the Darwinian theory of evolution,
while the latter exploits the emergent collective behavior of
groups of living organisms. In both cases, a population of randomly
created individuals that encode candidate solutions is
iteratively updated to find the optimal solution of the given
optimization problem. Regarding EC strategies, the population
is bred through the generations using a method to mimic natural
selection, together with two genetic operators: crossover
and mutation [15], [16]. For what concerns SI approaches, the
population is composed ofcooperating agents moving within a
bounded search space and interacting with each other.
Both EC and SI might struggle to solve some problems, as
the optimization process is impaired by the intrinsic characteristics
of the fitness landscape. Several research studies attempt to
improve the performance ofthe meta-heuristics by adding new
re-map the original search space onto a dilated search space [26].
To be more precise, DFs act with the aim of " compressing " the
regions characterized by poor fitness values and " expanding " the
most promising fitness landscape regions. These transformations
not only allow for improving the generation ofthe initial population
but also the convergence to the global optimum [26]. However,
DFs are problem-dependent and require prior knowledge
ofthe fitness landscape. Thus, to address this problem, an automaticmethod
calledGA-FSTPSO, based onGenetic Algorithms
(GAs) and Particle Swarm Optimization (PSO), was introduced
to evolve optimal DFs [27].
This article follows the latter line of research, focusing on
the limitations ofthe previous works on DFs and on the strategies
designed to automatically build DFs [27]. Specifically, this
work presents a novel strategy based on Genetic Programming
(GP) to identify optimal DFs, called GP4DFs. GP4DFs evolves
multiple DFs that apply a different transformation to each
dimension of the search space. This is particularly useful for
optimization problems where the global optimum is located in
different positions along the search space dimensions, as in the
case of the Parameter Estimation of the kinetic constants of
biochemical systems [7]. Besides, GP4DFs drastically reduces
the number offitness evaluations required by the optimization
process, allowing for an efficient and effective application to
problems characterized by a high number of dimensions, as in
many real-world optimization scenarios. The performance of
GP4DFs is analyzed and compared to GA-FSTPSO for the
optimization of DFs on a set of benchmark functions, including
those presented in [28].
II . Background
Given a D-dimensional search space S RD and an objective
function F : S ! R, the aim of any global optimization algorithm
is the identification of the solution x 2 S that minimizes
(i.e., FðxÞ FðxÞ8x 2 S; x 6¼ x) or maximizes (i.e.,
FðxÞ FðxÞ8x 2 S; x 6¼ x) the function F. For several
real-world optimization problems, the fitness landscape is
non-convex, non-separable, non-differentiable, and multimodal.
These characteristics make the global optimum
FEBRUARY 2023 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 23
IEEE Computational Intelligence Magazine - February 2023

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - February 2023
