Towards Rigorous Foundations for Metaheuristic Research
Thimershen Achary
1,2 a
, Anban W. Pillay
1,2 b
and Edgar Jembere
1,2 c
University of KwaZulu-Natal, Westville 4001, South Africa
Centre for AI Research (CAIR), South Africa
Keywords: Metaheuristics, Component-based View, Definition, Rigor.
Abstract: Several authors have recently pointed to a crisis within the metaheuristic research field, particularly the
proliferation of metaphor-inspired metaheuristics. Common problems identified include using non-standard
terminology, poor experimental practices, and, most importantly, the introduction of purportedly new
algorithms that are only superficially different from existing ones. In this paper, we argue that although
metaphors may be good sources of inspiration and creativity, being the only reason for publication is
insufficient. Instead, adopting a formal, mathematically sound representation of metaheuristics is a valuable
path to follow. We believe this will lead to more insightful research.
The recent past has seen an increase in research that
is critical of numerous trends and practices observed
in the field of metaheuristics (Aranha et al., 2021;
Fister jr et al., 2016; Molina et al., 2020; Sörensen,
2015; Stegherr et al., 2020; Tzanetos & Dounias,
2021). An influential study by (Sörensen, 2015)
points out several broad issues, including
irresponsible metaphor usage, poor experimental
practices, and misconceptions of what a metaheuristic
Others have lamented the poor quality and lack of
rigor and insights in published works (see
According to (Campelo & Aranha, 2021; Fister Jr et
al., 2016; Sörensen, 2015), this has severe
consequences for productivity, the credibility of the
field, and the capability to stimulate new, valuable
insights effectively.
In this paper, we review the issues raised by
various researchers, consider proposed solutions, and
argue that metaheuristic studies should adopt a
mathematically formulated metaheuristic definition
where the underlying philosophy is mindful of the
issues affecting the metaheuristic field. We also agree
with recent sentiments that metaphors are useful to
inspire creativity but are insufficient on their own. We
then propose a mindful and rigorous core
understanding of metaheuristics.
1.1 Metaheuristics
The term 'meta-heuristic' was coined by Glover in
(Glover, 1986), where the authors suggested that
Tabu Search could be viewed as a metaheuristic
"superimposed" on another heuristic. The suggestion
is that metaheuristics operate on a higher level than
Early definitions of the term metaheuristic were
critically analyzed in (Voß, 2001). These definitions
generally suggest that a metaheuristic is a higher-
level strategy that guides subordinate heuristics, with
some auxiliary constituents such as information for
the guiding process and intelligent combinations of
various exploration and exploitation concepts.
The meta-level is described as dealing with
applying control and strategy to a given domain
(Ostrowski & Schleis, 2008). In the context of
heuristics being the domain, metaheuristics can then
be defined as entities that apply control and strategy
to heuristics, as depicted in Figure 1. Metaheuristics
consists of a base plan, an integrated learning
component, and strategic heuristics. The base plan
Achary, T., Pillay, A. and Jembere, E.
Towards Rigorous Foundations for Metaheuristic Research.
DOI: 10.5220/0011552600003332
In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 151-157
ISBN: 978-989-758-611-8; ISSN: 2184-3236
Copyright © 2023 by SCITEPRESS Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
and integrated learning component are utilized only
in hyper-parameter tuning, while strategic heuristics
are required for all metaheuristic activities. The
strategic heuristics apply control and strategy to the
Figure 1: Metarules framework (Ostrowski & Schleis, 2008).
Metaheuristics are described in (Sörensen & Glover,
2013) as frameworks that can be used to derive
heuristic optimization algorithms and notes that, in
literature, the frameworks and the heuristic
optimization algorithms are both referred to as
metaheuristics. An elaboration of why the distinction
between framework and algorithm is essential when
discussing novelty can be found in (Lones, 2020); it
can be inferred that, often, a novelty at the algorithm
level is hardly a significant feat.
The rest of the paper is structured as follows: a
review of several criticisms of the field is given in
Section 2. Section 3 briefly reviews potential
solutions to these problems discussed in literature. A
proposal for instilling rigor in the metaheuristics
research space is given in Section 4. This is discussed
in Section 5, and Section 6 concludes.
Several authors have recently pointed to problems
afflicting metaheuristic research. This section
summarizes these issues.
Irresponsible metaphor usage, in the
metaheuristic field, is the use of sources of
inspiration, e.g., nature, physics, and human behavior,
to be the most, if not the only, pivotal aspect to justify
the algorithm as a "new" metaheuristic to the field
(Aranha et al., 2021; Sörensen, 2015). These works
usually include practices that obscure details by using
non-standard terminology (terminology specific to
the metaphor/inspiration used). Doing so adds to the
challenge of positioning the proposed contribution in
literature and may give the impression that the
research output is novel. Symptoms of this activity
are, according to (Aranha et al., 2021; de Armas et al.,
2021; Molina et al., 2020; Sörensen, 2015; Tzanetos
& Dounias, 2021), a flood of metaheuristics,
numerous cases of very similar/overlapping work,
lack of novelty, and according to (Molina et al., 2020)
instances where inspirational source and algorithm
behaviour are disconnected.
Researchers have also pointed to poor
experimental practices. Reports such as (Aranha et
al., 2021; Sörensen, 2015; Stegherr et al., 2020;
Tzanetos & Dounias, 2021) suggest unfair and biased
comparisons such as comparing new proposals to
older metaheuristics instead of state-of-the-art and
tweaking hyperparameters in favor of a metaheuristic
to lift its performance above the rest.
Comparative studies are not transparent enough,
resulting in difficulties in extending past studies and
existing data (Aranha et al., 2021; Sörensen, 2015). A
lack of proper motivation for selecting metaheuristics
to compare is common (Stegherr et al., 2020). There
is also a lack of rigorous data analytics (Sörensen,
2015). Competitive studies produce very little insight
and do not answer or aid in answering the how and
why (Birattari et al., 2003; Hooker, 1995), yet
comparative studies are still widely setup as
competitive ones (Campelo & Aranha, 2021;
Sörensen, 2015).
The proliferation of metaphor-inspired
metaheuristics is also a cause for concern. A GitHub
project called the Evolutionary Computational
Bestiary lists a vast and ever-growing number of bio-
inspired metaheuristics (with only a few exceptional
bio-inspired metaheuristics being exempt) (Campelo
& Aranha, 2021). The aforementioned project
opposes the flood of metaheuristics, especially the
creation of new bio-inspired metaheuristics. Articles
and other projects that criticize certain metaheuristic
research trends are listed, some of which are intended
to parody or ridicule the fact that these trends still
The above criticisms have not been universally
accepted. One such counter-argument is that
metaheuristics are currently being applied in various
domains from numerous disciplines and have also
been applied to real-world problems (Torres-Jiménez
& Pavón, 2014). The view that metaheuristic research
is of poor quality may very well be overly pessimistic
and aims to make capital out of flaws in research
techniques that are merely pragmatic. The pursuit of
ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications
being theoretically optimal has little benefit to the real
Also, the argument goes, there is a long history of
using nature to inspire the development of
metaheuristic algorithms. Thus, to reject work that
uses natural inspiration is to hinder creativity. The
researcher pool has a diverse skill set, i.e., not all
possess an advanced mathematical background, and
researchers have skills/talents which may lie more in
creativity than analytics. Therefore, the move to
abandon natural inspiration or inspirational sources,
in general, can be interpreted as a move to
discriminate against researchers that are more
creative than analytical.
To refute these arguments, we refer to a study by
(Ven & Johnson, 2006) that explores the relationship
between scholarly and practical knowledge. It
analyses ways in which the discrepancies between
these domains have been framed and discusses
methods to address this, such as a method of engaged
scholarship (proposed by the aforementioned study).
From the study, it can be understood that practical and
scholarly knowledge have different contexts and
objectives. Practical knowledge deals with specific
circumstances in certain scenarios, while scholarly
knowledge deals with viewing specific circumstances
as instances of a more general case to further
understand and explain how what is done works.
Reaping both benefits can be achieved through
methods of communication between both spaces.
This entails that the scholarly domain must be robust
so that new knowledge can be framed efficiently
amongst existing knowledge and communicated
effectively to practical domains and other scholarly
Recent studies have shown instances of scholarly
work claiming to be novel, but the novelty does not
stand up to scrutiny. Comparative studies have been
questioned regarding their transparency and choice of
experimental practices. The overloading of well-
known concepts with non-standard terminology is
creating confusion in literature. In summary, the
issues highlighted by several publications are
indicators that the metaheuristic research space falls
extremely short of ideal conditions for a scholarly
According to (Swan et al., 2015), expressing
metaheuristics via mathematical formulations
facilitates a rigorous definition of the term
metaheuristic. Some may criticize and label this
decision as systematically marginalizing creative
research because mathematical definitions often use
cryptic notation that may not be friendly to
researchers without an advanced mathematical
background and with a different skill set. However,
the benefits of using mathematical formulations
(more specifically, functional descriptions), as listed
and discussed in (Swan et al., 2015), include
promoting better communicability, reproducibility,
interoperability, facilitating automated metaheuristic
assembly, and promoting scientific advancement.
Therefore, using mathematical formulations does not
marginalize creative research; instead, it guides
The No Free Lunch theorem (Wolpert &
Macready, 1997) being a valid premise in the
argument for justifying the existence of a vast number
of metaheuristics in the research space, is viewed as
unclear in (Lones, 2020). The study also speculates
that the argument may have substance as the
performance of different optimizers varies when
subjected to different problems. However, a
discussion is presented in (Camacho‐Villalón et al.,
2022) that criticizes the aforementioned argument as
being based on a misunderstanding of the No Free
Lunch theorem for optimization and that the vast
number of published metaheuristics based on
metaphors are creating confusion in the research
space, leading it away from proper scientific goals.
Therefore, relying on the No Free Lunch theorem is
not advisable to support the creation of a novel
Several authors have not only given critical
commentary on the field but have also suggested
potential solutions.
The solutions to the metaheuristic research
quality issues require adoption by researchers so that
their impact, as argued in the respective research
publications, may influence the metaheuristic
research space. Increasing awareness about issues
associated with metaphor-based research is therefore
essential to stimulate the adoption of these solutions
(Campelo & Aranha, 2021), and it is a recurring
theme in many such publications, e.g., (Lones, 2020;
Sörensen, 2015; Stegherr et al., 2020; Tzanetos &
Dounias, 2021). Projects such as the Evolutionary
Computational Bestiary are also ways to raise
A component-based view of metaheuristics, as a
solution to the issues afflicting the metaheuristic
research space, is highlighted in (Sörensen, 2015).
This view suggests understanding metaheuristics as
sets of general concepts, accompanied by the decision
Towards Rigorous Foundations for Metaheuristic Research
to distinguish metaheuristics from the optimization
algorithms derived from them. Its widespread
adoption may help resolve several of the problems
discussed above. The component-based view of
metaheuristics deals with conceptualization at the
foundational layer, i.e., where definitions,
taxonomies, ontologies etc., are crucial.
Applying mathematical formulations to express
metaheuristics facilitates a rigorous definition of the
term metaheuristic (Swan et al., 2015). Several
definitions of the term metaheuristic incorporate
tuples. Tuples encapsulate the specifications, main
components, and sometimes structures that hold the
relationships between the specifications and
The study by (Wang, 2010) provides worded
definitions for the terms metaheuristic and
metaheuristic computing. The study provides a
rigorous definition of metaheuristic computing using
tuples, in which the elements are concept algebra
A tuple definition for population-based
metaheuristics is presented as part of the unified
framework for population-based metaheuristics
introduced in (Liu et al., 2011).
The work done in (Cruz-Duarte et al., 2020)
defines a metaheuristic as a map (expressible in terms
of three components: initializer, search operator, and
finalizer heuristics) from an arbitrary domain to a
feasible domain of an optimization problem.
As part of the proposed design of a software
framework to solve combinatorial optimization
problems presented in (Peres & Castelli, 2021), a
metaheuristic – actually an abstract metaheuristic – is
defined as a map from a domain of specifications
(encapsulated in a tuple) to a set of possible variations
of the metaheuristic.
Swan et al. (Swan et al., 2015) advocate for
metaheuristics to be described entirely in terms of
functions (which are essentially maps), in which
metaheuristics are parameterized by their
environment, state, and the environments of the
employed components. The environment, in this
sense, refers to information required during
execution, and the state refers to the solution in
chosen representation form. The component
heuristics are also parameterized with their
environment and state.
The component-based view proposed by
(Sörensen, 2015) is meritorious but has drawbacks if
not used properly (Achary & Pillay, 2022).
Definitions such as those presented by (Cruz-Duarte
et al., 2020; Liu et al., 2011) express metaheuristics
in terms of components, but as emphasized above, the
ambiguity present in the definitions by (Cruz-Duarte
et al., 2020) may lead to conflicting understandings.
The definition by (Liu et al., 2011) uses biological
terminology and thereby promotes the metaphor-
based philosophy of metaheuristics. However,
metaphor usage, non-standard terminology, and
natural inspiration have been criticized in literature,
indicating that the perspective used may nullify the
long-term advantages of using the component-based
The framework proposed in (Peres & Castelli,
2021) resolves this ambiguity by providing
mathematically formulated definitions of conceptual-
level and concrete-level metaheuristics. Both are
formulated as maps. The former maps from a tuple of
abstract specifications to a set of concrete heuristic
optimization algorithms, and the concrete heuristic
optimization algorithms map from their concrete
specifications to an optimal solution.
4.1 Towards a Rigorous Foundation
for Metaheuristic Research
Conducting meaningful metaheuristic research for
both the long and short term requires metaheuristic
research to adopt strong foundations and a rigorous
The study by (Campelo & Aranha, 2021)
summarizes some promising alternative approaches
to conducting research in metaheuristics rather than
relying on metaphor-based techniques. They propose
understanding metaheuristics as frameworks of semi-
independent modules that influence one or more
intrinsic algorithmic structures. This is similar to the
proposal made in (Sörensen, 2015) to see
metaheuristics as frameworks and not concrete
heuristic optimization algorithms. Defining
metaheuristics as functions is advocated in (Swan et
al., 2015), which also suggested a specific template
for expressing these functions. Describing metaphor-
based metaheuristics using standard terminology that
effectively describes similarities and differences
between metaheuristics is motivated in (Lones,
2020). Comparing metaheuristics with structure-wise
similarity metrics, which facilitates determining
special-case and general-case relationships between
metaheuristics, is made possible by the work in (de
Armas et al., 2021). Using existing taxonomies from
literature rigorously is facilitated by work done in
(Achary & Pillay, 2022).
Each of the above contributions has little overlap
and a strict scope. Using these contributions together
ECTA 2022 - 14th International Conference on Evolutionary Computation Theory and Applications
may be effective for establishing strong foundations
for metaheuristic research and stimulating good
quality, insightful research.
A rigorous foundation for metaheuristic research
that makes use of the contributions, advice, and
guidelines of existing literature is proposed below.
A philosophy of metaheuristics that is mindful of
the issues affecting the field is provided by (Sörensen
& Glover, 2013) and further explained in (Sörensen,
2015). In this view, metaheuristics are problem-
independent frameworks that provide a set of
guidelines to create heuristic optimization algorithms
and are not the heuristic optimization algorithms
Mathematical definitions are known to be
rigorous, and there are also added benefits to
expressing metaheuristics as functions (Swan et al.,
2015). Metaheuristics could be formulated as:
𝑀: 𝑆
Where M is an arbitrary metaheuristic and S is a
set of tuples of specifications. The metaheuristic M
has an influence on the tuple format, and a tuple of
the set S must contain at least one heuristic operator.
A is the set of heuristic optimization algorithms, each
of which the rules of M can construct using a certain
element of S. A proof-of-concept for the formulation
in (1) can be found in Section 4.2.
The format and values of the tuples in the set S
may be determined using the works of (Lones, 2020)
and (Achary & Pillay, 2022). The novelty and
influence of metaheuristics can be determined by
applying the work of (de Armas et al., 2021) to
metaheuristics defined in terms of (1).
This map formulation (1) aligns with the
component-based view, as it guides the researcher to
elucidate which components are variable in the
specification tuple, thus providing scope for
experiments in future research.
The restriction that an element of S must contain
at least one heuristic operator enforces the
component-based view and avoids scenarios where
hyper-parameter values are the only elements of a
specification tuple.
This map is very abstract and does not have many
restrictions on how one may specialize it with details.
Its intended use is to be a rigorous underlying
conceptualization of what a metaheuristic is when
proposing a concrete formulated definition for future
research; this underlying conceptualization enforces
alignment with the component-based view and
considers the insights, advice, suggestions, and
guidelines from existing literature on the problems
within the metaheuristic field.
4.2 Proof of Concept
The Genetic Algorithm (GA), Particle Swarm
Optimization (PSO), Bat Algorithm (BAT), and
Differential Evolution (DE) metaheuristics are used
to illustrate how the formulation in (1) could be used;
a description of each of the aforementioned
metaheuristics can be found in (Yang, 2020).
4.2.1 Genetic Algorithm
1. Substitute GA in place of M.
2. An element of S would then contain the initializer,
crossover operator, mutation operator, selector,
and terminating condition.
3. An element of A will be a resulting concrete
Genetic Algorithm.
4.2.2 Particle Swarm Optimization
1. Substitute PSO in place of M.
2. An element of S would then contain the initializer,
location update, velocity update, and terminating
3. An element of A will be a resulting concrete
Particle Swarm Optimization algorithm.
4.2.3 Bat Algorithm
1. Substitute BAT in place of M.
2. An element of S would then contain the initializer,
position update, velocity update, local search
technique, and terminating condition.
3. An element of A will be a resulting concrete Bat
4.2.4 Differential Evolution
1. Substitute DE in place of M.
2. An element of S would then contain the initializer,
crossover operator, mutation operator, and
terminating condition.
3. An element of A will be a resulting concrete
Differential Evolution algorithm.
The definition of metaheuristics adopted by a
researcher will significantly influence their
metaheuristic research.
A contributing factor to the proliferation of novel
metaheuristics is arguably the ambiguity of whether
metaheuristics are frameworks, concrete heuristic
Towards Rigorous Foundations for Metaheuristic Research
optimization algorithms, or both. The study in
(Sörensen, 2015) remarks that it is unfortunate that
the term "metaheuristic" is used for both general,
problem-independent, algorithmic frameworks and
concrete heuristic optimization algorithms derived
from these frameworks and further expresses that
metaheuristics are not algorithms, but they are each a
set of ideas, concepts, and operators from which
heuristic optimization algorithms can be derived.
The definitions presented in (Cruz-Duarte et al.,
2020; Liu et al., 2011; Voß, 2001; Wang, 2010) fail
to resolve this ambiguity. Difficulty in determining
the novelty of new proposals may result from this
ambiguity since a heuristic optimization algorithm
could be related to a few or many concepts, ideas, or
operators of a framework, garnished with a metaphor
and non-standard terminology, then published as a
novel metaheuristic.
Comparative studies of metaheuristics have
received criticism in the literature (Aranha et al.,
2021; Sörensen, 2015). A flaw that has been
highlighted is that the implementations of
metaheuristics, whose selections are poorly
motivated (Stegherr et al., 2020), are compared, and
the results could be misunderstood as representative
of the framework.
Using metaphors and natural sources of
inspiration has led to the creation of well-known,
influential, and disruptive contributions such as
Particle Swarm Optimization, Genetic Algorithm,
Simulated Annealing, and Ant Colony Optimization,
as indicated in (Camacho‐Villalón et al., 2022).
However, the incorporation of natural inspiration in
research must outweigh the cost.
Research into the trends of metaphor and
inspirational source usage (Aranha et al., 2021;
Campelo & Aranha, 2021; Fister jr et al., 2016;
Sörensen, 2015; Tzanetos & Dounias, 2021) has
shown that metaphors and non-standard terminology
introduce challenges when trying to frame
metaheuristics amongst existing literature. It
facilitates work similar to existing literature to be
published as novel work. Non-standard terminology
confuses readers and clouds the relevance and the link
of the phenomenon described by the terms to the
A flood of metaheuristics has been linked to
metaphor and inspiration source usage. Research by
(Molina et al., 2020) showed that there are many more
inspiration sources than algorithmic behaviors.
Hence, it can be said that inspiration source usage is
a heuristic, in the general sense, for creativity, similar
to the exploration of ideas. However, there is too
much exploration and not enough rigor. Since
metaphor/inspiration usage enhances creativity, it is
insufficient on its own; this analogy is similar to those
used in (Fister jr et al., 2016; Lones, 2020) with the
similar computational optimization terminology
Although various publications argue that new
novel metaheuristics are not needed at this point in
the field's timeline, if a metaheuristic is to be
published, it should be accompanied by a formulation
of the metaheuristic in the format of (1). M represents
the abstract pseudocode, ideas, and concepts that
make up the metaheuristic. The format of elements of
S will convey which components are variable, i.e.,
different concrete components can be substituted in
their respective placeholders, which is then passed to
M to create a concrete optimization algorithm of the
set denoted by A in the formulation.
In this study, it is argued that metaheuristics studies
should adopt a mathematically formulated
metaheuristic definition where the underlying
philosophy is mindful of the issues affecting the
metaheuristic field; in other words, adopt definitions
that sustain good quality research. Mathematical
formulated definitions are rigorous and leave less
room for vagueness that can lead to convenient
interpretations. Ambiguities in adopted or proposed
definitions can potentially allow choosing a
definition/perspective/interpretation of the shelf that
suits a requirement for publication, leading to low-
quality research. The underlying philosophy of the
mathematically formulated definition must be
mindful of issues affecting metaheuristic research to
prevent the definition from having the potential to
stimulate problematic trends.
This work takes the stance that inspiration source
usage is a good heuristic for creativity but is not
needed right now; it has the capacity to become
saturated, which is detrimental to the field.
Intensifying research on existing work would be a
better practice at present.
Increasing theoretical insight, better analytical
techniques, and solid foundations should be a top
priority of metaheuristic research.
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