Framework for identifying the equivalence between Nature-Inspired Metaheuristics

Iztok Fister; Žan Hozjan; Iztok Fister,; Damjan Strnad

cs.NEMar 30, 2026

Framework for identifying the equivalence between Nature-Inspired Metaheuristics

Authors: Iztok Fister, Žan Hozjan, Iztok Fister,, Damjan Strnad

Organizations: Faculty of Electrical Engineering and Computer Science, University of Maribor, Koroˇska cesta 46, Maribor, 2000, Slovenia.

Abstract

The domain of metaheuristic optimization has become vibrant due to a flood of new algorithms using a new nature-inspired metaphor but lacking clear methodological novelty. The Criticism behind the development of these algorithms has reached such an extent that the critics started to assert that all novel algorithms are only copies of already developed ones. In this study, we try to show that the situation is not so black and white. Therefore, we define a strong equivalence theorem for estimating the similarity between two nature-inspired metaheuristics, according to which two algorithms are equivalent if, and only if, the cosine similarity of their phenotypic and genotypic feature vectors, characterizing their behavior by searching for the optimal solutions, is above some threshold. On the theorem basis, a framework is developed for identifying the equivalence between nature-inspired metaheuristics. Extensive experimental work using the framework has shown that searching for conditions to achieve the high similarity of the more well-known nature-inspired metaheuristics is hard, or even not possible to achieve, in the limited computational environments.

Figures & tables

Figure 1: Workflow of the framework for identifying strong equivalence between NI metaheuristics.

Switch	Argument
$-\text{a}$ to $-\text{c}$	Various algorithm’s parameters
$-\text{g}$	Maximum number of generations
$-\text{d}$	Problem dimension
$-\text{n}$	Number of parameters
$-\text{r}$	Maximum number of independent runs
$-\text{s}$	Random generator seed

Table 1: Passing parameter.

Figure 2: Scheme of the framework for identifying the similarity of NI algorithms, where $Alg1$ denotes the control and $Alg2$ the controlled algorithm.

Algorithm	Representation	Variation	Selection	Population	Adaptation
Algorithm	of individuals	operator	scheme	model	(optional)
accPSO	Floating-point	Eq. ( 5 )	global best	1-population	n/a
accFA	Floating-point	Eq. ( 8 )	global best	1-population	n/a
PSO	Floating-point	Eq. ( 2 )	one-to-one	2-populations	n/a
FA	Floating-point	Eq. ( 7 )	global best	1-population	n/a
BA	Floating-point	Eq. ( 9 )	SA -selection 1 1 1 SA -selection Excluded when $A_{i}=1$ .	1-population	Eq. ( 13 ), GRW 2 2 2 GRW excluded when $r_{i}=0$ .

Table 2: Summary characteristics of the NI metaheuristics in the study.

Algorithm	Initial			Parameter control			Domain
Algorithm	1	2	3	1	2	3	1	2	3
accPSO	$\alpha_{0}$	$\beta_{0}$	n/a	FIX	FIX	n/a	$[0.1,1.0]$	$[0.1,1.0]$	n/a
accFA	$\alpha_{0}$	$\beta_{0}$	n/a	FIX	FIX	n/a	$[0.1,1.0]$	$[0.1,1.0]$	n/a
PSO	$w$	$c_{1}$	$c_{2}$	FIX	FIX	FIX	$[0.4,0.9]$	$[1.5,2.5]$	$[1.5,2.5]$
FA	$\alpha_{0}$	$\beta_{0}$	$\gamma$	DET	META	FIX	$[0.1,1.0]$	$[0.5,1.5]$	$[0.1,1.0]$
BA	$A_{0}$	$\gamma$	n/a	DET	META	n/a	$[0.7,1.2]$	$[0.1,0.3]$	n/a

Table 3: Parameter analysis of the observed NI metaheuristic algorithm.

Panel of Figure 3 — Figure 3: Simplified syntax diagrams of the variation operators used in the observed NI metaheuristic algorithms.

\langle\text{operator}\rangle::=\langle\text{base\_term}\rangle~"+"~\langle\text{term\_1}\rangle~["+"~\langle\text{term\_2}]\rangle

\langle\text{base\_term}\rangle::=[\langle\text{weight}\rangle]~[\langle\text{individual\_current}\rangle|\langle\text{global\_best}\rangle]

\langle\text{term\_1}\rangle::=\langle\text{coefficient\_1}\rangle~[\langle\text{random\_term}\rangle|\langle\text{global\_best\_term}\rangle|\langle\text{FA\_neighbor\_term}\rangle|\langle\text{BA\_term}\rangle]

\langle\text{coefficient\_1}\rangle::=[\langle\text{fixed}\rangle|\langle\text{stochastic}\rangle|\langle\text{deterministic}\rangle|\langle\text{metaphor-based}\rangle]

\langle\text{random\_term}\rangle::="("~\langle\text{epsilon}\rangle~"-"~\langle\text{one\_half}\rangle~")"

\langle\text{global\_best\_term}\rangle::="("~\langle\text{global\_best}\rangle~"-"~\langle\text{individual\_current}\rangle~")"

\langle\text{FA\_neighbor\_term}\rangle::="("~\langle\text{individual\_better}\rangle~"-"~\langle\text{individual\_current}\rangle")"

Table 4: Simplified BNF notation of the variation operator syntax.

$\langle\text{weight}\rangle$	$\text{Parameter}~w$
$\langle\text{fixed}\rangle$	No parameter control
$\langle\text{stochastic}\rangle$	$\text{Parameter control involves randomness (e.g., }\epsilon_{1}\cdot c_{1}\text{ or }\epsilon_{2}\cdot c_{2}$ )
$\langle\text{deterministic}\rangle$	Deterministic parameter control (e.g., $\alpha$ in FA)
$\langle\text{metaphor-based}\rangle$	Metaphor-based parameter control (e.g., $\beta(r)$ in FA, $\bar{r}$ in BA)
$\langle\text{epsilon}\rangle$	$\text{Random number drawn from uniform distribution}~\epsilon\in\mathcal{U}(0,1)$
$\langle\text{mean\_pulse\_rate}\rangle$	Average pulse rate

Table 5: Meaning of the undefined non-terminals.

Table 6: Characteristics of the PC and the correlation coefficient interpretation.

Role	Algorithm	Metric	Min	Max	Mean	StDev	Median
Control	accPSO	$\alpha$	0.4999	0.5001	0.5000	$<0.0001$	0.5000
		$\beta$	0.2000	0.2000	0.2000	0.0000	0.2000
		Fitness	-0.1197	-0.0134	-0.0618	0.0269	-0.0577
Controlled	accFA	$\alpha$	0.4999	0.5001	0.5000	$<0.0001$	0.5000
		$\beta$	0.2000	0.2000	0.2000	0.0000	0.2000
		Fitness	-0.1197	-0.0134	-0.0618	0.0269	-0.0577

Table 7: The results of the Meta-DE by parameter setting analysis in CS-1.

Algorithm	Parameter	$Sim_{\cos}$	$Sim_{\text{SMAPE}}$	$\rho$	$Sim_{\text{kNN}}$	$Sim_{\text{SVM}}$	$Sim_{\text{RF}}$
accFA	Min	1.0000	1.0000	1.0000	n/a	n/a	n/a
	Max	1.0000	1.0000	1.0000	n/a	n/a	n/a
	Mean	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
	StDev	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	Median	1.0000	1.0000	1.0000	n/a	n/a	n/a

Table 8: The results of the statistical analysis according to similarity metrics for identifying the strong equivalence between the accPSO and accFA in CS-1.

Role	Algorith.	Metric	Min	Max	Mean	StDev	Median
Control	accPSO	$\alpha$	0.4999	0.5001	0.5000	$\mathbf{<0.0001}$	0.5000
		$\beta$	0.2000	0.2000	0.0000	0.2000	0.2000
		Fitness	-0.1197	-0.0134	-0.0618	0.0269	-0.0577
Controlled	PSO	$w$	0.4740	0.7760	0.6322	0.0780	0.6251
		$c_{1}$	1.5000	1.8624	1.5511	0.1084	1.5000
		$c_{2}$	1.5000	2.5000	2.3575	0.2193	2.4712

Table 9: The results of the Meta-DE by parameter setting analysis in CS-2.

Algorithm	Parameter	$Sim_{\cos}$	$Sim_{\text{SMAPE}}$	$\rho$	$Sim_{\text{kNN}}$	$Sim_{\text{SVM}}$	$Sim_{\text{RF}}$
PSO	Min	0.9674	0.9997	0.9673	n/a	n/a	n/a
	Max	0.9818	0.9999	0.9819	n/a	n/a	n/a
	Mean	0.9756	0.9998	0.9756	0.0000	0.0000	0.0000
	StDev	0.0038	$\mathbf{<0.0001}$	0.0038	0.0000	0.0000	0.0000
	Median	0.9764	0.9998	0.9764	n/a	n/a	n/a
FA	Min	0.1410	0.9991	0.1351	n/a	n/a	n/a

Table 10: The results of the statistical analysis according to the similarity metrics for identifying the strong equivalence using the accPSO as the control algorithm in CS-2.

$\langle\text{oper\_accPSO}\rangle::=\langle\text{base\_term}\rangle~"+"~\langle\text{fixed}\rangle\langle\text{random\_term}\rangle~"+"~\langle\text{fixed}\rangle\langle\text{global\_best\_term}\rangle$
$\langle\text{oper\_accFA}\rangle::=\langle\text{base\_term}\rangle~"+"~\langle\text{fixed}\rangle\langle\text{global\_best\_term}\rangle~"+"~\langle\text{fixed}\rangle\langle\text{random\_term}\rangle$
$\langle\text{oper\_PSO}\rangle::=\langle w\rangle\langle\text{base\_term}\rangle~"+"~\langle\text{stochastic}\rangle\langle\text{global\_best\_term}\rangle~"+"~\langle\text{stochastic}\rangle\langle\text{local\_best\_term}\rangle$
$\langle\text{oper\_FA}\rangle::=\langle\text{base\_term}\rangle"+"\langle\text{metaphor\_based}\rangle\langle\text{FA\_neighbor\_term}\rangle"+"\langle\text{deterministic}\rangle\langle\text{random\_term}\rangle$
$\langle\text{oper\_BA\_exploration}\rangle::=\langle\text{base\_term}\rangle~"+"~\langle\text{stochastic}\rangle\langle\text{global\_best\_term}\rangle$
$\langle\text{oper\_BA\_exploitation}\rangle::=\langle\text{base\_term}\rangle~"+"~\langle\text{metaphor\_based}\rangle\langle\text{epsilon\_normal}\rangle$

Table 11: The comparison of the variation operators from the set

\mathbf{NIA}^{(\text{CS-2})}

Role	Algorit.	Metric	Min	Max	Mean	StDev	Median
Control	PSO	$w$	0.4954	0.5000	0.4999	0.0004	0.5000
		$c_{1}$	1.5000	1.5002	1.5000	$\mathbf{<0.0001}$	1.5000
		$c_{2}$	1.5000	1.5106	1.5001	0.0009	1.5000
		Fitness	$<-0.0001$	$<-0.0001$	$\mathbf{<-0.0001}$	$<0.0001$	$<-0.0001$
Controlled	accPSO	$\alpha$	0.1696	1.0000	0.9027	0.2185	1.0000
		$\beta$	0.1301	0.1936	0.1513	0.0109	0.1502

Table 12: The results of the Meta-DE by parameter setting analysis in CS-3.

Algorithm	Parameter	$Sim_{\cos}$	$Sim_{\text{SMAPE}}$	$\rho$	$Sim_{\text{kNN}}$	$Sim_{\text{SVM}}$	$Sim_{\text{RF}}$
accPSO	Min	0.9549	0.9997	0.9561	n/a	n/a	n/a
	Max	0.9924	0.9999	0.9924	n/a	n/a	n/a
	Mean	0.9862	0.9999	0.9863	0.0000	0.0000	0.0000
	StDev	0.0045	0.0000	0.0044	0.0000	0.0000	0.0000
	Median	0.9869	0.9999	0.9869	n/a	n/a	n/a
FA	Min	0.1541	0.9991	0.1499	n/a	n/a	n/a

Table 13: The results of the statistical analysis according to the similarity metrics for identifying the strong equivalence between the PSO as control algorithm in CS-3.

Panel of Figure 4 — Figure 4: Population descriptive metrics.

Panel of Figure 5 — Figure 5: Individual descriptive metrics.

Role	Algorit.	Metric	Min	Max	Mean	StDev	Median
Control	BA	$A$	1.0000	1.0000	1.0000	0.0000	1.0000
		$\gamma$	0.1000	0.1805	0.1148	0.0199	0.1041
		Fitness	-1.5632	-0.3465	-1.0069	0.2497	-1.0332
Controlled	accPSO	$\alpha$	1.0000	1.0000	1.0000	0.0000	1.0000
		$\beta$	0.3911	0.9629	0.7815	0.0569	0.7869
		Fitness	-0.2243	-0.0557	-0.1375	0.0341	-0.1412

Table 14: The results of the Meta-DE by parameter setting analysis in CS-4.

Algorithm	Parameter	$Sim_{\cos}$	$Sim_{\text{SMAPE}}$	$\rho$	$Sim_{\text{kNN}}$	$Sim_{\text{SVM}}$	$Sim_{\text{RF}}$
accPSO	Min	0.8605	0.9996	0.8607	n/a	n/a	n/a
	Max	0.9575	0.9998	0.9573	n/a	n/a	n/a
	Mean	0.9069	0.9997	0.9050	0.0000	0.0000	0.0000
	StDev	0.0142	0.0000	0.0147	0.0000	0.0000	0.0000
	Median	0.9064	0.9997	0.9042	n/a	n/a	n/a
PSO	Min	0.6173	0.9993	0.6108	n/a	n/a	n/a

Table 15: The results of the statistical analysis according to the similarity metrics for identifying the strong equivalence between the BA as the control algorithm in CS-4.

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