An innovative complex-valued encoding black-winged kite algorithm for global optimization

Abstract

The black-winged kite algorithm (BKA) constructed on the black-winged kites’ migratory and predatory instincts is a revolutionary swarm intelligence method that integrates the Leader tactic with the Cauchy variation procedure to retrieve the expansive appropriate convergence solution. The essential BKA exhibits marginalized resolution efficiency, inferior assessment precision, and stagnant sensitive anticipation. To foster aggregate discovery intensity and advance widespread computational efficacy, an innovative complex-valued encoding BKA (CBKA) is presented to resolve the global optimization. The complex-valued encoding manipulates the dual-diploid configuration to encode the black-winged kite, and the actual and fictitious portions are inserted into the BKA, which transforms dual-dimensional encoding into a single-dimensional manifestation. With the inherent parallelism and consistency, the actual and fictitious portions are renewed separately for each search agent, which reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources, and fosters collaboration efficiency. The CBKA not only showcases abundant flexibility and compatibility to accomplish supplementary advantages and sharpen resolution precision but also incorporates localized exploitation and universal exploration to forestall exaggerated convergence and cultivate desirable solutions. The function evaluations, engineering layouts, and adaptive infinite impulse response system identification are executed to certify the suitability and affordability of the CBKA. The experimental results manifest that the computational accomplishment and convergence productivity of the CBKA are superior to those of other comparison algorithms, the CBKA delivers noteworthy stabilization and resilience to explore superior assessment precision and swifter convergence efficiency.

Introduction

The optimized premium is to carryout the layout regulations and accomplish the appropriate largest or lowest amount within an extensive selection of pertinent restrictions through employing the most acceptable customized variables of the recognition framework. Numerous multi-model, multi-objective, large-scale, uncertain, and exacerbated analytical features are observed in real-world optimization scenarios. The conventional optimization procedures incorporate mixed-integer programming, dynamic programming, Newton’s approach, gradient collapse, quadratic programming, and conjugated gradient, which exhibit some weaknesses: (1) Excessive consumption of the mathematical structure. The goal-oriented solution could be divergent if the prerequisites of first- and second-order differentiability are not fulfilled. (2) Sensitive initial solution. A substantial inaccuracy in the initial location selection will culminate in ineffective precision of mathematical and numerical outcomes. (3) Ineffectual to tackle the multifaceted, combinational, and huge-scale framework. The conventional optimization procedures emphasize inadequate elimination efficiency, instructive resource garbage, sensitive anticipation stagnation, frequent uncharacteristic convergence, multidimensional amplification, and inferior assessment precision.

Motivated by unforeseen occurrences or sophisticated predatory conduct, metaheuristic algorithms (MAs) are indifferent to a selection of starting solutions and are not fixated on concaveness and convolution. MAs inherit the more sophisticated progressive information of the privileged individual to investigate the calculation region, which reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources and fosters collaboration efficiency, exhibits easy-to-implement structure, and displays courageous self-organization and intelligence. The MAs are arranged into four categories according to the numerous sources of motivation.

1. Swarm intelligence (SI)

   Intelligent organizations that manifest self-sustaining action are collectively referred to as SI, which is characterized as an intelligent decision-making behavior that emerges from collaboration between individuals and surroundings. Individuals within an affiliation adhere to a fundamental code action with no predominant centralized management between organizations. The collaborative intelligence of the entire population eventually originates from individual interaction, such as puma optimizer (PO)¹, elk herd optimizer (EHO)², spider wasp optimization (SWO)³, walrus optimizer (WO)⁴, coati optimization algorithm (COA)⁵, sand cat swarm optimization (SCSO)⁶, horned lizard optimization algorithm (HLOA)⁷, GOOSE algorithm (GOOSE)⁸, artificial gorilla troops optimizer (GTO)⁹, mountain gazelle optimizer (MGO)¹⁰, african vultures optimization algorithm (AVOA)¹¹, greater cane rat algorithm (GCRA)¹², secretary bird optimization algorithm (SBOA)¹³, arctic puffin optimization (APO)¹⁴, blood-sucking leech optimizer (BSLO)¹⁵, Eel and grouper optimizer (EGO)¹⁶, frilled lizard optimization (FLO)¹⁷, giant armadillo optimization (GAO)¹⁸. SI refines an inefficient solution or create an alternative one in approximating the most appropriate solution contingent on the searchable resolution information and the repeatable evolutionary approach. SI not only exhibits spectacular flexibility and adaptability to exploit the pursuit productivity and advance computational accurateness but also amalgamates investigation and extraction to enrich population pluralism and reinforce the global alternative solutions. SI exhibits attractive consistency and resilience to tackle the multifaceted, combinational and huge-scale difficulties.

2. Evolutionary algorithms (EAs)

   EAs emphasize self-sustaining, self-adapting, and self-studying attributes and are motivated by natural biological advancement. Reproduction, mutation, competitiveness, and selection all contribute to biological progression. The EAs cope with the optimization challenges through genetic alteration, recombination, and choice, such as liver cancer algorithm (LCA)¹⁹, coronavirus mask protection algorithm (CMPA)²⁰, gooseneck barnacle optimization (GBO)²¹, nutcracker optimization algorithm (NOA)²², altruistic population algorithm (APA)²³, anti-coronavirus optimization (ACVO)²⁴, water optimization algorithm (WOA)²⁵, poplar optimization algorithm (POA)²⁶, starling murmuration optimizer (SMO)²⁷, plant competition optimization (PCO)²⁸, remora optimization algorithm (ROA)²⁹, water flow optimizer (WFO)³⁰, differential evolution (DE)³¹, snow ablation optimizer (SAO)³², gradient-based optimizer (GBO)³³. EAs exhibit attractive stability and robustness to promote the effectiveness of accomplishing multifaceted issues. EAs utilize a variety of sophisticated heuristic procedures to broaden the resolution productivity and alleviate the computing charges, which harmonizes the localized extraction and universal exploration to restrict anticipation stagnation and bolster evaluation precision. EAs has strong reliability and feasibility to administer the massive amount of data. EAs are attempted to expedite the preparation and evaluation of the huge-scale datasets, which can capture valuable analytical information and bolster practicability and parallelism.

3. Physics/mathematics-based algorithms.

   Physics/mathematics-based algorithms are precipitated by the inevitable physical/mathematical theorems or occurrences, which typically symbolizes the paramount principles of physical/mathematical procedures in the partnerships between search individuals in the procedure actualization, such as Kepler optimization algorithm (KOA)³⁴, numeric crunch algorithm (NCA)³⁵, exponential distribution optimizer (EDO)³⁶, elastic deformation optimization algorithm (EDOA)³⁷, geometric octal zones distance estimation (GOZDE)³⁸, young’s double-slit experiment (YDSE)³⁹, arithmetic optimization algorithm (AOA)⁴⁰, integrated optimization algorithm (IOA)⁴¹, atomic orbital search (AOS)⁴², triangulation topology aggregation optimizer (TTAO)⁴³, newton–raphson-based optimizer (NRBO)⁴⁴, simulated annealing (SA)⁴⁵, gravitational search algorithm (GSA)⁴⁶, sinh cosh optimizer (SCHO)⁴⁷, Chernobyl disaster optimizer (CDO)⁴⁸. Physics/mathematics-based algorithms exhibit strong adaptability and stability to regulate the issue’s ambiguity. These algorithms integrate fuzziness-based logic, resilient optimization, uncontrollable probabilistic instruction, and reinforced training to promote universal discovery precision and upgrade design adaptability. They utilize theoretical mathematical concepts to employ algorithmic convergence, operational complexity, and problematic solvability.

4. Human-based algorithms

   Human-based algorithms, which incorporate algorithms motivated by both physical and non-physical human interactions like contemplating and social actions, are the ultimate type of MAs that have been explored, such as football team training algorithm (FTTA)⁴⁹, love evolution algorithm (LEA)⁵⁰, partial reinforcement optimizer (PRO)⁵¹, human memory optimization ⁵², musical chairs algorithm (MCA) ⁵³, tactical unit algorithm (TUA)⁵⁴, alpine skiing optimization (ASO)⁵⁵, search in forest optimizer (SIFO)⁵⁶, criminal search optimization algorithm (CSOA)⁵⁷, competitive search algorithm(CSA)⁵⁸, hunter–prey optimizer (HPO)⁵⁹, archerfish hunting optimizer (AHO)⁶⁰, hunger games search (HGS)⁶¹, human felicity algorithm (HFA)⁶², group learning algorithm (GLA)⁶³. Human-based algorithms furnish innovative solutions and competitive advantage in scientific and industrial contexts, incentivizing academics to allocate additional resources to synthesizing innovative methodologies. These algorithms receive formidable consistency and endurance from recognizing supplemental advantages and precocious convergence and prioritizing investigation and utilization to advance convergence frequency and bolster estimated precision.

Zhang et al. juxtaposed a black-winged kite algorithm based on logistic chaotic mapping with an osprey optimization algorithm to address the function evaluations and engineering layouts, and this algorithm exhibited large-scale discovery and small-scale extraction to foster aggregate discovery intensity and advance widespread computational efficacy⁶⁴. Ma et al. explored the black-winged kite algorithm with a good point set, nonlinear convergence factor, and adaptive t-distribution to address the robot parallel gripper design, this algorithm exhibited abundant adaptability and versatility to determine a more stable evaluation accuracy⁶⁵. Xue et al. integrated the black-winged kite algorithm with artificial rabbit optimization to address the function optimization, and this algorithm exhibited abundant sustainability and versatility to forestall exaggerated convergence and locate the appropriate solution⁶⁶. Zhou et al. integrated the black-winged kite algorithm with sine cosine guidelines to address function optimization. This algorithm exhibited delightful reliability and adaptability, enriching the detection information capacity and promoting global convergence performance⁶⁷. Rasooli et al. established the black-winged kite algorithm to address the clustering, and this algorithm exhibited suitability and affordability to foster aggregate discovery intensity, advance widespread computational efficacy, and retrieve the universal adequate solution⁶⁸.

Although the altered versions of the black-winged kite algorithm (BKA) demonstrate outstanding reliability and flexibility to promote assessment precision and collaboration efficiency, they remain insufficient in reconciling localized exploitation and universal exploration. The no-free-lunch (NFL) theorem speculates that no single search methodology could successfully address the entirety of optimization difficulties. The BKA is constructed on the black-winged kites’ migratory and predatory instincts that integrate the Leader tactic with the Cauchy variation procedure to retrieve the universal adequate solution⁶⁹. The basic BKA exhibits limitations, such as marginalized resolution efficiency, sensitive anticipation stagnation, sluggish collaboration speed, inferior assessment precision, and inadequate investigation and extraction. The innovative complex-valued encoding technique is integrated into the BKA to foster aggregate discovery intensity and advance widespread computational effectiveness, which incorporates a dual-diploid configuration to encode every black-winged kite and translates dual-dimensional encoding to single-dimension manifestation. The main contributions are summarized below: (1) The complex-valued encoding black-winged kite algorithm (CBKA) is established to address global optimization. (2) The complex-valued encoding reinforces population pluralism, restricts discovery stagnation, extends the identification zone, promotes estimation excellence, advances information resources, fosters collaboration efficiency, and exhibits remarkable parallelism and consistency. (3) The CBKA is compared with various comparison approaches that contain the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, BSLO, EGO, FLO, GOOSE, YDSE, SCHO, SWO, GAO and BKA. The CBKA is tested against the function evaluations, engineering layouts, and infinite impulse response (IIR) system identification. (4) The CBKA not only emphasizes formidable flexibility and sustainability to renew the actual and fictitious portions of each black-winged kite but also reconciles localized exploitation and universal exploration to forestall exaggerated convergence and locate the appropriate solution. In addition, the CBKA showcases abundant adaptability and versatility to reap supplementary advantages and explore superior assessment precision and swifter convergence efficiency.

The article is partitioned into the following components. Section “Black-winged kite algorithm (BKA)” reveals the BKA. Section “Complex-valued encoding black-winged kite algorithm (CBKA)” articulates the CBKA. Section “Simulation evaluation and result interpretation for benchmark functions” portrays comparative experiments and result analysis. Section “CBKA for adaptive infinite impulse response system identification” portrays the comparative experiments and result analysis. Section “CBKA for classical engineering design” portrays the comparative experiments and result analysis. Section “Impact analysis” showcases the impact analysis of the CBKA. Section “Conclusion and future exploration” summarizes the conclusion of future research.

Black-winged kite algorithm (BKA)

The BKA integrates the Leader tactic with the Cauchy variation procedure to foster aggregate discovery intensity and retrieve the appropriate computational solution, which not only captures the black-winged kites’ migratory and predatory instincts but also meticulously mimics the fabulous adaptability to alter the surrounding circumstances and target locations.

Initialization population

The population is haphazardly initialized, and a matrix is stipulated as:

1

where showcases the population magnitude, showcases the problematic dimension, showcases jth dimension of ith individual. The is stipulated as:

2

where showcases an integer in , and showcase the lower and upper limitations, and .

The most appropriate leader is stipulated as:

3

4

Assaulting behavior

Black-winged kites are predators of tiny pastureland creatures and parasites; they swiftly descend and strike after silently monitoring their prey and regulating their wings and tail angles relative to motion velocity. Figure 1(a) articulates a black-winged kite safeguarding equilibrium and hovering. Figure 1(b) articulates a black-winged kite wafting towards the prey at breakneck speed. Figure 2(a) articulates a black-winged kite remaining hovering and foreseeing an assault. Figure 2(b) articulates a black-winged kite remaining hovering and foraging for prey.

Fig. 1.

(a) Safeguarding equilibrium and hovering, (b) Wafting towards the prey at breakneck speed.

Fig. 2.

(a) Remaining hovering and foreseeing an assault, (b) Remaining hovering and foraging for prey.

The assaulting behavior is stipulated as:

5

6

where and showcase the positions of ith black-winged kite in jth dimension, , , showcases the iteration termination.

Migration behavior

The sophisticated behavior of bird migration is restricted by multiple environmental conditions incorporating food accessibility and humidity. Migration is to acclimatize to seasonal fluctuations, and the numerous black-winged kites migrate form the northwest to the southeast during winter to pursue superior living circumstances and materialistic resources. If the anticipated fitness of the particular species is lower than that of the haphazard species, the leader will allocate to step down and revert to the migrating species. Conversely, the leader will instruct the search population until it accomplishes the detection destination. This procedure continuously designates exemplary leaders to guarantee a productive migration. Figure 3 articulates the strategic alterations of the leading black-winged kite.

Fig. 3.

The strategic alterations of the leading black-winged kite.

The migration behavior is stipulated as:

7

8

where showcases the leading scorer, showcases the current position, showcases the accidental position, showcases the Cauchy mutation.

The single-dimensional Cauchy is a continuous stochastic distribution with two metrics, which is stipulated as:

9

where , , the probability density fitness is stipulated as:

10

Algorithm 1.

Algorithm 1 yields the pseudocode of BKA.

Complex-valued encoding black-winged kite algorithm (CBKA)

The exclusive chromosomes of natural biological tissues are constructed of double-stranded or multi-stranded structures, and the CBKA characterizes a pair of genotypes to enrich the complexity and diversity of the biological data. The actual and fictitious portions are correlated with the actual and fictitious genes⁷⁰. The is stipulated as:

11

where and showcase the actual and fictitious portions.

Table 1 portrays the chromosome structure of CBKA.

Table 1.

Chromosome structure of CBKA.

Individual
actual portion
fictitious portion
chromosome structure

Initialization CBKA population

The definition interval is , and modules and arguments are randomly generated.

12

13

The complex values are stipulated as:

14

Updating positions of CBKA

Attacking behavior

(1) Refresh the actual portion:

15

(2) Refresh the fictitious portion:

16

Migration behavior

(1) Refresh the actual portion:

17

(2) Refresh the fictitious portion:

18

Calculating the fitness value

The CBKA manipulates actual and fictitious portions to encode the black-winged kite, the encoding conversion region is transformed into actual and fictitious solutions, and the fitness value is stipulated as:

19

20

where showcases the altered actual argument.

The solution procedure of CBKA

The CBKA emphasizes formidable flexibility and sustainability to forestall exaggerated convergence and locate the appropriate solution, which reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources, fosters collaboration efficiency, and exhibits remarkable parallelism and consistency. Algorithm 2 yields the pseudocode of CBKA. Figure 4 articulates the flowchart of CBKA.

Fig. 4.

Convergence curves of the CBKA and compared algorithms for resolving the benchmark functions.

Algorithm 2.

CBKA

Computational complexity

Computational complexity is adopted to measure the time and space resources consumed by a procedure when remedying huge-scale troublesome challenges. This subsection will investigate the computational complexity of the CBKA to emphasize sustainability and productivity.

Time complexity: the CBKA embodies three essential actions: initialization, estimating fitness, and refreshing the black-winged kite’s location. In CBKA, showcases population magnitude, showcases iteration termination, and showcases problematic dimension. (1) The complexity is related to the initialization approach and issue size. Initialization symbolizes establishing prospective solutions, embarking on parameters, and initializing other necessitate procedures to advance the discovery potential and optimization productivity and time complexity of the initialization equals . (2) Estimating fitness is attempted to validate the practicality and quality of potential solutions, which necessitates sophisticated computation and experimental validation, the time complexity of estimating fitness equals . (3) Refreshing black-winged kite’s location executes a neighborhood discovery predation and complex-valued encoding strategy to revise the black-winged kites’ locations and furnish alternative solutions, the time complexity of the refreshing black-winged kite’s location equals . Therefore, the total time complexity of the CBKA is equal .

Space complexity: the CBKA not only showcases abundant flexibility and compatibility to accomplish supplementary advantages and sharpen resolution precision but also incorporates localized exploitation and universal exploration to forestall exaggerated convergence and cultivate desirable solutions. Space complexity is the supplemental data storage area, the stored alternative solutions, associated intermediate outcomes, auxiliary ephemeral variables, and inevitable discovery- and extraction-related data layouts contributing to CBKA’s space complexity utilization. In CBKA, showcases population magnitude, showcases iteration termination, and showcases problematic dimension. Therefore, the space complexity of the CBKA is equal .

Simulation evaluation and result interpretation for benchmark functions

Experimental setup

Numerical experiments are conducted on a Windows 10 machine with an Intel Core i7-8750H 2.2 GHz CPU, a GTX1060, and 8 GB memory.

Benchmark functions

Benchmark functions involve three variations: unimodal functions , multimodal functions , and fixed-dimension multimodal functions . The unimodal functions do not exhibit local optimal, and the purpose is to furnish an associated benchmark for monitoring the exploitation and search localization of metaheuristic algorithms. These functions have a particular guiding significance in quantifying the advantages and drawbacks of the algorithms and maintaining the attention concentrated on establishing the expansive optimal solution without being disturbed by false peaks. The multimodal functions preserve many locally optimum solutions, and the purpose is to furnish a fantastic foundation for investigating the entirety of the exploration and worldwide discovery of metaheuristic algorithms. The fixed-dimension multimodal functions endure fewer local optimal solutions as compared to the multimodal functions, and the purpose is to furnish an appropriate metric for quantifying the optimization efficiency of metaheuristic algorithms in harmonizing large-scale exploration and small-scale exploitation. Table 2 portrays the benchmark functions.

Table 2.

Benchmark functions.

Benchmark functions	Dim	Range	fmin
	30	[− 100,100]	0
	30	[− 10,10]	0
	30	[− 100,100]	0
	30	[− 100,100]	0
	30	[− 30,30]	0
	30	[− 100,100]	0
	30	[− 1.28,1.28]	0
	30	[− 5.12,5.12]	0
	30	[− 32,32]	0
	30	[− 600,600]	0
	30	[− 50,50]	0
	30	[− 50,50]	0
	2	[− 65,65]	0.998
	4	[− 5,5]	0.000307
	2	[− 5.12,5.12]	− 1
	2	[− 2,2]	3
	6	[0,1]	− 3.32
	4	[0,10]	− 10.1532
	4	[0,10]	− 10.4029
	4	[0,10]	− 10.5364
	2		− 1
	2	[− 100,100]	− 1
	10	[− 10,10]	0

Parameter settings

The CBKA is contrasted with GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA to emphasize practicality and accessibility. Certain representative empirical variables that are extracted from the source manuscripts serve as the control parameters. The portrayed regulation variables of each approach are stipulated as:

GTO: precarious value , precarious values , precarious value , fixed value , fixed value .

MGO: precarious values , precarious values .

PO: fixed value , fixed value , fixed value , fixed value , fixed value , fixed value .

AVOA: fixed value , fixed value , fixed value , fixed value , fixed value , fixed value .

GCRA: precarious value , fixed value , precarious value .

HLOA: hue angle , fixed value , precarious value , precarious value .

WO: precarious value , precarious values , , precarious values , precarious value , precarious value , standard deviation , fixed value , precarious value .

SBOA: precarious value , precarious value , fixed value , fixed value , precarious value , precarious value , fixed value .

NRBO: fixed factor , precarious value , precarious value , precarious value , precarious value .

APO: fixed value , fixed value .

EHO: precarious value , precarious value , precarious value .

BKA: precarious value , fixed value , precarious value , Cauchy mutation , fixed value , fixed value .

CBKA: precarious value , fixed value , precarious value , Cauchy mutation , fixed value , fixed value .

Simulation evaluation and result interpretation

For approaches, , and . Best, Worst, Mean and Std showcase the optimal score, worst score, mean score, and standard deviation.

Table 3 portrays the comparative solutions of the benchmark functions. Twelve metaheuristic algorithms are used as comparison methods to resolve the function evaluations and ensure the dependability and superiority of the CBKA. The optimal score (Best), worst score (Worst), mean score (Mean), and standard deviation (Std) are regarded as the most comprehensive and standardized assessment indications to track the stability and robustness of each method. The optimal score is inextricably linked to the fitness score of multi-model, multi-objective, large-scale, and uncertain issues, which is to recognize the lowest or highest score in the entire fixed search area. The global optimal score represents the most effective search agent of all candidate solutions, which infinitely overlaps the position close to the prey during the foraging and capturing operations of the entire detection population. The optimal score highlights the detection efficiency and exploitation accuracy. The worst score means that the worst candidate solution is obtained in the independent operation results of a certain algorithm. The gap between the worst and the mean scores highlights that the comparison algorithm produces a slower discovery efficiency and poorer execution accuracy to fall into the local optimality and premature convergence, which can indirectly highlight the stability and feasibility. The mean score is the arithmetic average value obtained by calculating and evaluating the population size, maximum iteration, and independent operation. The mean score clearly highlights stability, robustness, overall search efficiency, and global detection accuracy. The standard deviation is a statistic that measures the degree of deviation of various possible results form the expectation in the probability distribution, which reflects the fluctuation of the value in the data set relative to the mean score. The smaller the standard deviation, the smaller the data dispersion and more stable the data changes. The larger the standard deviation, the greater the data dispersion and the more unstable the data changes. For unimodal functions , the optimal scores, worst scores, mean scores and standard deviations of CBKA, GTO, AVOA, and GCRA remain at the same magnitude and consistent for functions , , and . The evaluation accuracy of the CBKA is superior to those of the MGO, PO, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA can reasonably deploy large-scale discovery and small-scale extraction to foster aggregate discovery intensity and advance widespread computational efficacy. The CBKA demonstrates strong stability and reliability. The evaluation accuracy of the CBKA has been augmented astronomically in juxtaposition to BKA. The CBKA manipulates a dual-diploid organization to encode the black-winged kite, reinforce population pluralism, restrict discovery stagnation, extend identification area, promote estimation excellence, advance information resources, and foster collaboration efficiency. For , the evaluation accuracy of the CBKA has been slightly enhanced, the CBKA has the weakest difference between the worst score and the mean score. The CBKA has the strength and reliability to achieve the exact optimal scores. The CBKA has the smallest standard deviation, showcasing abundant adaptability and versatility to determine a more stable evaluation accuracy. For and , the calculation magnitude and evaluation accuracy of the CBKA are superior to those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA utilizes the dual-diploid organization of the complex-valued encoding to transform the black-winged kite into an individual with actual and fictitious portions, expand the detection area, and avoid local optimality. For multimodal functions , the optimal scores, worst scores, mean scores ues, and standard deviations of the CBKA, GTO, MGO, PO, AVOA, GCRA, HLOA, WO, NRBO, and BKA remain at the same magnitude and consistent for . The evaluation accuracy of the CBKA is superior to those of the SBOA, APO, and EHO. The CBKA showcases abundant sustainability and versatility to forestall exaggerated convergence and locate the appropriate solution. For , the calculation magnitude and evaluation accuracy of the CBKA, GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, and BKA are the same and consistent, the comparative solutions of the CBKA outperform those of the APO and EHO. The CBKA exhibits delightful reliability and adaptability to enrich the detection information capacity and promote global convergence performance. For , the optimal scores, worst scores, mean scores, and standard deviations of the CBKA, GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, and BKA are the global exact solutions, the evaluation accuracy of the CBKA is superior to those of the APO and EHO. The CBKA exhibits suitability and affordability to foster aggregate discovery intensity, advance widespread computational efficacy, and retrieve an adequate universal solution. For and , the evaluation accuracy of the CBKA has been slightly enhanced compared to the BKA, and the CBKA has the weakest difference between the optimal, worst, and mean scores. The CBKA showcases the smallest standard deviation, and the CBKA receives exceptional consistency and endurance to promote exploration efficiency and disrupt anticipation stagnation. For fixed-dimension multimodal , the optimal, worst, and mean scores of the CBKA, MGO, and APO remain at the same magnitude, and the global exact solutions for and . The evaluation accuracy of the CBKA is superior to those of the GTO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, EHO, and BKA. The CBKA exhibits a relatively smaller standard deviation and maintains instructive supremacy and stabilization to recognize universal solutions. For and , the calculation magnitude and evaluation accuracy of the CBKA, GTO, MGO, HLOA, WO, SBOA, NRBO, APO, and BKA are the same and consistent, the comparative solutions of the CBKA outperform those of the PO, AVOA, GCRA and EHO. The CBKA exhibits suitability and affordability to explore a superior assessment precision and a swifter convergence rate. The variation in computational magnitude and evaluation accuracy between these algorithms is subtle. The CBKA has smaller standard deviations, indicating that the CBKA deploys large-scale exploration and small-scale exploitation to enhance overall search efficiency and improve optimization stability. For , the evaluation accuracy of the CBKA is superior to those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA, the CBKA has strong reliability and robustness to locate the exact assessment precision. For , and , the optimal scores of the GTO, MGO, PO, AVOA, HLOA, WO, SBOA, NRBO, APO, EHO, BKA, and CBKA remain at the same magnitude and consistent, but the worst scores, mean scores, and standard deviations of the CBKA outperform those of the comparison procedures. The CBKA reconciles localized exploitation and universal exploration to forestall exaggerated convergence and locate the appropriate solution. For and , all procedures utilize their own global search properties and location update strategies to obtain the exact solutions, and the CBKA showcases abundant adaptability and versatility to reap supplementary advantages and explore superior assessment precision. For , the computational magnitude and evaluation accuracy of the CBKA are superior to those of the MGO, PO, GCRA, HLOA, WO, SBOA, APO, EHO, and BKA, the CBKA maintains excellent equilibrium and endurance to enhance the exploration efficiency and locate the advance widespread computational efficacy.

Table 3.

Comparative solutions of benchmark functions.

Function	Result	GTO	MGO	PO	AVOA	GCRA	HLOA	WO	SBOA	NRBO	APO	EHO	BKA	CBKA
	Best	0	1.5E-180	0	0	0	0	0	0	0	3.45E-10	2.66E-22	3.3E-219	0
	Worst	0	4.1E-165	0	0	0	0	0	0	0	1.55E-08	1.90E-18	1.5E-166	0
	Mean	0	2.3E-166	0	0	0	0	0	0	0	2.30E-09	1.57E-19	4.9E-168	0
	Std	0	0	0	0	0	0	0	0	0	2.96E-09	4.56E-19	0	0
	Best	0	3.3E-102	5.9E-277	0	0	4.6E-295	4.0E-197	8.1E-189	0	7.31E-06	2.49E-14	2.7E-111	0
	Worst	0	2.60E-95	3.4E-267	0	0	3.3E-250	1.9E-161	2.7E-164	2.4E-305	5.89E-05	5.16E-11	8.90E-99	0
	Mean	0	1.70E-96	2.4E-268	0	0	1.6E-251	6.4E-163	9.2E-166	1.2E-306	2.30E-05	3.39E-12	5.1E-100	0
	Std	0	5.51E-96	0	0	0	0	3.1E-162	0	0	1.25E-05	9.60E-12	2.00E-99	0
	Best	0	7.46E-33	0	0	0	0	0	1.2E-241	0	7.70E-07	2.626421	1.8E-218	0
	Worst	0	3.64E-20	0	0	0	0	0	6.0E-206	0	3.09E-05	40.65992	1.2E-183	0
	Mean	0	1.40E-21	0	0	0	0	0	2.0E-207	0	7.94E-06	12.04016	3.9E-185	0
	Std	0	6.64E-21	0	0	0	0	0	0	0	8.22E-06	8.628604	0	0
	Best	0	3.72E-64	5.9E-276	0	0	1.3E-271	7.3E-191	3.0E-148	4.7E-308	0.010068	3.896559	3.0E-108	0
	Worst	0	1.03E-52	1.2E-268	0	0	1.0E-246	1.0E-156	1.1E-130	3.1E-299	0.046604	22.69124	5.92E-82	0
	Mean	0	3.44E-54	4.5E-270	0	0	4.0E-248	3.3E-158	3.6E-132	1.2E-300	0.024490	9.496622	1.97E-83	0
	Std	0	1.87E-53	0	0	0	0	1.8E-157	2.0E-131	0	0.009466	3.663021	1.08E-82	0
	Best	6.47E-09	0	3.91E-05	2.27E-07	1.22E-09	0.005377	2.97E-05	22.81122	26.35627	0.000520	2.678482	24.79555	0
	Worst	1.92E-05	2.39E-29	25.40717	1.42E-05	2.02E-05	28.70675	0.237591	23.57569	28.81243	26.05845	141.7184	28.93170	3.10E-29
	Mean	4.47E-06	1.46E-30	23.05609	2.85E-06	4.16E-06	25.80643	0.022344	23.26998	27.70786	11.88652	39.99899	26.35369	1.32E-30
	Std	5.45E-06	5.28E-30	6.282092	2.77E-06	5.67E-06	8.744028	0.050907	0.213851	0.799264	12.92475	33.96517	1.152647	5.71E-30
	Best	7.72E-18	2.25E-29	6.56E-11	1.73E-09	3.10E-11	8.00E-06	8.19E-08	1.04E-15	1.619463	9.97E-09	6.22E-22	3.38E-05	1.34E-26
	Worst	2.31E-14	2.54E-18	1.44E-08	2.02E-08	5.79E-07	0.000246	0.000417	5.08E-13	2.754444	2.68E-07	3.24E-19	6.036753	1.33E-21
	Mean	3.44E-15	8.47E-20	2.07E-09	5.27E-09	7.40E-08	7.14E-05	6.12E-05	6.31E-14	2.208682	6.08E-08	4.86E-20	0.799383	1.04E-22
	Std	6.07E-15	4.64E-19	2.89E-09	3.73E-09	1.37E-07	5.46E-05	8.94E-05	1.13E-13	0.295708	5.47E-08	8.78E-20	1.740813	2.98E-22
	Best	9.03E-07	1.47E-05	8.64E-06	2.70E-06	8.52E-07	1.14E-05	5.72E-06	5.97E-06	1.37E-06	0.007746	0.018150	1.78E-06	2.75E-07
	Worst	0.000193	0.000662	0.000229	0.000343	0.000134	0.000314	0.000469	0.000358	0.000248	0.024298	0.115852	0.000334	4.47E-05
	Mean	3.99E-05	0.000141	6.41E-05	5.45E-05	4.85E-05	9.66E-05	0.000136	0.000153	6.94E-05	0.013823	0.058306	7.34E-05	1.61E-05
	Std	4.16E-05	0.000133	5.10E-05	6.92E-05	3.52E-05	7.46E-05	0.000122	8.56E-05	6.60E-05	0.004052	0.026607	6.94E-05	1.15E-05
	Best	0	0	0	0	0	0	0	0	0	20.08312	15.91934	0	0
	Worst	0	0	0	0	0	0	0	6.005041	0	155.9933	58.70249	0	0
	Mean	0	0	0	0	0	0	0	0.200168	0	73.71032	30.47889	0	0
	Std	0	0	0	0	0	0	0	1.096365	0	35.97173	10.13477	0	0
	Best	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	6.18E-06	1.86E-11	8.88E-16	8.88E-16
	Worst	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	2.45E-05	4.736201	8.88E-16	8.88E-16
	Mean	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	8.88E-16	1.13E-05	1.876198	8.88E-16	8.88E-16
	Std	0	0	0	0	0	0	0	0	0	4.61E-06	1.013469	0	0
	Best	0	0	0	0	0	0	0	0	0	7.57E-10	0	0	0
	Worst	0	0	0	0	0	0	0	0	0	0.012321	0.127386	0	0
	Mean	0	0	0	0	0	0	0	0	0	0.000411	0.020849	0	0
	Std	0	0	0	0	0	0	0	0	0	0.002250	0.028999	0	0
	Best	6.03E-18	1.57E-32	5.76E-12	8.47E-11	1.75E-12	2.44E-07	5.87E-10	8.66E-18	0.094215	1.90E-10	2.01E-23	1.96E-06	1.57E-32
	Worst	6.38E-14	1.57E-32	1.41E-09	8.50E-10	3.68E-08	0.103671	5.96E-06	8.99E-09	0.344426	3.82E-09	3.031747	0.460621	5.02E-12
	Mean	6.07E-15	1.57E-32	1.43E-10	3.01E-10	3.06E-09	0.003458	3.56E-07	4.26E-10	0.197062	1.46E-09	0.474842	0.033007	1.67E-13
	Std	1.50E-14	5.57E-48	2.61E-10	1.93E-10	7.11E-09	0.018927	1.08E-06	1.76E-09	0.066425	1.04E-09	0.850770	0.091725	9.17E-13
	Best	5.58E-18	1.35E-32	3.17E-11	2.78E-10	4.82E-12	5.49E-06	2.44E-08	4.28E-15	1.308839	3.87E-09	1.05E-20	0.742207	1.35E-32
	Worst	2.49E-09	1.35E-32	6.18E-09	1.41E-08	3.57E-07	1.874620	1.79E-05	0.197740	2.882751	0.010987	3.597465	2.451350	6.66E-22
	Mean	8.31E-11	1.35E-32	1.23E-09	2.11E-09	2.60E-08	0.073584	3.76E-06	0.019991	1.954286	0.000733	0.495947	1.346790	3.45E-23
	Std	4.55E-10	5.57E-48	1.42E-09	2.53E-09	6.57E-08	0.341755	4.34E-06	0.047508	0.431302	0.002788	1.003017	0.434012	1.37E-22
	Best	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004
	Worst	0.998004	0.998004	0.998004	1.992031	0.998004	12.67051	0.998004	0.998004	12.67051	0.998004	3.968250	0.998004	0.998004
	Mean	0.998004	0.998004	0.998004	1.064272	0.998004	6.186815	0.998004	0.998004	1.651634	0.998004	1.361954	0.998004	0.998004
	Std	0	1.93E-16	0	0.252193	2.55E-12	4.212248	2.90E-16	0	2.190709	0	0.712300	5.83E-17	0
	Best	0.000307	0.000307	0.000307	0.000307	0.000419	0.000307	0.000307	0.000307	0.000307	0.000307	0.000307	0.000307	0.000307
	Worst	0.001223	0.000307	0.001223	0.000342	0.001674	0.020363	0.000737	0.020363	0.020363	0.000307	0.001212	0.020363	0.000307
	Mean	0.000399	0.000307	0.000399	0.000309	0.001611	0.003919	0.000331	0.001736	0.003711	0.000307	0.000841	0.001962	0.000307
	Std	0.000279	5.37E-18	0.000279	6.21E-06	0.000254	0.007497	7.96E-05	0.005071	0.007578	1.95E-19	0.000307	0.005023	2.02E-19
	Best	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	Worst	-1	-1	-0.93625	-1	-1	-1	-1	-1	-1	-1	-0.93625	-1	-1
	Mean	-1	-1	-0.99787	-1	-1	-1	-1	-1	-1	-1	-0.99362	-1	-1
	Std	0	0	0.011640	0	1.26E-10	0	0	0	0	0	0.019453	0	0
	Best	3	3	3	3	3.033880	3	3	3	3	3	3	3	3
	Worst	3	3	3	3.000004	32.68463	3	3	3	3	3	3	3	3
	Mean	3	3	3	3.000001	17.26016	3	3	3	3	3	3	3	3
	Std	1.30E-15	1.35E-15	1.38E-15	8.57E-07	11.21137	1.07E-14	6.83E-15	1.25E-15	2.10E-15	1.81E-15	1.56E-15	1.26E-15	1.66E-15
	Best	-3.32200	-3.32200	-3.32200	-3.32200	-2.76541	-3.32200	-3.32200	-3.32200	-3.32200	-3.32200	-3.32200	-3.32200	-3.32200
	Worst	-3.20310	-3.20310	-3.20310	-3.20310	-1.16984	-3.08668	-3.03515	-3.20310	-3.09417	-3.20310	-3.20310	-3.12916	-3.32200
	Mean	-3.27840	-3.25066	-3.26255	-3.25462	-2.03807	-3.26759	-3.24902	-3.27444	-3.24090	-3.31407	-3.26651	-3.29575	-3.32200
	Std	0.058273	0.059241	0.060463	0.059923	0.463088	0.067481	0.071609	0.059241	0.077155	0.030164	0.060328	0.054838	4.73E-15
	Best	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532	-10.1532
	Worst	-10.1532	-10.1532	-2.63047	-10.1532	-10.1487	-2.63047	-10.1532	-10.1532	-9.71179	-10.1532	-2.63047	-10.1532	-10.1532
	Mean	-10.1532	-10.1532	-9.90244	-10.1532	-10.1526	-9.14953	-10.1532	-10.1532	-10.1383	-10.1532	-7.57178	-10.1532	-10.1532
	Std	6.85E-15	6.08E-15	1.373456	4.06E-14	0.001099	2.600697	7.44E-13	6.62E-15	0.080549	7.23E-15	3.498603	5.54E-15	6.68E-15
	Best	-10.4029	-10.4029	-10.4029	-10.4029	-10.4028	-10.4029	-10.4029	-10.4029	-10.4029	-10.4029	-10.4029	-10.4029	-10.4029
	Worst	-10.4029	-10.4029	-3.72430	-10.4029	-10.4008	-1.83759	-10.4029	-10.4029	-8.51057	-10.4029	-10.4029	-10.4029	-10.4029
	Mean	-10.4029	-10.4029	-10.1803	-10.4029	-10.4025	-8.52768	-10.4029	-10.4029	-10.3301	-10.4029	-10.4029	-10.4029	-10.4029
	Std	8.73E-16	8.08E-16	1.219347	2.82E-14	0.000503	3.463578	3.57E-12	1.23E-15	0.346642	1.55E-15	1.28E-15	1.48E-15	8.08E-16
	Best	-10.5364	-10.5364	-10.5364	-10.5364	-10.5363	-10.5364	-10.5364	-10.5364	-10.5364	-10.5364	-10.5364	-10.5364	-10.5364
	Worst	-10.5364	-10.5364	-2.80663	-10.5364	-10.5355	-1.67655	-10.5364	-10.5364	-3.35462	-10.5364	-2.42173	-3.83543	-10.5364
	Mean	-10.5364	-10.5364	-9.83202	-10.5364	-10.5361	-7.52129	-10.5364	-10.5364	-10.0108	-10.5364	-9.77207	-10.3130	-10.5364
	Std	2.36E-15	2.42E-15	2.154953	3.91E-14	0.000283	4.045844	1.86E-11	1.98E-15	1.604531	1.78E-15	2.342062	1.223427	1.23E-15
	Best	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	Worst	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	Mean	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	Std	0	0	0	0	2.46E-07	0	0	0	0	0	0	0	0
	Best	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	Worst	-1	-1	-0.99028	-1	-1	-1	-1	-1	-1	-1	-0.99028	-1	-1
	Mean	-1	-1	-0.99968	-1	-1	-1	-1	-1	-1	-1	-0.99223	-1	-1
	Std	0	0	0.001774	0	1.32E-10	0	0	0	0	0	0.003953	0	0
	Best	0	0	1.2E-273	0	8.24E-14	1.1E-274	2.0E-211	1.7E-239	0	1.03E-16	9.8E-248	8.2E-114	0
	Worst	0	3.47E-18	1.9E-260	0	1.83E-05	3.5E-256	2.0E-172	2.29E-25	0	7.02E-12	6.38E-15	6.62E-87	0
	Mean	0	1.16E-19	6.2E-262	0	1.09E-06	2.3E-257	6.8E-174	7.63E-27	0	6.58E-13	3.90E-16	2.21E-88	0
	Std	0	6.33E-19	0	0	3.63E-06	0	0	4.18E-26	0	1.47E-12	1.16E-15	1.21E-87	0

To enhance clarity and relevance, we have reduced the number of iterations and re-plotted the graps accordingly for the benchmarks. Figure 4 portrays the convergence curves of the CBKA and compares algorithms for resolving the benchmark functions. The convergence curves show the compared algorithms’ assessment precision and faster convergence efficiency. A higher convergence efficiency means that the compared algorithm exhibits high optimization efficiency and high convergence performance to avoid search stagnation. A lower calculation accuracy means that the compared algorithm has excellent detection breadth and exploitation efficiency for determining the global optimal feasible solution. For unimodal functions , the convergence productivity and assessment precision of the CBKA outperforms those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA exhibits delightful reliability and adaptability to enrich the detection information capacity and promote global convergence performance. The CBKA features strong stability and parallelism to avoid search stagnation and obtain higher convergence accuracy. For multimodal functions , the optimal scores, worst scores, mean scores ues, and standard deviations of the CBKA are superior to those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA exhibits remarkable parallelism and consistency to restrict discovery stagnation, extend identification area, and foster collaboration efficiency in terms of convergence productivity and assessment precision. For fixed-dimension multimodal , the CBKA exhibits suitability and affordability to explore a superior assessment precision and swifter faster convergence rate, the variation in computational magnitude and evaluation accuracy. Compared with GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA, the CBKA features instructive superiority and reliability to cultivate better convergence productivity and superior assessment precision. The CBKA showcases abundant flexibility and versatility to sharpen resolution precision and incorporates localized exploitation and universal exploration to cultivate desirable solutions.

Figure 5 portrays the boxplots of the CBKA and compares algorithms for resolving the benchmark functions. The standard deviation is a statistic that measures the degree of deviation of various possible results form the expectation in the probability distribution, which reflects the fluctuation of the value in the data set relative to the mean score. The standard deviation is inextricably linked to the dispersion of sample data, which exhibits stability and robustness. A lower standard deviation means that the compared algorithm has strong effectiveness and feasibility in advancing information resources and balances exploration and exploitation to locate the appropriate solution. For unimodal functions , the CBKA utilizes the dual-diploid organization of the complex-valued encoding to transform the black-winged kite into an individual with actual and fictitious portions, expand the detection area, avoid local optimality, and enhance calculation magnitude and evaluation accuracy. The standard deviations and stability of the CBKA are superior to those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA features fantastic durability and adaptability to advance inherent parallelism and locate the appropriate solution. For multimodal functions , the CBKA showcases abundant sustainability and versatility to forestall exaggerated convergence and discover the proper solution. Compared with GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA, the CBKA features more minor standard deviations and more substantial stability. The CBKA attributes admirable consistency and endurance to promote inherent parallelism and enhance local exploitation capability. For fixed-dimension multimodal , the standard deviations and stability of the CBKA are superior to those of the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, and BKA. The CBKA emphasizes formidable flexibility and sustainability to renew each black-winged kite’s actual and fictitious portions to advance widespread computational efficacy and forestall exaggerated convergence. The CBKA recognizes the supplementary advantages of BKA and the actual and fictitious portions to mitigate the marginalized resolution efficiency, inferior assessment precision, and sensitive anticipation stagnation, which reconciles localized exploitation and universal exploration to forestall exaggerated convergence and locate the appropriate solution.

Fig. 5.

Boxplots of the CBKA and compared algorithms for resolving the benchmark functions.

Wilcoxon rank-sum test is executed to ascertain if there is an instructive distinction between CBKA and other procedures⁷¹. is an instructive distinction, is no instructive distinction, and N/A is “not applicable”. Table 4 portrays the comparative solutions of the Wilcoxon rank-sum test.

Table 4.

Comparative solutions of Wilcoxon rank-sum test.

Function	GTO	MGO	PO	AVOA	GCRA	HLOA	WO	SBOA	NRBO	APO	EHO	BKA
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12
	4.11E-12	9.80E-03	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12	4.11E-12
	3.02E-11	2.46E-02	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	5.49E-11	3.02E-11
	1.50E-02	7.38E-10	5.09E-06	3.37E-04	1.39E-06	6.01E-08	5.46E-09	9.76E-10	2.77E-05	3.02E-11	3.02E-11	8.84E-07
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	3.02E-11
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	3.02E-11
	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	5.76E-11	1.10E-02
	1.45E-10	1.10E-02	6.48E-12	6.48E-12	8.92E-12	6.48E-12	6.48E-12	1.19E-10	6.48E-12	6.48E-12	7.14E-11	6.48E-12
	3.16E-12	8.15E-03	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12	3.16E-12
	N/A	5.54E-03	N/A	1.44E-09	1.21E-12	5.76E-11	9.71E-12	3.34E-11	3.14E-07	3.69E-11	2.78E-03	1.61E-02
	2.55E-02	2.69E-11	9.47E-07	2.69E-11	2.69E-11	2.96E-11	2.69E-11	2.69E-11	1.91E-07	3.02E-11	4.91E-11	2.81E-08
	3.07E-07	1.45E-11	1.71E-04	1.45E-11	4.25E-02	1.45E-11	1.13E-03	N/A	N/A	2.14E-02	8.14E-04	N/A
	2.44E-04	3.65E-05	1.39E-05	2.25E-11	2.25E-11	7.29E-11	2.21E-11	7.19E-03	3.80E-08	2.69E-11	7.15E-09	6.94E-05
	2.41E-03	8.58E-03	2.78E-03	1.96E-11	1.59E-11	1.59E-11	1.59E-11	2.99E-03	1.59E-11	6.07E-03	8.81E-05	1.59E-11
	3.99E-03	1.07E-02	7.96E-03	1.09E-11	1.14E-11	1.13E-11	1.84E-11	7.95E-05	3.79E-10	6.18E-04	2.26E-02	4.75E-05
	7.63E-08	N/A	2.79E-03	6.03E-12	6.32E-12	6.28E-12	6.31E-12	3.05E-02	5.60E-06	4.18E-05	1.61E-02	8.93E-04
	6.15E-03	1.12E-04	3.63E-11	1.40E-11	1.45E-11	1.44E-11	1.45E-11	5.32E-04	2.87E-06	1.90E-05	5.27E-03	4.87E-03
	2.95E-11	1.59E-11	4.39E-03	1.59E-11	1.21E-12	5.80E-07	1.14E-11	1.85E-10	1.14E-11	6.00E-05	1.14E-11	4.10E-03
	6.39E-07	6.32E-12	1.13E-08	6.32E-12	8.12E-08	6.32E-12	1.58E-04	5.95E-05	1.21E-12	N/A	4.67E-10	1.59E-11
	2.48E-08	5.77E-11	1.21E-12	2.25E-11	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.66E-11	1.21E-12	1.13E-12	1.21E-12

CBKA for adaptive infinite impulse response system identification

Adaptive infinite impulse response system identification

The adaptive infinite impulse response (IIR) system potentially incorporates a multifaceted error structure, and accurately acquiring the trustworthy filter coefficients for system simulation remains sophisticated. The CBKA is incorporated to resolve the IIR system identification, and the fundamental objective is to establish the most advantageous modulating coefficients, mitigate the mean square error (MSE) between an unanticipated system’s input and the IIR system’s output, and recognize an appropriate transfer function that corresponds to the unanticipated system. Figure 6 articulates the adaptive IIR system identification via CBKA.

Fig. 6.

The adaptive IIR system identification via CBKA.

The input and output is stipulated as:

21

where showcases the feedforward order, showcases the feedback, showcases the pole factor, showcases the zero factor. The is stipulated as:

22

The discrepancy between unanticipated system and the IIR system is . The MSE is stipulated as:

23

where showcases the input sample number, showcases the coefficient factor, .

CBKA-based adaptive IIR system identification

Algorithm 3 emphasizes the CBKA-based adaptive IIR system identification.

Algorithm 3.

CBKA-based adaptive IIR system identification

Parameter settings

The CBKA is contrasted with BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO and BKA to emphasize the practicality and accessibility. Certain representative empirical variables that are extracted from the source manuscripts serve as the control parameters. The portrayed regulation variables of each approach are stipulated as:

BSLO: fixed value , fixed value , fixed value , fixed value , fixed value , fixed value .

EGO: precarious value , precarious value , precarious value .

FLO: precarious value , fixed value .

GOOSE: precarious value , precarious value , precarious value , precarious value .

HLOA: hue angle , fixed value , precarious value , precarious value .

TTAO: precarious value , precarious value .

WO: precarious value , precarious values , , precarious values , precarious value , precarious value , standard deviation , fixed value , precarious value .

YDSE: wavelength , distance between two slits , distance between the barrier and the projection screen , distance between light source and barrier , constant value .

SCHO: precarious value , fixed value , precarious value , fixed value , fixed value , fixed value , fixed value , fixed value , fixed value .

SWO: fixed value , fixed value , precarious value , precarious value .

GAO: precarious value , fixed value .

BKA: precarious value , fixed value , precarious value , Cauchy mutation , fixed value , fixed value .

CBKA: precarious value , fixed value , precarious value , Cauchy mutation , fixed value , fixed value .

Simulation evaluation and result interpretation

The CBKA emphasizes formidable flexibility and sustainability to renew the actual and fictitious portions of each black-winged kite and reconciles localized exploitation and universal exploration to forestall exaggerated convergence and locate the appropriate solution.

For case 1, each methodology promotes a first-order IIR filter to discern a second-order structure, the unanticipated system and IIR filter are stipulated as:

24

25

For case 2, each methodology promotes a second-order IIR filter to discern a second-order structure, the unanticipated system and IIR filter are stipulated as:

26

27

For case 3, each methodology promotes a higher-order IIR filter to discern a higher-order structure, the unanticipated system and IIR filter are stipulated as:

28

29

Table 5 portrays each approach’s experimental results (MSE) for cases 1, 2, and 3. Table 6 portrays each approach’s experimental results (average estimation parameters) for cases 1, 2, and 3. The CBKA is incorporated to resolve the IIR system identification, and the fundamental objective is to establish the most advantageous modulating coefficients, mitigate the mean square error (MSE) between an unanticipated system’s input and the IIR system’s output, and recognize an appropriate transfer function that corresponds to the unanticipated system. For each approach, , and . Best, Worst, Mean, and Std are regarded as the most comprehensive and conventional assessment metrics for recognizing the stability and robustness of each method. For case 1, the optimal score, worst score, mean score, standard deviation, and average estimation parameters of CBKA are superior to those of the BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO and BKA. The CBKA utilizes large-scale discovery and small-scale extraction to foster aggregate discovery intensity, advance widespread computational efficacy, and retrieve an adequate universal solution. The CBKA demonstrates strong stability and reliability to restrict discovery stagnation and achieve the exact optimal score. For case 2, the optimal scores of CBKA and BKA remain consistent and at the same magnitude. The worst score, mean score, and standard deviation of CBKA have been significantly enhanced compared to the BKA. The computational magnitude, evaluation accuracy, and average estimation parameters of the CBKA are superior to those of the BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO and BKA. The CBKA manipulates a dual-diploid organization to encode the black-winged kite, reinforce population pluralism, restrict discovery stagnation, extend identification area, promote estimation excellence, advance information resources, and foster collaboration efficiency. For case 3, compared with the BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO, and BKA, the CBKA has showcases abundant adaptability and versatility to achieve the better optimal score, worst score, mean score, standard deviation, and average estimation parameters. The CBKA utilizes the dual-diploid organization of the complex-valued encoding to transform the black-winged kite into an individual with actual and fictitious portions to enrich the detection information capacity and promote global convergence performance. The CBKA not only showcases abundant predictability and versatility to reap additional advantages and sharpen resolution precision but also incorporates localized extraction and universal utilization to forestall exaggerated convergence and cultivate desirable solutions.

Table 5.

Experimental results (MSE) of each approach for different cases.

Cases	Result	BSLO	EGO	FLO	GOOSE	HLOA	TTAO	WO	YDSE	SCHO	SWO	GAO	BKA	CBKA
Case 1	Best	0.009619	0.011792	0.010877	0.010071	0.009702	0.011797	0.009907	0.010108	0.010165	0.012263	0.011892	0.009214	0.009145
	Worst	0.020216	0.019464	0.017904	0.019437	0.018869	0.017062	0.019894	0.013176	0.020697	0.028699	0.018868	0.011791	0.011436
	Mean	0.014151	0.015811	0.014361	0.012890	0.012814	0.014766	0.013090	0.012003	0.014128	0.018372	0.015393	0.010813	0.010508
	Std	0.003457	0.001659	0.002071	0.002813	0.002371	0.001277	0.002726	0.000783	0.003374	0.004110	0.002051	0.000632	0.000522
Case 2	Best	3.90E-18	0.019511	0.008645	6.43E-09	4.22E-05	1.54E-26	2.68E-09	3.34E-09	6.96E-06	0.012970	0.019014	0	0
	Worst	0.254200	0.218795	0.240145	0.296689	0.248417	7.71E-05	0.193884	1.89E-06	0.248278	0.429215	0.238565	4.79E-07	7.74E-30
	Mean	0.093332	0.099097	0.166015	0.087159	0.056197	7.23E-06	0.010959	2.88E-07	0.147983	0.149503	0.152798	1.65E-08	2.67E-31
	Std	0.095738	0.057353	0.070880	0.113079	0.066993	1.88E-05	0.037428	3.78E-07	0.092816	0.103427	0.068690	8.74E-08	1.41E-30
Case 3	Best	0.009810	0.012654	0.019347	0.000396	0.008047	0.002295	0.009524	0.003971	0.001311	0.022960	0.025003	0.001376	4.35E-05
	Worst	0.020286	0.036350	0.043832	0.028945	0.079879	0.016452	0.036851	0.026148	0.073240	0.113430	0.043899	0.042917	0.010862
	Mean	0.014171	0.020770	0.031369	0.003752	0.048457	0.007182	0.024878	0.012225	0.033006	0.067567	0.033446	0.014957	0.001628
	Std	0.003353	0.005317	0.005181	0.005631	0.020911	0.003521	0.005693	0.005951	0.022947	0.018361	0.004580	0.009655	0.002589

Table 6.

Experimental results (average estimation parameters) of each approach for different cases.

Cases	Result	BSLO	EGO	FLO	GOOSE	HLOA	TTAO	WO	YDSE	SCHO	SWO	GAO	BKA	CBKA
Case 1		0.542458	0.318714	0.339855	0.582849	0.667811	0.904038	0.597943	0.916925	0.533434	0.737912	0.218789	0.867937	0.911457
		-0.15241	-0.01958	-0.07276	-0.18917	-0.22987	-0.31236	-0.13789	-0.27828	-0.15183	-0.22145	-0.06522	-0.27495	-0.28481
Case 2		-0.77123	-0.95025	-0.41910	-0.75968	-1.02974	-1.40170	-1.30803	-1.40008	-0.43169	-1.38518	-0.49819	-1.40004	-1.40000
		0.184355	0.175280	0.087197	0.372299	0.314144	0.491543	0.424788	0.490056	0.012500	0.488388	0.099516	0.490040	0.490000
		0.629255	0.938298	0.228870	0.678497	0.767260	0.997413	0.960312	0.999801	0.428387	0.929266	0.271766	0.999915	0.998991
Case 3		0.407971	0.044818	0.571741	0.071850	-0.04235	-0.00297	0.691073	0.006687	-0.05245	-0.00645	0.443501	-0.01587	0.107773
		-0.09455	0.028528	0.558930	-0.21465	-0.05206	-0.37043	0.660711	-0.29537	-0.04282	-0.07825	0.437121	-0.15919	-0.22174
		-0.20427	0.064564	0.564440	-0.03334	-0.04315	-0.02191	0.672803	-0.03962	-0.03690	0.070357	0.414343	0.055081	-0.0084
		0.053098	-0.01152	0.544305	-0.48601	-0.05995	-0.43280	0.592081	-0.25636	-0.05829	0.065729	0.402791	-0.12678	-0.51181
		-0.13546	0.713472	0.602154	0.982854	0.350901	0.859356	0.798099	0.977551	0.654671	0.650681	0.498056	0.743585	0.977775
		-0.25513	0.048506	0.552431	0.077448	-0.01815	-0.00359	0.624065	-0.00876	0.027253	-0.01436	0.408190	0.001201	0.088317
		0.128562	0.070804	0.571683	0.135751	0.101738	0.081952	0.664480	0.068004	0.135691	0.120896	0.465989	0.089398	0.130274
		0.316745	0.035232	0.545284	0.010769	-0.06340	-0.03118	0.605366	-0.00025	-0.05008	0.045962	0.429138	0.019532	0.019495
		0.016512	0.119900	0.573087	-0.16379	0.044414	0.030244	0.653779	0.061685	0.049006	0.090746	0.472957	0.162096	-0.15249

Wilcoxon rank-sum test is executed to ascertain if there is an instructive distinction between CBKA and other procedures⁷¹. is an instructive distinction, is no instructive distinction, and N/A is “not applicable”. Table 7 portrays the comparative solutions of the Wilcoxon rank-sum test.

Table 7.

Results of the p-value Wilcoxon rank-sum test on the different cases.

Result	BSLO	EGO	FLO	GOOSE	HLOA	TTAO	WO	YDSE	SCHO	SWO	GAO	BKA
Case 1	3.09E-06	3.02E-11	6.70E-11	4.12E-06	3.08E-08	3.02E-11	1.70E-08	7.77E-09	1.43E-08	3.02E-11	3.02E-11	3.27E-02
Case 2	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	6.48E-12	4.43E-03
Case 3	4.50E-11	3.02E-11	3.02E-11	3.99E-04	4.08E-11	2.19E-08	3.34E-11	4.62E-10	1.61E-10	3.02E-11	3.02E-11	1.07E-09

Figure 7 portrays the convergence curves of the CBKA and compares algorithms for resolving the IIR system identification. For different cases 1, 2, and 3, the convergence productivity and assessment precision of the CBKA are superior to those of the BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO, and BKA. The CBKA showcases abundant adaptability and versatility to forestall exaggerated convergence and determine a more stable evaluation accuracy. Figure 8 portrays Boxplots of the CBKA and compares algorithms for resolving the IIR system identification. For different cases 1, 2, and 3, the standard deviations and stability of the CBKA are superior to those of the BSLO, EGO, FLO, GOOSE, HLOA, TTAO, WO, YDSE, SCHO, SWO, GAO, and BKA. The CBKA receives exceptional consistency and endurance to promote exploration efficiency and disrupt anticipation stagnation. The actual and fictitious portions are inserted into the BKA that converts the dual-dimensional encoding region to the single-dimension manifestation region, and this is refreshed separately for each search agent that exhibits inherent parallelism and reliability, which reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources, fosters collaboration efficiency, exhibits remarkable parallelism and consistency. The CBKA has strong stability and reliability to foster aggregate discovery intensity and advance widespread computational efficacy.

Fig. 7.

Convergence curves of the CBKA and compared algorithms for resolving the IIR system identification.

Fig. 8.

Boxplots of the CBKA and compared algorithms for resolving the IIR system identification.

CBKA for classical engineering design

To emphasize scalability and adaptability, the CBKA is administered to remedy the engineering layouts: speed reducer⁴, gear train⁷², multiple disc clutch brake⁵⁹, and rolling element bearing⁷³.

Speed reducer layout

The foremost objective is to alleviate the combined weight as articulated in Fig. 9, which showcases seven evaluation elements: facial breadth , teeth magnitude , teeth size , first shaft distance , second shaft distance , first shaft diameter , and second shaft diameter . The mathematical scheme is stipulated as:

Fig. 9.

Speed reducer layout.

Consider

30

Minimize

31

Subject to

32

33

34

35

36

37

38

39

40

41

42

Variable range

43

Table 8 portrays the comparative solutions of the speed reducer layout. The CBKA ascertains the most appropriate convergence fitness with the evaluation elements . The extraction productivity and assessment precision of the CBKA are superior to those of other procedures, and the CBKA reconciles localized exploitation and universal exploration to forestall exaggerated convergence and locate the appropriate solution.

Table 8.

Comparative solutions of speed reducer layout.

Algorithm	Optimal values for variables							Optimal cost
SA74	3.4979	0.7	17	7.9205	7.9513	3.3518	5.2853	3004.85
EHO74	3.4889	0.7782	23.2193	7.849	8.1021	3.5603	5.2459	73,504.7
GOA74	3.5126	0.7033	17.2246	7.9131	7.9627	3.6567	5.2784	3169.32
TEO74	3.4261	0.7	17.6222	7.7408	7.9775	3.4145	5.2758	3595.59
MPA75	3.50669	0.7	17	7.380933	7.815726	3.357847	5.286768	3001.288
TLBO75	3.508755	0.7	17	7.3	7.8	3.46102	5.2892113	3030.563
BWO6	3.58	0.72	18.28	7.73	7.73	3.43	5.28	3417.1535
RUN5	3.507125	0.7	17	7.307812	7.8078	3.356534	5.29705	3004.852
SMA5	3.512233	0.7	17	7.388958	7.8236	3.363121	5.29507	3007.596
MSA5	3.505551	0.7	17	8.308882	7.8079	3.357677	5.29502	3012.079
MBO5	3.514048	0.7	17	7.418166	7.8239	3.363347	5.29508	3009.238
INFO5	3.514301	0.7	17	7.307301	7.8078	3.466456	5.29752	3036.931
CPA5	3.525688	0.7	17	8.378957	7.8078	3.372258	5.29702	3035.367
BOA76	3.5239	0.7003	17.0088	8.0962	8.004	3.4048	5.3286	3061.6
AO77	3.49688	0.7	17	8.10828	7.8	3.37081	5.28578	3008.168
HOA77	3.56008	0.7	17	7.34912	7.8	3.49325	5.28415	3058.577
ES73	3.506163	0.700831	17	7.460181	7.962143	3.3629	5.309	3025.005
SBS73	3.506122	0.700006	17	7.549126	7.85933	3.365576	5.289773	3008.981
SCSO69	3.5	0.7	17	7.32	8.029658	3.350294	5.286794	3001.69686
MDO4	3.5	0.7	17	7.3	7.67	3.542	5.246	3019.583365
CBKA	3.50281	0.7	17	7.3	7.71535	3.35071	5.28668	2995.5868

Gear train layout

The foremost objective is to alleviate the combined cost as articulated in Fig. 10, which showcases four evaluation elements: gears teeth magnitude , , and . The mathematical scheme is stipulated as:

Fig. 10.

Gear train layout.

Consider

44

Minimize

45

Variable range

46

Table 9 portrays the comparative solutions of gear train layout. The CBKA ascertains the most appropriate convergence fitness with the evaluation elements . The extraction productivity and assessment precision of the CBKA are superior to those of other procedures, and the CBKA can reasonably deploy large-scale discovery and small-scale extraction to foster aggregate discovery intensity and advance widespread computational efficacy.

Table 9.

Comparative solutions of gear train layout.

Algorithm	Optimal value for variables				Optimal cost
ICA78	43	16	19	49	2.70E-12
BBO78	53	26	15	51	2.31E-11
NNA78	49	16	19	43	2.70E-12
WSA78	43	16	19	49	2.70E-12
KOA34	44	20	16	50	2.700857E-12
FLA34	44	16	20	49	2.700857E-12
COA34	23	14	12	48	9.92158E-10
RUN34	44	17	19	49	2.700857E-12
SMA34	52	30	13	53	2.307816E-11
DO34	49	16	19	44	2.700857E-12
POA34	44	17	19	49	2.700857E-12
PDO79	48	17	22	54	2.70E-12
DMOA79	49	19	16	43	2.70E-12
AOA79	49	19	19	54	2.70E-12
SSA79	49	19	19	49	2.70E-12
SCA79	49	19	34	49	2.700857E-12
GMO72	43	19	16	49	2.700857E-12
GBO43	53	13	20	34	2.3078E-11
TTAO43	43	16	19	49	2.70E-12
WO4	43	16	19	43	2.700857E-12
CBKA	50	32	16	47	2.4277E-18

Multiple disc clutch brake layout

The foremost objective is to alleviate the combined weight as articulated in Fig. 11, which showcases five evaluation elements: disc depth , inner radius , outer radius , activation force , and friction surface magnitude . The mathematical scheme is stipulated as:

Fig. 11.

Multiple-disc clutch brake layout.

Consider

47

Minimize

48

Subject to

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

Table 10 portrays the comparative solutions of multiple-disc clutch brake layout. The CBKA ascertains the most appropriate convergence fitness with the evaluation elements . The extraction productivity and assessment precision of the CBKA are superior to those of other procedures, and the CBKA incorporates localized exploitation and universal exploration to forestall exaggerated convergence and cultivate desirable solutions.

Table 10.

Comparative solutions of multiple-disc clutch brake layout.

Algorithm	Optimal value for variables					Optimal cost
TLBO80	70	90	1	810	3	0.313657
MFO81	70	90	1	910	3	0.313656
MVO82	70	90	1	910	3	0.313656
CMVO82	70	90	1	910	3	0.313656
WCA83	70	90	1	910	3	0.313656
PVS84	70	90	1	980	3	0.31366
MBFPA85	70	90	1	600	2	0.2352424579
GOA86	71	92	1	835	3	0.3355146
EOBL-GOA86	70	90	1	984	3	0.31365661053
ABC87	70	90	1	790	3	0.313657
CS87	70	90	1	810	3	0.3136566
GSA87	72	92	2	815	3	0.3175771
AEO87	70	90	1	810	3	0.3136566
AHA88	70	90	1	840	3	0.3136566
HBO89	70	90	1	1000	3	0.3136566
HGS61	70	90	1	1000	3	0.313657
MRFO90	70	90	1	835	3	0.3136566
GA90	72	92	1	918	3	0.321498
DE90	71	92	1	835	3	0.3355146
RSO91	70	90	1	810	3	0.313657
CBKA	70	90	1	440	2	0.2352

Rolling element bearing layout

The foremost objective is to alleviate the combined weight as articulated in Fig. 12, which showcases ten evaluation elements: pitch diameter (), ball diameter (), number of balls (), inner (), and outer (), raceway curvature coefficients, , , , , . The mathematical scheme is stipulated as:

Fig. 12.

Rolling element bearing layout.

Consider

68

Minimize

69

Subject to

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

Variable range

85

86

87

88

89

Table 11 presents the comparative solutions for rolling element bearing layout. The CBKA ascertains the most appropriate convergence fitness with the evaluation elements . The extraction productivity and assessment precision of the CBKA are superior to those of other procedures, and the CBKA exhibits suitability and affordability to foster aggregate discovery intensity, advance widespread computational efficacy, and retrieve the universal adequate solution.

Table 11.

Comparative solutions of rolling element bearing layout.

Algorithm	Optimal value for variables					Optimal cost
TLBO80	125.7191	21.42559	11	0.515	0.515
	0.424266	0.633948	0.3	0.068858	0.799498	81,859.74
CSA92	125	21.418	11.356	0.515	0.515
	0.4	0.7	0.3	0.02	0.612	85,201.641
GSA92	125	20.854	11.149	0.515	0.517
	0.5	0.618	0.3	0.02	0.624	82,276.941
HHO93	125	21	11.09207	0.515	0.515
	0.4	0.6	0.3	0.050474	0.6	83,011.88
RSA93	125.1722	21.29734	10.88521	0.515253	0.517764
	0.41245	0.632338	0.301911	0.024395	0.6024	83,486.64
RSO94	125	21.41769	10.94027	0.515	0.515
	0.4	0.7	0.3	0.02	0.6	85,069.021
STOA95	125	21.4189	10.94113	0.515	0.515
	0.4	0.7	0.3	0.02	0.6	85,067.983
TSA96	125	21.4175	10.941	0.51	0.515
	0.4	0.7	0.3	0.02	0.6	85,070.08
EPO97	125	21.4189	10.94113	0.515	0.515
	0.4	0.7	0.3	0.02	0.6	85,067.983
ESA97	125	21.4175	10.94109	0.51	0.515
	0.4	0.7	0.3	0.02	0.6	85,070.085
SSA97	125	20.77562	11.01247	0.515	0.515
	0.5	0.61397	0.3	0.05004	0.61001	82,773.982
WCA83	125.721167	21.4233	11.00103	0.515	0.515
	0.401514	0.659047	0.300032	0.040045	0.6	85,538.480
WOA24	125.100734	21.4233	10.95119	0.515	0.515
	0.4	0.7	0.314216	0.02	0.6	85,265.167
ACVO24	125.70959	21.4232997	11.000104	0.515	0.515
	0.48352698	0.61821897	0.3002753	0.02	0.6478817	85,533.4103
MBA87	125.7153	21.4233	11	0.515	0.515
	0.488805	0.627829	0.300149	0.097305	0.646095	85,535.9611
HBO89	125.7227184	21.4233	11	0.515	0.515
	0.438476	0.699998	0.3	0.047532	0.601081	85,533.18
HPO59	125	21.875	10.777	0.515	0.515
	0.4	0.7	0.3	0.029	0.6	83,918.4925
MGA73	125.718	21.8745119	10.7770658	0.51500082	0.51500299
	0.405908353	0.65558802	0.30000415	0.07754492	0.6	83,912.87983
CGO73	125	21.875	10.777009	0.515	0.515
	0.4	0.64620052	0.3	0.050152445	0.6	83,918.49253
EVO73	125.7190556	21.4255902	10.6955328	0.515	0.515
	0.463182936	0.6999265	0.3	0.063431519	0.604213108	81,859.7415974
CBKA	125.3527	21.5041	11	0.515	0.515
	0.4013	0.7014	0.3	0.07481	0.6103	85,539.302

Impact analysis

The CBKA is constructed on the black-winged kites’ migratory and predatory instincts and integrates the Leader tactic with the Cauchy variation procedure to retrieve the expansive appropriate convergence solution. The CBKA efficiently resolves the function evaluations, engineering layouts, and IIR system identification for the following aspects of impact analysis. First, the CBKA exhibits some advantages of simple algorithm framework construction, few control variables, high computational efficiency, easy approach fusion, good information exchange, strong parallelism and feasibility, superior exploration and exploitation, strong stability, and robustness. Second, the Cauchy variation enhances global exploration efficiency, avoids premature convergence, and improves the calculation accuracy. The leader tactic accelerates exploitation efficiency, promotes directional search, and improves convergence speed. Third, the complex-valued encoding strategy manipulates a dual-diploid organization to encode the black-winged kite, the actual and fictitious portions are inserted into the BKA that converts the dual-dimensional encoding region to the single-dimension manifestation region. This strategy reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources, and fosters collaboration efficiency. To summarize, the CBKA not only showcases abundant adaptability and versatility to reap supplementary advantages and sharpen resolution precision but also incorporates localized exploitation and universal exploration to achieve superior assessment precision and a swifter convergence rate.

Conclusion and future exploration

This paper establishes the CBKA to resolve the function evaluations, engineering layouts, and adaptive IIR system identification. The purpose is to attain the minimal available solutions of function evaluations, the satisfactory computational expenditures of engineering layouts, and the most advantageous modulating coefficients and mitigate the mean square error (MSE) between an unanticipated system’s input and the IIR system’s output. The BKA is constructed on the black-winged kites’ migratory and predatory instincts that integrate the Leader tactic with the Cauchy variation procedure to foster aggregate discovery intensity and retrieve the appropriate computational solution. The basic BKA exhibits marginalized resolution efficiency, inferior assessment precision, and stagnant sensitive anticipation. The innovative complex-valued encoding strategy is inserted into the BKA to foster aggregate discovery intensity and advance widespread computational efficacy. The CBKA manipulates a dual-diploid organization to encode the actual and fictitious portions of each black-winged kite and translates the dual-dimensional encoding region to the single-dimension manifestation region, which reinforces population pluralism, restricts discovery stagnation, extends identification area, promotes estimation excellence, advances information resources, fosters collaboration efficiency, exhibits remarkable parallelism and consistency. The CBKA showcases abundant sustainability and versatility to forestall exaggerated convergence, reconciling localized exploitation and universal exploration to locate the appropriate solution. The CBKA is compared with the GTO, MGO, PO, AVOA, GCRA, HLOA, WO, SBOA, NRBO, APO, EHO, BSLO, EGO, FLO, GOOSE, TTAO, YDSE, SCHO, SWO, GAO and BKA. The experimental results show that the CBKA exhibits suitability and affordability to explore a superior assessment precision and a swifter convergence rate.

Future research on CBKA will focus on the following three aspects: (1) We will introduce more streamlined extraction strategies (e.g., golden-sine, multi-population, quasi-opposition-based learning, Brownian motion, intelligent perception, S-shaped escape energy, Tent chaotic map, elite pool, differential evolution), distinguishable encoding formats (e.g., quantum, discrete, binary, integer, hybridized encodings), and hybrid approaches (e.g., genetic algorithms or differential evolution) to reap supplementary advantages and explore a superior assessment precision and swifter convergence efficiency. (2) A more extensive application of real-world datasets will improve the research relevance, such as neural network layouts (e.g., convolutional neural networks, recurrent neural networks, generative adversarial networks, graph neural networks, long short-term memory, Hopfield networks, deconvolutional networks, and recurrent neural networks), image processing (e.g., sophisticated patterns, feature tracking, stereo matching, real-world images), multilevel thresholding segmentation methods (Tsallis entropy, Renyi entropy, cross-entropy, fuzzy entropy, and Otsu’s method). (3) Due to the wide distribution of agriculture and forestry crops, the geographical dispersion, and the inconvenience of planting and maintenance in the Dabie Mountains in Anhui Province, the CBKA will utilize data collection, infrared detection, image recognition, big data intelligent analysis and decision-making, intelligent precision plant protection equipment to achieve agricultural intelligent perception and detection, agricultural data intelligent processing, and agricultural equipment intelligent control.

Author contributions

Chengtao Du: Conceptualization, Methodology, Formal analysis, Resources, Data curation, Project administration, Writing – original draft and Funding acquisition. Jinzhong Zhang: Conceptualization, Methodology, Software, Data curation, Formal analysis, Writing – original draft and Funding acquisition. Jie Fang: Software, Validation, Investigation, Data curation, Visualization, Supervision, Writing – review & editing and Funding acquisition.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. All data generated or analyzed during this study are included directly in the text of this submitted manuscript. There are no additional external files with datasets.

Declarations

Competing interests

The authors declare no competing interests.