Quantum Computing and Adaptive Mechanism-Based Bounty Hunter Optimizer for Numerical Optimization and Bankruptcy Prediction

Abstract

To improve the optimization performance of the original Bounty Hunter Optimizer (BHO) in complex search environments, this paper proposes a quantum computing and adaptive mechanism-based BHO, named QCAMBHO. The proposed algorithm integrates three complementary strategies: quantum-computing-enhanced initialization, adaptive Lévy flight, and an adaptive differential operator. These mechanisms are designed to improve population diversity, strengthen global exploration, and enhance later-stage exploitation. The performance of QCAMBHO is evaluated on the CEC2017 and CEC2022 benchmark test suites. Experimental results show that QCAMBHO achieves competitive or superior optimization performance compared with several advanced algorithms in terms of convergence accuracy, stability, and robustness. Ablation experiments further confirm the positive contribution of each strategy and the synergistic effect of their integration. To examine its practical applicability, QCAMBHO is further used to optimize the key parameters of Kernel Extreme Learning Machine (KELM), and a QCAMBHO-KELM model is constructed for enterprise bankruptcy prediction. The results show that QCAMBHO-KELM achieves better overall classification performance than BHO-KELM and other comparison models across multiple evaluation metrics, including accuracy, Matthews correlation coefficient, sensitivity, specificity, precision, recall, and F1-score. These findings indicate that QCAMBHO not only provides an effective optimizer for complex numerical problems but also offers a promising decision-support tool for improving the accuracy and reliability of enterprise bankruptcy early warning.

1. Introduction

Enterprise bankruptcy prediction is a fundamental problem in financial early warning, credit risk management, and capital market supervision. Its research value lies not only in identifying the risk status of individual firms, but also in its spillover effects on supply chain stability, the asset security of financial institutions, and regional economic resilience. Recent systematic reviews have generally pointed out that with increasing global economic uncertainty, increasingly complex business environments, and the rapid accumulation of multi-source data, bankruptcy prediction has gradually evolved from a traditional financial analysis problem into a typical data-driven risk identification problem [1,2,3,4]. Meanwhile, the scarcity of public datasets, highly imbalanced sample distributions, and inconsistencies in data quality and evaluation criteria among different studies continue to limit the comparability and external validity of model results [5,6]. Nevertheless, bankruptcy prediction research is still built on the analytical frameworks established by several foundational studies, including early identification based on financial ratios [7], the Z-score model based on multiple discriminant analysis [8], the ZETA model that extends the scope of corporate default identification [9], and the probabilistic prediction framework based on conditional Logit [10]. These classic studies defined the basic form of the bankruptcy prediction problem, while recent reviews further indicate that developing more robust and forward-looking prediction algorithms for complex business environments has become a central direction for the continuous development of this field [1,2,3].

From the perspective of methodological development, bankruptcy prediction models have gradually expanded from traditional statistical discrimination methods to a multi-level technical system that includes machine learning, ensemble learning, deep learning, and hybrid optimization. In classical studies, Altman [8] and Ohlson [10] represent two basic routes, namely discriminant analysis and probabilistic classification. Shumway [11] introduced the hazard model into bankruptcy prediction and emphasized the importance of dynamic default modeling. Chava and Jarrow [12] further demonstrated that industry effects and time-varying information can significantly improve predictive ability. With the development of artificial intelligence, neural network methods were systematically summarized by Atiya [13], hybrid and ensemble soft computing frameworks were reviewed in [14], and systematic comparisons of machine learning methods can be found in [15,16,17], which summarized the overall trend of this field from traditional models to machine learning and deep learning. In recent years, research attention has further shifted to model construction under high-dimensional, imbalanced, and heterogeneous data scenarios. Study [3] pointed out that missing value processing, outlier control, class imbalance, and feature selection have become key factors affecting model robustness. Studies [1,2] further emphasized that advanced machine learning methods generally show stronger modeling potential, but their performance is highly dependent on data construction, variable systems, and experimental design. From the perspective of datasets, ref. [6] indicated that bankruptcy prediction has expanded from relying only on financial indicators to incorporating market variables, macroeconomic variables, relational data, and textual disclosure information. Study [5] showed through large-scale experiments that, in real imbalanced scenarios, if the model design ignores the ability to identify minority-class samples, relying only on overall accuracy may lead to misleading conclusions. Against this background, optimization algorithms have begun to be deeply embedded in the bankruptcy prediction modeling process and have been widely used for feature selection, parameter optimization, and hybrid model design. Li and Sun [18] conducted a comparative study of machine learning methods for bankruptcy risk prediction and showed that the integration of machine learning classifiers with metaheuristic optimizers can further improve prediction accuracy. Naved et al. [19] also constructed several hybrid bankruptcy prediction models by combining Histogram Gradient Boosting Classification with different optimization algorithms, indicating that optimization-driven machine learning models are effective for identifying financial distress. From a broader perspective, Pazouki et al. [20] reviewed the application of artificial intelligence and digital technologies in finance and emphasized that AI-driven methods have become increasingly important for improving operational efficiency, risk identification, and decision support in financial systems. Representative studies include the GA-LSTM time-series financial distress prediction model proposed in [21], the genetic-algorithm-optimized autoencoder ensemble method proposed in [22], the GA-optimized XGBoost and ANN hybrid framework proposed in [23], and the improved particle swarm optimization wrapper feature selection method combined with tree boosting ensemble models proposed in [24]. Therefore, bankruptcy prediction research has gradually shifted from the comparison of single classifiers to a collaborative modeling paradigm of “prediction model and optimization mechanism”.

As an important technical basis of the above collaborative modeling paradigm, metaheuristic algorithms provide a general and flexible search framework for high-dimensional, nonlinear, and combinatorial optimization problems. Recent classification studies generally suggest that metaheuristic algorithms can be divided into several main categories according to their sources of inspiration, such as evolutionary algorithms, swarm intelligence algorithms, physics-inspired algorithms, and human-behavior-inspired algorithms. Study [25] provided a systematic summary from the perspectives of bibliometrics and classification, while [26] further pointed out that, compared with the natural metaphors used in algorithm naming, the search behavior itself and the design of operators can better reflect the essential differences among algorithms. In terms of representative algorithms, the evolutionary computation route includes the genetic algorithm [27] and differential evolution [28]. The swarm intelligence route includes ant colony optimization [29,30,31], particle swarm optimization [32,33], and the artificial bee colony algorithm [34], and has later been extended to cuckoo search [35], grey wolf optimizer [36], whale optimization algorithm [37], and red-billed blue magpie optimizer [38]. Metaheuristic algorithms are widely used in many fields, such as image segmentation [39], UAV path planning [40], engineering applications [41], point cloud registration [42], etc. [43]. For feature selection tasks, studies [44,45] showed that metaheuristic algorithms have attracted wide attention mainly because they can maintain a relatively effective balance between global exploration and local exploitation, and can be well adapted to wrapper-based evaluation, discrete encoding, and complex constraints. Therefore, introducing metaheuristic algorithms into bankruptcy prediction is not simply to add an external optimizer to a classification model. Instead, it uses their search mechanisms to improve the construction of feature subsets, the configuration of hyperparameters, and the overall collaborative design of the prediction process [21,23,24].

The Bounty Hunter Optimizer (BHO) is a recently proposed hunting-inspired metaheuristic algorithm. According to its original paper [46], the algorithm is inspired by the cooperative tracking, regional screening, and dynamic task scheduling behaviors of bounty hunters. It attempts to enhance population diversity and improve the dynamic balance between exploration and exploitation through decentralized position updating, the Explorpolis screening mechanism, and the quantum probabilistic rotation selection mechanism. However, as emphasized in [26], the natural metaphor of a new algorithm is not sufficient to constitute methodological innovation by itself. Its effectiveness still needs to be demonstrated through rigorous comparative experiments, statistical tests, and stable performance in specific application scenarios. Therefore, the key point of introducing BHO into bankruptcy prediction research does not lie in the novelty of its name or bio-inspired mechanism, but in examining whether it can provide more reproducible and robust performance improvements than genetic algorithms, particle swarm optimization, and other swarm intelligence methods in tasks such as high-dimensional financial feature selection, classification model parameter optimization, and imbalanced sample identification.

Although existing meta-heuristic algorithms and their improved variants have shown promising performance in numerical optimization and financial prediction tasks, several research gaps remain. First, the original BHO still relies heavily on the quality of the initial population, and random or quasi-random initialization may lead to uneven spatial distribution and insufficient population diversity in complex high-dimensional search spaces. Second, the rough search process of BHO mainly depends on conventional random perturbation, which may limit its ability to perform long-distance exploration and escape from local optima when solving multimodal and hybrid optimization problems. Third, during the later stage of iteration, the population tends to concentrate around currently promising regions, but the original BHO lacks an additional adaptive refinement mechanism to strengthen local exploitation and maintain effective population recombination. Finally, in bankruptcy prediction, the predictive performance of KELM is highly dependent on the selection of key parameters, while existing studies have not sufficiently explored how an enhanced BHO framework can be used to optimize KELM for improving enterprise bankruptcy early-warning performance. These gaps motivate the development of a quantum computing and adaptive mechanism-based BHO, which aims to improve initialization quality, enhance global exploration, strengthen later-stage exploitation, and further verify its practical value in bankruptcy prediction.

To this end, this paper proposes an improved Bounty Hunter Optimizer based on quantum computing and adaptive mechanisms, named the Quantum Computing and Adaptive Mechanism-Based Bounty Hunter Optimizer (QCAMBHO). While retaining the cooperative tracking and dynamic search mechanisms of the original BHO, the proposed algorithm improves the optimization process from three aspects: population initialization, global perturbation, and later-stage exploitation. These improvements aim to achieve a coordinated enhancement of exploration and exploitation. The main contributions of this paper are as follows:

(1)

A quantum-computing-enhanced initialization strategy is introduced. Through qubit encoding and a quantum rotation mechanism, the distribution diversity of the initial population is improved, allowing candidate solutions to cover the search space more sufficiently. This helps reduce the sensitivity of the original BHO to the quality of the initial population and provides a higher-quality starting point for subsequent global search.

(2)

An adaptive Levy flight strategy is designed. To address the problems that the original BHO may suffer from in complex multimodal problems, such as rapid shrinkage of the search range and insufficient ability to escape from local optima, Levy flight perturbation is introduced. An adaptive mechanism dynamically adjusts the perturbation intensity so that the algorithm maintains strong global exploration capability in the early stages of iteration and gradually improves local search stability in the later stage. This further improves the optimization efficiency of the algorithm in complex search spaces.

(3)

An adaptive differential operator is constructed. To further improve the later-stage exploitation and fine-grained search ability of the algorithm, the idea of differential evolution is introduced in the later stage of iteration. New candidate solutions are generated by using difference information among individuals, and the differential perturbation intensity is adaptively adjusted according to the search process. This strategy helps enhance information interaction among individuals, alleviates the insufficient convergence accuracy of the algorithm in the later stage, and improves the stability of solutions.

(4)

Dual validation is conducted through numerical optimization and bankruptcy prediction. First, the numerical optimization performance of QCAMBHO is systematically evaluated on the CEC2017 and CEC2022 benchmark test functions. Its optimization capability is verified using the Wilcoxon rank-sum test, Friedman mean rank test, and convergence curve analysis. Furthermore, QCAMBHO is combined with the Kernel Extreme Learning Machine (KELM) to construct the QCAMBHO-KELM bankruptcy prediction model, which is used to optimize the key parameters of KELM. The predictive performance of the proposed model in financial classification tasks is evaluated using Accuracy, MCC, Sensitivity, Specificity, Precision, Recall, and F1-score.

The remainder of this paper is organized as follows. Section 2 reviews related studies on bankruptcy prediction, metaheuristic optimization algorithms, and their applications in machine learning model optimization. Section 3 introduces the original BHO algorithm and then describes the proposed QCAMBHO algorithm and its three improvement strategies in detail. Section 4 presents the numerical experimental settings, benchmark function test results, and statistical analysis. Section 5 constructs the QCAMBHO-KELM bankruptcy prediction model and empirically evaluates its classification performance through different evaluation metrics. Section 6 summarizes the main conclusions of this study and discusses possible directions for future research.

2. Related Work

Bankruptcy prediction research was initially mainly based on financial ratios and statistical discrimination methods. Study [7] early verified the predictive value of financial ratios in identifying corporate failure, providing an empirical basis for subsequent research. Study [8] further proposed the Z-score model and introduced multiple discriminant analysis into bankruptcy prediction, promoting a shift in this field from single-ratio judgment to multivariate integrated modeling. Subsequently, study [10] used the Logit model to describe the probability of corporate bankruptcy, which gradually formed a basic paradigm dominated by statistical learning. These early studies laid an important theoretical and methodological foundation for bankruptcy prediction. However, statistical models usually rely on relatively strict assumptions regarding variable distribution, linear separability, and functional form. When corporate financial distress is driven by nonlinear interactions among financial indicators, macroeconomic shocks, and firm-specific heterogeneity, traditional statistical methods may have limited ability to capture complex decision boundaries. Later reviews generally suggest that, with the development of machine learning methods, neural networks, support vector machines, ensemble learning, and deep learning have gradually become important technical routes in this field. Nevertheless, there is still no single model that consistently dominates under different datasets, feature systems, and evaluation criteria [2,16,47]. This indicates that bankruptcy prediction remains a data-dependent and model-sensitive task, and model optimization continues to be an important research issue.

In response to the limitations of traditional statistical models and standalone classifiers, metaheuristic algorithms have been introduced into bankruptcy prediction as effective tools for feature selection, parameter tuning, and model structure optimization. Ant colony optimization was proposed in [29], particle swarm optimization in [33], differential evolution in [28], the artificial bee colony algorithm in [34], and cuckoo search in [35]. Although these classical swarm intelligence and evolutionary optimization methods were initially designed mainly for general optimization problems, their global search ability and strong model adaptability enabled them to be quickly introduced into bankruptcy prediction tasks. Compared with manual parameter setting or exhaustive grid search, metaheuristic algorithms can search a wider solution space and are more suitable for nonlinear, high-dimensional, and non-convex optimization problems. However, early applications of metaheuristic algorithms in bankruptcy prediction were often limited to a supporting role, mainly improving existing classifiers rather than developing a complete optimizer-driven prediction framework.

Earlier related studies mostly used metaheuristic algorithms as auxiliary optimizers for classification models. Study [48] combined the genetic algorithm with support vector machine to perform feature subset selection and parameter optimization at the same time, thereby improving bankruptcy prediction performance. Subsequently, study [49] further developed the GA-SVM hybrid framework, and the results showed that the genetic algorithm could effectively improve the classification performance of support vector machine in bankruptcy prediction. Study [50] systematically discussed variable selection in bankruptcy prediction from the perspective of feature selection and pointed out that feature subset construction has a key influence on prediction performance. Study [51] combined particle swarm optimization with support vector machine for parameter setting and feature screening, and the results showed that PSO-SVM outperformed traditional grid search and several existing models in corporate financial distress identification. These studies demonstrate the usefulness of metaheuristic algorithms in enhancing classifier performance. However, most of them focus on classical classifiers such as SVM, and the optimization target is usually restricted to feature subsets or a small number of hyperparameters. The interaction between optimizer design, classifier parameter search, and prediction robustness has not been sufficiently explored.

Recent studies have further extended this research line by embedding metaheuristic algorithms more deeply into the complete modeling process of bankruptcy prediction. Study [24] proposed an improved particle swarm optimization-based wrapper feature selection method and combined it with tree boosting ensemble models for financial distress prediction. The results showed that the improved particle swarm optimization method could alleviate premature convergence and local optimum problems, and the constructed ensemble framework showed better robustness in both short-term and long-term prediction scenarios. Study [52] proposed a multiple heterogeneous self-paced ensemble learning framework and used particle swarm optimization to jointly search model parameters and ensemble weights. This study considered high-dimensional features, class imbalance, and parameter optimization at the same time, indicating that recent research has shifted from single-point optimization to collaborative design for the entire prediction pipeline. Nevertheless, these studies also reveal a common challenge: as the prediction framework becomes more complex, the optimizer needs to maintain stronger global exploration ability, more stable convergence behavior, and better adaptability to different data structures.

In the direction of genetic algorithms, Literature [21] proposed a GA-LSTM hybrid model and applied it to time-series financial distress prediction. This method uses the genetic algorithm to search the number of LSTM layers, the number of neurons, the batch size, and the number of training epochs, thus achieving joint optimization of the temporal model structure and training parameters. The results showed that this method outperformed traditional machine learning models and the unoptimized LSTM model in two-year-ahead prediction tasks. Literature [23] proposed a genetic-algorithm-optimized XGBoost and ANN hybrid model for bankruptcy prediction on imbalanced Polish datasets. This study found that the genetic algorithm can not only perform feature selection, but also optimize model hyperparameters. In its experimental setting, retaining all features while optimizing parameters achieved better results, suggesting that the role of metaheuristic algorithms in bankruptcy prediction is not always reflected in reducing feature dimensionality. From the perspective of anomaly detection, another study [22] proposed a genetic-algorithm-optimized autoencoder ensemble method, which identifies bankrupt firms in highly imbalanced data by optimizing the reconstruction error threshold and ensemble structure. This study indicates that the application scenarios of genetic algorithms have expanded from traditional supervised classification to unsupervised or semi-supervised risk identification frameworks. A study [53] combined genetic-algorithm-based support vector machine feature selection with stacking ensemble learning and constructed a multi-classifier fusion model on SMOTE-processed data, which also reflects the trend of jointly using genetic algorithms and ensemble learning. However, genetic algorithm-based methods may still suffer from relatively high computational cost, sensitivity to encoding schemes, and unstable search efficiency in complex parameter spaces. Therefore, improving the search mechanism of metaheuristic algorithms remains necessary.

In addition to particle swarm optimization and genetic algorithms, ant colony optimization and other new swarm intelligence algorithms have also been continuously introduced into bankruptcy prediction tasks. Literature [54] proposed an improved Ant Miner method based on the Hellinger distance and applied it to bankruptcy prediction for Malaysian Shariah-compliant securities companies. This method embeds ant colony optimization into the rule generation process and enhances minority-class identification ability through skew-insensitive heuristic information. The results showed that it outperformed the traditional Ant Miner and several tree-based models in imbalanced data environments. A study [55] combined the tuna swarm optimization algorithm with XGBoost for corporate financial risk early warning and improved the predictive performance of the model by searching for the optimal parameter combination. Study [56] proposed an improved RIME optimization algorithm and applied it to feature selection in bankruptcy prediction, further indicating that new swarm intelligence algorithms are being continuously used for feature search tasks. These studies enrich the methodological toolbox for bankruptcy prediction, but they also suggest that newly proposed optimizers must be carefully evaluated in terms of population diversity, convergence stability, and the balance between exploration and exploitation before being applied to financial prediction tasks.

With the increasing application of deep learning in financial risk identification, more studies have begun to adopt dual-optimization or multi-stage optimization frameworks. A study [57] proposed a hybrid model combining extended osprey optimization with a Bayesian neural network. In this model, osprey optimization is used for feature subset selection, while multi-verse optimization is used for parameter optimization of the Bayesian neural network. Thus, feature selection and model training are placed in different optimization stages. Another study found in the literature [58] proposed a financial distress prediction framework based on an attention-based bidirectional gated recurrent unit. The Flying Fox Optimizer is used for feature selection, and the Coati Optimization Algorithm is used for parameter fine-tuning, so as to improve prediction accuracy for imbalanced data. Another study in the literature [59] further used the Elephant Herding Lion Optimizer for dimensionality reduction and the Pied Kingfisher Optimizer to optimize the attention-GRU classifier. This reflects the recent tendency to connect multiple new optimizers within the same deep learning framework. However, multi-stage frameworks usually introduce more algorithmic components and higher computational complexity. In addition, when multiple optimizers are used simultaneously, it becomes more difficult to identify which mechanism contributes to the final performance improvement. This highlights the need for a relatively compact but effective optimizer that can improve search quality without excessively increasing modeling complexity.

Recent related studies also show that the improvement in bankruptcy prediction performance does not come entirely from metaheuristic algorithms themselves. It is often closely related to class imbalance processing, data construction methods, and the selection of evaluation metrics. Study [60] proposed a cost-sensitive XGBoost model, which improved minority-class identification without relying on resampling. This indicates that even without introducing metaheuristic algorithms, modeling designs for class imbalance may significantly improve bankruptcy detection ability. A study [5] has experimentally compared imbalanced learning algorithms on several public datasets and pointed out that hybrid ensemble and undersampling methods are usually more robust than models that only pursue high overall accuracy. Literature [6] further indicated that public bankruptcy prediction datasets differ greatly in sample size, firm type, feature source, and imbalance degree, and that data quality largely determines the upper limit of model performance. Therefore, existing studies are more suitable for comparison from the perspective of how optimization algorithms are embedded into the prediction process, rather than drawing conclusions only based on the best accuracy in a single experiment. For bankruptcy prediction models, it is also necessary to consider evaluation indicators such as sensitivity, specificity, MCC, and F1-score, rather than relying solely on overall accuracy.

Overall, existing research shows that metaheuristic algorithms have become an important tool for improving bankruptcy prediction models. Early studies mainly focused on genetic algorithms and particle swarm optimization, with emphasis on feature selection and parameter optimization for classifiers such as support vector machines. Recent studies have further expanded to ensemble learning, time-series deep models, anomaly detection models, and multi-stage hybrid frameworks, while continuously introducing new swarm intelligence algorithms. However, several limitations remain. First, many existing methods still rely on conventional optimizers whose search performance may be weakened by premature convergence, insufficient population diversity, or limited late-stage exploitation ability. Second, although complex hybrid models can improve prediction accuracy, they may increase computational burden and reduce interpretability. Third, the optimization of KELM parameters in bankruptcy prediction still requires a more effective optimizer with strong global exploration and stable local refinement capability. Therefore, this study proposes QCAMBHO and integrates it with KELM for enterprise bankruptcy prediction. By introducing quantum-computing-enhanced initialization, adaptive Lévy flight, and an adaptive differential operator, the proposed method aims to improve initialization quality, enhance global exploration, strengthen later-stage exploitation, and provide a more robust optimization framework for bankruptcy early warning.

3. Materials and Methods

3.1. An Overview of BHO

The Bounty Hunter Optimizer is a swarm-intelligence-based metaheuristic optimization algorithm proposed by Yu et al. [46]. It is inspired by the behaviors of bounty hunter organizations during target pursuit, including intelligence collection, task deployment, collaborative division of labor, and encirclement redistribution. Unlike traditional position update methods that rely on the population mean, centroid, or a centralized combination of leaders for guidance, BHO constructs a decentralized search framework. In this framework, individual updates mainly depend on local differences, random perturbations, and representative reference solutions, such as the global best solution, the group best solution, and the individual historical best solution, rather than the average position of the population. This design can effectively alleviate center bias and premature population collapse. Meanwhile, BHO further introduces the Explorpolis rule and the Quantum Probability Rotation Selection (QPRS) mechanism to enhance population diversity and achieve a dynamic balance between exploration and exploitation. Overall, BHO mainly includes several key steps, such as oversampling initialization, clustering-based grouping, adaptive search mode switching, fine search, coarse search, Explorpolis screening, inferior individual redistribution, and QPRS-based inter-group selection. The mathematical formulation of the original BHO in this subsection follows the notation of Yu et al. [46], with several symbols slightly reorganized for consistency.

(1)

Clustering Initialization

Let the population size be $N$ and the problem dimension be $D$. BHO first generates 10N candidate solutions in the search space to form an oversampled candidate pool $X_o$:

$$X_o=\left[\begin{array}{c}{X}_{o_1}\\ X_{o_2}\\ ⋮\\ X_{o_{10N}}\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{x}}_{o_{\mathrm{1,1}}}& {\mathrm{x}}_{o_{\mathrm{1,2}}}& \dots & {\mathrm{x}}_{o_{1,D}}\\ {\mathrm{x}}_{o_{\mathrm{2,1}}}& {\mathrm{x}}_{o_{\mathrm{2,2}}}& \dots & {\mathrm{x}}_{o_{2,D}}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{x}}_{o_{10N,1}}& x_{o_{10N,2}}& \dots & {\mathrm{x}}_{o_{10N,D}}\end{array}\right]$$

(1)

On this basis, K-means clustering is used to divide $X_o$ into $N$ clusters, and the centroid of each cluster is taken as the initial population $X$:

$$X=\left[\begin{array}{c}{X}_1\\ X_2\\ ⋮\\ X_N\end{array}\right]=\left[\begin{array}{cccc}{x}_{1,1}& x_{1,2}& \dots & x_{1,D}\\ x_{2,1}& x_{2,2}& \dots & x_{2,D}\\ ⋮& ⋮& \ddots & ⋮\\ x_{N,1}& x_{N,2}& \dots & x_{N,D}\end{array}\right]$$

(2)

Subsequently, according to the spatial positional relationships among individuals, the population is further divided into $N_{group}$ search groups to improve initial coverage and reduce redundant sampling.

In the standard BHO, the initial population is usually generated by uniform random initialization. Although this method is simple to implement and has low computational cost, it may easily cause uneven distribution, individual aggregation, and insufficient diversity in complex high-dimensional optimization problems. These problems weaken the early global exploration ability of the algorithm, make the search process prematurely biased toward local regions, and delay the algorithm from entering an effective optimization state.

(2)

Adaptive Search Mode Switching

In each iteration, BHO adaptively assigns search roles according to the relative ranking of each individual within its subgroup. Let $s_i^k\in \left\{0,\left.1\right\}\right.$ denote the search mode of the i-th individual at the k-th iteration, where $s_i^k=1$ indicates fine search and $s_i^k=0$ indicates coarse search. Let $R_{h,i}^k$ denote the ranking of the individual within its subgroup, and let $r_h^k$ be the mode-switching threshold. Then, we have:

$$s_i^k=1\{R_{h,i}^k\le r_h^k\}$$

(3)

To allow the search process to adaptively adjust the ratio between exploration and exploitation according to the current optimization state, BHO uses a self-feedback mechanism to update the threshold $r_h^k$:

$$r_h^k=\left\{\begin{array}{ll}0.5,& k=1\\ r_h^{k-1}+{\displaystyle \sum _{i\in M_t}}{score}_i^{k-1}-{\displaystyle \sum _{i\in M_r}}{score}_i^{k-1},& k>1\end{array}\right.$$

(4)

where $M_t$ and $M_r$ denote the sets of individuals currently in the fine search mode and the coarse search mode, respectively. $c_i^k$ denotes the individual update score, which is defined as follows:

$$c_i^k=\left\{\begin{array}{ll}-{\displaystyle \frac{1}{N\cdot k_{max}}},& F_i^k\le F_i^{k-1}\\ {\displaystyle \frac{1}{N\cdot k_{max}}},& F_i^k>F_i^{k-1}\end{array}\right.$$

(5)

where $F_i^k$ is the objective function value of the $i$-th individual at the $k$-th iteration, and $k_{max}$ is the maximum number of iterations. This mechanism allows the algorithm to enhance exploration when the search stagnates and strengthen exploitation when a better region is found.

(3)

Fine Search Strategy

For individuals with higher rankings, BHO adopts a fine search strategy and jointly uses three reference solutions, namely the global best solution, the group best solution, and the individual historical best solution, for local exploitation. The position update formula is given as follows:

$$X_j^{cand}=X_{i,j}^k+m_i^k\left[c_{1_i}^k\cdot rand\cdot \left(X_{best,j}^k-X_{i,j}^k\right)+c_{2_i}^k\cdot rand\cdot \left(X_{best,j}^{{k,team}_l}-X_{i,j}^k\right)+c_{3_i}^k\cdot rand\cdot \left(X_{best,j}^{{k,team}_l}-X_{i,j}^k\right)\right]$$

(6)

where $j=\mathrm{1,2}\dots D.$ $rand\sim U\left(\mathrm{0,1}\right)$ denotes an independently generated random number for each term or dimension. $X_j^{cand}$ denotes the candidate solution generated by the current fine or rough search operation. $X_{best}^k$ denotes the global best solution at the k-th iteration, $X_{best,j}^{{k,team}_l}$ denotes the current best solution of the $l$-th group, and $X_i^{best}$ denotes the historical best solution of the $i$-th individual. $c_{1_i}^k$, $c_{2_i}^k$, and $c_{3_i}^k$ are the adaptive reference factors associated with the global best, group best, and individual historical best solutions, and $m_i^k$ is the movement factor used to scale the overall step size. Furthermore, let the mean values of the three reference weights be defined as follows:

$${\overline{c}}_i^k={\displaystyle \frac{c_{1_i}^k+c_{2_i}^k+c_{3_i}^k}{3}}$$

(7)

Here, ${\overline{c}}_i^k$ is only the average of the three adaptive reference factors. The actual updated parameters are $c_{j_i}^k\left(j=\mathrm{1,2},3\right)$, which correspond to the weights associated with the global best, group best, and individual historical best positions, respectively. Following the original BHO formulation, the update of $c_{j_i}^k$ is performed according to the previous value $c_{j_i}^{k-1}$, the relative rank ${R_h}_i^k$, and the comparison between $c_{j_i}^{k-1}$ and ${\overline{c}}_i^{k-1}$. Then, the weight update rule given in the original paper can be organized as follows using ${R_h}_i^k$. Following the original BHO formulation, $c_i^k$ denotes the update score of the $i$-th individual at the $k$-th iteration, which is used to measure whether the individual obtains an improvement after position updating. In this paper, $c_i^k$ follows the same meaning as the individual update score defined in Equation (5), and it is used to adaptively adjust the reference factors in the fine search stage:

$$c_i^k=\left\{\begin{array}{c}\begin{array}{ll}\mathit{min}\left\{c_i^{k-1}+{\displaystyle \frac{{R_h}_i^k}{10}},0.8\right\},& c_i^k\ge \overline{c_i^k}, {R_h}_i^k\le 0.2\\ \mathit{max}\left\{c_i^{k-1}-{\displaystyle \frac{{R_h}_i^k}{10}},0.2\right\},& c_i^k<\overline{c_i^k}, {R_h}_i^k\le 0.2\\ c_i^k,& 0.2<{R_h}_i^k<0.8 \end{array}\\ \begin{array}{ll}\mathit{min}\left\{c_i^{k-1}+{\displaystyle \frac{{R_h}_i^k}{10}},0.8\right\},& c_i^k\ge \overline{c_i^k}, {R_h}_i^k\ge 0.8\\ \mathit{max}\left\{c_i^{k-1}-{\displaystyle \frac{{R_h}_i^k}{10}},0.2\right\},& c_i^k<\overline{c_i^k}, {R_h}_i^k\ge 0.8\end{array}\end{array}\right.$$

(8)

(4)

Rough Search Strategy

For individuals with lower rankings, BHO adopts a rough search strategy to enhance global exploration ability. First, a reference solution $X_r^{{k,team}_l}$ is randomly selected from the corresponding group. If this reference solution is better than the current individual, the individual moves toward it. Otherwise, the individual moves away from it to expand the search range. The update formula is given as follows:

$$X_{i,j}^{k,rough}=\left\{\begin{array}{ll}{X}_{i,j}^k+rand\cdot \left(X_{r,j}^{{k,team}_l}-X_{i,j}^k\right),& F_i^k\le F_i^{{k,team}_l}\\ X_{i,j}^k-rand\cdot \left(X_{r,j}^{{k,team}_l}-X_{i,j}^k\right),& F_i^k>F_i^{{k,team}_l}\end{array}\right.$$

(9)

where $rand$, ${rand}_1$ and ${rand}_2$ are independent random numbers uniformly distributed in $\left[\mathrm{0,1}\right]$. In Equation (9), $rand$ controls the movement amplitude toward or away from the randomly selected reference solution $X_r^{{k,team}_l}$. In Equation (10), ${rand}_1$ controls the radius of the local random diffusion around $X_i^{k,rough}$, while ${rand}_2$ determines the positive or negative perturbation direction. After obtaining the rough moving position, a local random diffusion operation is further performed:

$$X_j^{cand}=\left\{\begin{array}{ll}{X}_{i,j}^{k,rough}+m_i^k\cdot {rand}_1\cdot X_{i,j}^{k,rough},& {rand}_2>0.5\\ X_{i,j}^{k,rough}-m_i^k\cdot {rand}_1\cdot X_{i,j}^{k,rough},& {rand}_2\le 0.5\end{array}\right.$$

(10)

This mechanism can effectively enhance the spatial diffusion ability of individuals and reduce the risk of falling into local optima.

(5)

Explorpolis Rule

When the new solution $X_i^{k}test$ is not directly better than the old solution, BHO does not simply discard it. Instead, the Explorpolis rule is introduced to determine whether this solution has structural value for retention. In the Explorpolis rule, the elite solution refers to a solution that has been admitted into the elite set $S$ according to the current fitness ordering and diversity criterion. Let $X^{cand}$ denote the candidate solution generated after fine search or rough search, and let $X^{elite}\in S$ denote an admitted elite solution. The normalized distance between them is calculated as:

$$TD\left(X^{cand},X^{elite}\right)={\displaystyle \frac{\sqrt{{{\displaystyle {\sum }_{j=1}^D}\left(\frac{X_j^{cand}-X_j^{elite}}{U_j-L_j}\right)}^2}}{D}}$$

(11)

where $U_j$ and $L_j$ denote the upper and lower bounds of the $j$-th dimension, respectively. This formulation is equivalent to the distance definition in the original BHO, but the notation is rewritten to distinguish the candidate solution from the admitted elite solution more clearly.

Furthermore, let $S\left\{x_i\right\}$ denote the current elite solution set. Then, the nearest normalized distance from the candidate solution to the elite set is defined as follows:

$$TD(X^{cand},S)=\underset{X^{\mathit{elite}}\in \mathrm{S}}{\min }TD\left(X^{cand},X^{elite}\right)$$

(12)

where $S$ denotes the elite set admitted by the Explorpolis rule, $X^{cand}$ denotes the candidate solution generated by the current update, and $X^{elite}$ denotes an elite solution in $S$. $U=\left(U_1,\dots ,U_D\right)$ and $L=\left(L_1,\dots ,L_D\right)$ denote the upper and lower bound vectors of the search space, respectively. Therefore, $TD(X^{cand},S)$ represents the nearest normalized Euclidean distance from the candidate solution to the admitted elite set.

If the candidate solution is better than the current solution, it is directly accepted. If it is not better than the current solution, but there is no solution in the elite set that is both better and closer, the candidate solution can still be accepted to maintain structural diversity in the population. This mechanism can suppress homogeneous accumulation of solutions and thereby improve global search ability.

(6)

Inferior Individual Redistribution and QPRS Mechanism

When an individual remains at the bottom of its subgroup for several consecutive iterations, it may indicate that the individual has fallen into an inefficient search region. For this reason, BHO sets a worst-performance counting threshold, denoted as MPPC. When the continuous inferior performance of an individual reaches this threshold, it is reassigned to a better group. After redistribution, the update formula of the individual is given as follows:

$$X_{i,j}^{k,test}=X_{i,j}^k+m_i^k\left[c_{1_i}^k\cdot rand\cdot \left(X_{best,j}^k-X_{i,j}^k\right)+c_{2_i}^k\cdot rand\cdot (X_{best,j}^{k,{team}_{new}}-X_{i,j}^k)+c_{3_i}^k\cdot rand\cdot (X_{better,j}^{best,{team}_{new}}-X_{i,j}^k)\right]$$

(13)

where $X_{best,j}^{k,{team}_{new}}$ is the current best solution of the new group, and $X_{better,j}^{best,{team}_{new}}$ is the historical best solution of another member in the new group that is better than the current individual. Following the original BHO design, $X_{better}^{best,{team}_{new}}$ denotes the historical best position of another individual in the new group whose performance is better than that of the reassigned individual. The historical best solution is used instead of simply selecting the first current individual that is better than $i$, because it provides more stable and accumulated search experience in the new group. This design allows the reassigned individual to quickly adapt to a promising region while avoiding excessive dependence on a single instantaneous position.

To achieve smooth inter-group selection, BHO further adopts the Quantum Probability Rotation Selection (QPRS) mechanism. Let each candidate group be represented by a qubit, whose probability amplitudes are (α, β), satisfying:

$${\alpha }^2+{\beta }^2=1$$

(14)

In QPRS, ${\alpha }^l$ and ${\beta }^l$ denote the probability amplitudes of group $l$. satisfying ${\left({\alpha }^l\right)}^2+{\left({\beta }^l\right)}^2=1$. At initialization, the amplitude ${\alpha }^{1,\mathrm{l}}$ is calculated from the normalized cumulative contribution of group $l$:

$${\alpha }^{1,\mathrm{l}}={\displaystyle \frac{C^{1,l}}{\sum C^{1,l}}}$$

(15)

Then, ${\beta }^{1,\mathrm{l}}$ is obtained from the normalization condition as:

$${\beta }^{1,\mathrm{l}}=\sqrt{1-{\left({\alpha }^{1,\mathrm{l}}\right)}^2}$$

(16)

In subsequent iterations, ${\alpha }^l$ and ${\beta }^l$ are continuously updated by the quantum rotation gate. Therefore, $\alpha$ in Equation (14) denotes the general amplitude state, while ${\alpha }^{1,\mathrm{l}}$ denotes its initial value for group $l$.

In Equation (15), $C^{k,l}$ denotes the cumulative contribution of the l-th group at the k-th iteration, defined as the sum of the cumulative contributions of its members, and it decays over iterations with a fading factor $\rho$, it is updated as follows:

$$C^{k,l}=\left\{\begin{array}{ll}0,& k=0\\ (1-\rho )C^{k,l}+{\displaystyle \sum _{i\in B}}{c}_i^k,& k>0\end{array}\right.$$

(17)

where $\rho \in \left[\mathrm{0,1}\right]$ is the contribution decay factor, and $B$ denotes the set of members in this group. The individual contribution is defined as follows:

$$c_i^k=\left\{\begin{array}{ll}0,& k=0\\ 1-{\displaystyle \frac{F_i^k}{\sum F_i^k}},& k>0\end{array}\right.$$

(18)

Then, the probability amplitudes are updated using a quantum rotation gate:

$$\left[\begin{array}{c}\alpha \prime \\ \beta \prime \end{array}\right]=U_{\beta }(\theta )\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{cc}\mathit{cos}\theta & \mathit{sin}\theta \\ \mathit{sin}\theta & \mathit{cos}\theta \end{array}\right]\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]$$

(19)

where $\theta$ is the rotation angle. QPRS replaces the traditional discrete roulette-wheel selection with continuous probability amplitude updating, making the selection pressure smoother and more controllable.

The rotation angle $\theta$ is not fixed manually. Following the original BHO, it is determined according to the QPRS rotation-angle update rules, which depend on the group selection state, the comparison between the current group contribution and the best group contribution, and the signs of ${\alpha }^l{\beta }^l$. The possible values of $\theta$ include $0, +0.01\pi , -0.01\pi and \pm 0.01\pi$. The initial amplitudes ${\alpha }^{1,l}$ and ${\beta }^{1,l}$ are calculated from the group contribution and the normalization condition ${\alpha }^2+{\beta }^2=1$, and then they are updated iteratively through the rotation gate.

(7) Step-Size Shrinkage Mechanism

To balance global exploration in the early stage and local exploitation in the later stage, BHO adopts an iteration-dependent step-size decay mechanism:

$$m_i^k=1-{\displaystyle \frac{k}{2k_{max}}}\times {\displaystyle \frac{F_i^k}{\sum F_i^k}}$$

(20)

As the number of iterations increases, the step size gradually decreases, and the search radius continuously shrinks. As a result, the algorithm has stronger exploration ability in the early stage and becomes more focused on fine exploitation around promising regions in the later stage.

The main characteristics of BHO are reflected in the following three aspects. First, it adopts a decentralized update strategy, which avoids the center bias caused by mean-guided search. Second, it preserves structurally complementary solutions through the Explorpolis rule, thereby enhancing population diversity. Third, it uses QPRS instead of traditional roulette-wheel selection to achieve smoother inter-group migration and more controllable selection pressure. Therefore, BHO can maintain population dispersion while achieving stable convergence, showing strong ability to solve complex optimization problems. The overall workflow of the original BHO algorithm is illustrated in Figure 1.

3.2. Proposed QCAMBHO

Although the original BHO has a certain ability to balance global exploration and local exploitation and shows good application potential for continuous optimization problems, it still has several limitations when dealing with complex high-dimensional and multimodal optimization problems. Specifically, the random initialization method may easily lead to uneven initial population distribution and insufficient population diversity. The random perturbation mechanism in the rough search stage lacks adaptive adjustment of the search step size, making it difficult to balance global exploration and local exploitation effectively. In the later stage of the algorithm, as the population gradually gathers around promising regions, the differences among individuals continuously decrease. This may result in premature convergence and local stagnation, thereby affecting the convergence accuracy and overall optimization performance of the algorithm. To address these limitations of the original BHO in terms of initialization quality, search step-size control, and the ability to escape from local optima in the later stage, this paper proposes an improved BHO algorithm that integrates quantum-computing-enhanced initialization, an adaptive Levy flight strategy, and an adaptive differential operator. The proposed algorithm aims to improve population diversity, enhance global exploration ability, and improve fine search performance in the later stage.

3.2.1. Quantum-Computing-Enhanced Initialization Strategy

To further improve the spatial distribution quality of the initial population, this study introduces a quantum-computing-enhanced initialization strategy after the original BHO initialization process. The purpose of this strategy is to enrich the initial search information, reduce the dependence on purely random sampling, and provide more diverse and promising starting points for subsequent optimization.

In quantum computation theory, a quantum bit can be represented as a superposition of two basic states:

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

(21)

where $|0⟩$ and $|1⟩$ denote the two basis states of a qubit, and $\alpha$ and $\beta$ are the corresponding probability amplitudes:

$$|{\alpha |}^2+|\beta {|}^2=1$$

(22)

This means that, before observation, the system does not correspond to only a single deterministic state, but can contain information from multiple potential states at the same time. Inspired by this idea, this paper first generates the initial population according to the standard BHO method. Let the population size be N, the problem dimension be D, and the position of the i-th individual in the j-th dimension be denoted as $x_{ij}$. Then, the initial position can be expressed as follows:

$$x_{ij}=l_j+r_{ij}\left(u_j-l_j\right),i=\mathrm{1,2},\dots ,N,j=\mathrm{1,2},\dots ,D,r_{ij}\in \left[\mathrm{0,1}\right]$$

(23)

where $l_j$ and $u_j$ denote the lower and upper bounds of the search space in the (j)-th dimension, respectively, and $r_{ij}\in$ [0, 1] is a uniformly distributed random number.

After obtaining the standard initial population, this paper further maps the individual positions from the classical solution space to the quantum angle space. Specifically, for each position component $x_{ij}$, its corresponding quantum angle is constructed as follows:

$${\theta }_{ij}=arccos\left(2r_{ij}-1\right)=\mathit{arccos}\left(2{\displaystyle \frac{x_{ij}-l_j}{u_j-l_j}}-1\right)$$

(24)

When $r_{ij}\in$ [0, 1], the term $2r_{ij}-1$ always lies in [−1, 1], which ensures the validity of the inverse cosine mapping in Equation (24). Through the above mapping, the original individual can be transformed from classical position encoding into quantum phase parameters, thereby providing a basis for generating diversified candidate solutions. Then, quantum-enhanced individuals are constructed using trigonometric mapping relationships. In this paper, the following two mapping forms are adopted:

$${\tilde{x}}_{ij}^{\left(1\right)}={\displaystyle \frac{1}{2}}\left(1+\mathit{cos}{\theta }_{ij}\right)(u_j-l_j)+l_j$$

(25)

$${\tilde{x}}_{ij}^{\left(2\right)}={\displaystyle \frac{1}{2}}\left(1+\mathit{sin}{\theta }_{ij}\right)(u_j-l_j)+l_j$$

(26)

where ${\tilde{x}}_{ij}^{\left(1\right)}$ and ${\tilde{x}}_{ij}^{\left(2\right)}$ denote the candidate positions generated by the cosine mapping and sine mapping, respectively. Compared with traditional random initialization, this mechanism can derive alternative solutions from the original individuals, thereby significantly expanding the distribution range of the initial population.

To further enhance the perturbation ability in the quantum space, this paper also introduces a quantum rotation gate mechanism to update the quantum angle, which is formulated as follows:

$$\left[\begin{array}{c}\cos {\theta }_{ij}^{new}\\ \sin {\theta }_{ij}^{new}\end{array}\right]=\left[\begin{array}{cc}\cos \mathsf{\Delta }\theta & -\sin \mathsf{\Delta }\theta \\ \sin \mathsf{\Delta }\theta & \cos \mathsf{\Delta }\theta \end{array}\right]\left[\begin{array}{c}\cos {\theta }_{ij}\\ \sin {\theta }_{ij}\end{array}\right]$$

(27)

where Δθ denotes the rotation angle, which controls the perturbation intensity in the quantum angle space. The lower-right element of the rotation matrix is $+\cos \mathsf{\Delta }\theta$, which corresponds to the standard two-dimensional quantum rotation gate. After the rotation perturbation, new candidate individuals can be generated again using the above sine and cosine mappings, which enhances the dispersion and coverage of the initial population. Finally, the original population and the enhanced individuals generated by quantum mapping are merged and sorted according to their fitness values. The top $N$ superior individuals are retained as the final initial population of BHO.

It should be noted that the proposed quantum-computing-enhanced initialization strategy only modifies the population construction stage of BHO. After the enhanced candidate pool is generated and ranked, the top $N$ individuals are selected as the initial population. The subsequent search process still follows the original BHO framework, except for the adaptive mechanisms introduced in the rough search and later-stage exploitation phases. Therefore, this strategy improves the starting quality and diversity of the population without changing the basic population size and iterative structure of BHO.

3.2.2. Adaptive Levy Flight Strategy

In the original BHO, the Rough Search stage mainly relies on conventional random perturbation to perform a preliminary scan of the search space. Although this method has a certain degree of randomness, its step-size variation is relatively simple. In complex high-dimensional and multimodal problems, it may easily appear as disordered fluctuations within local neighborhoods, with insufficient cross-region search ability. Meanwhile, random perturbation lacks adaptive adjustment with different iteration stages, making it difficult to achieve a smooth transition from global exploration to local exploitation.

Levy flight is a random walk mechanism with a heavy-tailed distribution. It can perform small-step movements in most cases while generating long-distance jumps with a low probability. Therefore, it has been widely used in metaheuristic optimization to enhance global search ability. The individual step size is controlled by the adaptive Levy weight to simulate the exploration and exploitation intensities at different stages of the algorithm. The Levy weight can be expressed as follows:

$$w_j={\displaystyle \frac{\left|u_j\right|}{{\left|v_j\right|}^{1/\beta }}}$$

(28)

where $u_j\sim N(0,{\sigma }^2),v_j\sim N\left(\mathrm{0,1}\right)$

$$\sigma ={\left({\displaystyle \frac{\mathsf{\Gamma }\left(\frac{1+\beta }{2}\right)\cdot \beta \cdot 2^{\frac{\beta -1}{2}}}{\mathsf{\Gamma }(1+\beta )\cdot \sin \left(\frac{\pi \beta }{2}\right)}}\right)}^{1/\beta }$$

(29)

where $\mathsf{\Gamma }(\cdot )$ denotes the Gamma function, and $\beta$ is a key parameter that controls the shape of the Levy distribution. Unlike the fixed step-size mechanism, $\beta$ is dynamically adjusted during the iteration process in this strategy, and its expression is given as follows:

$$\beta ={\beta }_{base}+0.5\sin \left({\displaystyle \frac{\pi t}{2T}}\right)$$

(30)

where ${\beta }_{base}$ is the initial parameter, $t$ denotes the current iteration number, and $T$ denotes the maximum number of iterations. This dynamic adjustment mechanism makes the algorithm tend to use larger jump steps in the early search stage to enhance global exploration. In the later search stage, it gradually shifts to smaller steps to improve local exploitation accuracy. Thus, this mechanism can achieve a natural transition from larger exploratory steps to more precise descending steps. The comparison between the standard Levy flight and the adaptive Levy flight is illustrated in Figure 2.

In the proposed QCAMBHO, the adaptive Lévy flight strategy is embedded into the Rough Search stage of the original BHO, rather than being introduced as an additional independent search phase. To make this modification consistent with the notation used in Section 3.1 (4), let $X_i^k=\left(X_{i,1}^k,\cdots X_{i,D}^k\right)$ denote the $i$-th individual at the $k$-th iteration, and let $X_{r_i}^k$ denote the reference solution randomly selected from the same subgroup. In the original Rough Search strategy, the individual moves toward $X_{r_i}^k$ when the reference solution is better than the current individual; otherwise, it moves away from $X_{r_i}^k$ to expand the search range. QCAMBHO retains this search direction mechanism and only replaces the conventional uniform random perturbation term with an adaptive Lévy perturbation.

Specifically, the adaptive Lévy weight of the $j$-th dimension at the $k$-th iteration is denoted as $L_{i,j}^k$, which is generated according to Equations (28)–(30). Then, the Rough Search update in Equation (9) can be modified as follows:

$$X_{i,j}^{k,rough}={\{}_{X_{i,j}^k+L_{i,j}^k\left(X_{i,j}^k-X_{r_i,j}^k\right), \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}}^{X_{i,j}^k+L_{i,j}^k\left(X_{r_i,j}^k-X_{i,j}^k\right), \mathrm{i}\mathrm{f} f\left(X_{r_i}^k\right)<f\left(X_i^k\right) }$$

(31)

where $j=\mathrm{1,2},\dots ,D$. This formulation keeps the original Rough Search logic of BHO unchanged: a better reference individual guides the current individual, while an inferior reference individual induces an opposite movement to enlarge the search region. The main difference is that the movement amplitude is no longer controlled by a simple uniformly distributed random number, but by the adaptive Lévy weight $L_{i,j}^k$.

After obtaining $X_{i,j}^{k,rough}$, the local diffusion step in Equation (10) is also rewritten using the same notation. The candidate solution is generated as:

$$X_{i,j}^{k,test}=X_{i,j}^{k,rough}+L_{i,j}^k\cdot {rand}_{2,j}\cdot \left(X_{i,j}^{k,rough}-X_{i,j}^k\right)$$

(32)

where ${rand}_{2,j}\sim U\left(\mathrm{0,1}\right)$. This step preserves the local random diffusion operation of the original Rough Search, but adaptively scales the diffusion intensity through $L_{i,j}^k$. Therefore, the proposed adaptive Lévy flight strategy modifies only the perturbation amplitude of the original Rough Search stage, while the reference-solution selection, movement direction, candidate generation, boundary handling, and greedy selection rules of BHO remain unchanged.

Through this modification, the Rough Search stage of BHO is transformed from a uniformly random perturbation process into an adaptive heavy-tailed search process. In the early stage of iteration, the Lévy perturbation can generate relatively larger jumps with a certain probability, which helps individuals move across different regions of the search space and reduces the risk of premature convergence. In the later stage, the adaptive parameter gradually weakens the perturbation intensity, allowing the algorithm to perform more stable local exploitation around promising regions. Thus, the proposed strategy improves the exploration capability of the original Rough Search stage while maintaining the basic search structure of BHO.

3.2.3. Adaptive Differential Operator

In the original BHO, as the iteration proceeds, population individuals gradually concentrate around the current promising regions, making the algorithm more inclined toward local exploitation in the later stage. However, when the population prematurely shrinks to a suboptimal region, individual differences and search directions gradually become similar. This may lead to a lack of effective cross-region jumping ability and easily cause search stagnation in complex multimodal or composite functions. Therefore, this paper introduces an adaptive differential operator in the later stage of the algorithm and embeds the mutation, crossover, and greedy selection mechanisms of differential evolution into the BHO framework to enhance the later-stage search activity.

For the individual $X_i^t$ in the $t$-th generation, three mutually different individual indices $r_1,r_2,r_3$ are first randomly selected from the current population, satisfying $r_1\ne r_2\ne r_3\ne i$. Then, the mutation vector is constructed as follows:

$$M_i^{t+1}=X_{r_1}^t+F\left(X_{r_2}^t-X_{r_3}^t\right)$$

(33)

$$F_m=0.9-0.5\times {\displaystyle \frac{iter}{maxIter}}$$

(34)

where $F$ is the differential scaling factor, which is used to adjust the amplitude of the differential perturbation. The above equation corresponds to the differential mutation formula in the reference. Its core idea is to use the difference information among individuals in the population to generate new search directions, thereby overcoming the directional monotonicity caused by traditional local updating. After obtaining the mutation vector, a crossover operation is further performed to construct the trial vector $u_i^{t+1}$. The update rule for its $j$-th dimension is given as follows:

$$u_{i,j}^{t+1}=\left\{\begin{array}{ll}{m}_{i,j}^{t+1},& if ran{d}_j<CR or j=j_{rand}\\ x_{i,j}^t,& otherwise\end{array}\right.$$

(35)

$$CR=0.2+0.7\times {\displaystyle \frac{iter}{maxIter}}$$

(36)

where CR $\in$ [0, 1] is the crossover probability, and $j_{rand}$ is a randomly selected dimension index used to ensure that the trial vector inherits at least one dimension from the mutation vector. This mechanism allows the new individual to retain part of the effective information of the original individual while absorbing the new structure generated by differential mutation, thereby improving the diversity and recombination ability of candidate solutions. This equation corresponds to the crossover formula in the reference. After the trial vector is constructed, its fitness value is evaluated, and a greedy selection mechanism is used to retain the better individual. For a minimization problem, the selection rule can be written as follows:

$$X_i^{t+1}=\left\{\begin{array}{ll}{U}_i^{t+1},& iff\left(U_i^{t+1}\right)\le f\left(X_i^t\right)\\ X_i^t,& otherwise\end{array}\right.$$

(37)

That is, when the trial vector is better than the original individual, it replaces the original individual. Otherwise, the original solution remains unchanged. This strategy ensures that the differential recombination process does not destroy existing high-quality solutions. Instead, it further improves the population while maintaining its stability. This equation corresponds to the greedy selection rule in the reference.

In the improved BHO proposed in this paper, the adaptive differential operator is arranged in the later stage of the algorithm to enhance fine search ability and alleviate local stagnation. It should be emphasized that the DE operator is not executed simultaneously with the Explorpolis rule or the QPRS mechanism. In each iteration, QCAMBHO first completes the regular BHO update procedure, including fine search or rough search, Explorpolis-based candidate retention, inferior individual redistribution, QPRS-based group selection, boundary handling, and greedy selection. After the updated population is obtained, if the iteration enters the later stage, the adaptive DE operator is applied to the current population as a post-update refinement step. Therefore, the adaptive differential operator is not a modification of the inferior individual redistribution or QPRS mechanism in Part (6) of Section 3.1. Instead, it is an additional later-stage refinement module executed after the regular BHO update process has been completed. In this way, the later-stage search of BHO no longer relies only on continuous local exploitation, but introduces a structural recombination process based on population difference information. In an abstract form, the later-stage enhanced update can be written as follows:

$$X_i^{t+1}=\mathrm{D}\mathrm{E}(X_i^t,X_{r_1}^t,X_{r_2}^t,X_{r_3}^t,F,CR)$$

(38)

where DE(·) denotes the differential evolution operator composed of mutation, crossover, and greedy selection. If the adaptive mechanism is further considered, CR or F can also be designed as dynamically changing parameters with the iteration process. In this way, the algorithm can maintain stronger perturbation ability at the beginning and later stages and further enhance fine exploitation ability before termination.

The adaptive differential operator uses difference information among individuals to construct new search directions and integrates mutation information with the original individual structure through the crossover mechanism, thereby generating more promising candidate solutions. This recombination process can improve population diversity and provide cross-region jumping ability for the algorithm, which compensates for the limited continuous local updating ability of the original BHO in the later stage. Meanwhile, the greedy selection mechanism ensures that new solutions are retained without reducing the population quality, thereby further improving the later-stage optimization efficiency and stability of the algorithm. The pseudocode of QCAMBHO is shown in Algorithm 1.

Algorithm 1 QCAMBHO

Initialize population X by the standard BHO scheme via Equation (22) Enhance the initial population using quantum mapping and rotation via Equations (23)–(26) Evaluate fitness of all individuals and determine $X_{best}$ Set t = 1 while t ≤ T do        Divide the population into elite individuals and explorer individuals        for each individual $X_i$ do               if $X_i$ is elite then                     Update $X_i$ by the original BHO exploitation mechanism               else                     Replace the random disturbance in Rough Search with adaptive Levy flight                     Compute adaptive Levy weight via Equations (27)–(29)                     Update $X_i$ in Rough Search via Equations (30)–(31)               end if               Apply boundary control and greedy selection        end for        if t enters the late stage then               Compute adaptive DE parameters F and CR via Equations (33) and (35)               for each individual $X_i$ do                     Generate mutant vector via Equation (32)                     Generate trial vector via Equation (34)                     Update $X_i$ by greedy selection via Equation (36)               end for        end if        Update $X_{best}$ and record Convergence(t); t = t + 1 end while return $X_{best}$$, F_{best}$, Convergence

3.3. Time Comparison Analysis of BHO and QCAMBHO

Execution time analysis was further conducted to evaluate the computational cost of QCAMBHO compared with the original BHO. Under the same experimental settings, including a population size of 30, a maximum iteration number of 100, and 30 independent runs on the CEC2017 30D test suite, QCAMBHO requires more running time than BHO on all 30 benchmark functions. All experiments are conducted on a computer equipped with an Intel Core i5-1135G7 2.40 GHz processor and 16 GB of memory. The experimental platform is MATLAB 2021a. As shown in

Table 1. Execution time comparison between BHO and QCAMBHO on CEC2017 (30D).

Algorithm	F1	F2	F3	F4	F5	F6
BHO	5.0032	4.4125	3.9555	4.2645	4.0743	5.3194
QCAMBHO	5.8721	5.4189	4.7558	5.2820	5.9945	5.9778
Algorithm	F7	F8	F9	F10	F11	F12
BHO	4.3566	4.2716	4.3799	4.4405	4.1313	4.5465
QCAMBHO	5.7317	6.1479	5.3691	4.8209	5.0039	5.5258
Algorithm	F13	F14	F15	F16	F17	F18
BHO	4.3542	4.9157	4.2170	3.8678	4.6570	4.5148
QCAMBHO	4.8265	5.3637	5.0689	5.4596	5.8412	5.6948
Algorithm	F19	F20	F21	F22	F23	F24
BHO	5.6267	4.5516	4.5398	5.1230	5.2220	5.0927
QCAMBHO	6.7014	5.8220	5.8578	6.7054	6.1835	6.2793
Algorithm	F25	F26	F27	F28	F29	F30
BHO	1.9526	2.3578	2.9199	2.3202	2.0432	2.3255
QCAMBHO	2.5756	3.1621	3.5482	2.9974	2.5686	3.0160

, QCAMBHO requires more running time than BHO for all 30 functions. On average, the execution time increases from 4.1252 s for BHO to 5.1191 s for QCAMBHO, corresponding to an increase of approximately 24.09%. This additional cost mainly results from the quantum-computing-enhanced initialization, adaptive Lévy flight perturbation, and later-stage adaptive differential operator.

Nevertheless, the increase in computational cost is moderate. Considering that QCAMBHO obtains better optimization accuracy and stability on most benchmark functions, the additional running time represents an acceptable trade-off between computational cost and solution quality. Therefore, the proposed improvement mechanisms enhance the search performance of BHO at the cost of a limited increase in execution time.

4. Numerical Experiments

This section mainly investigates the convergence characteristics of QCAMBHO and systematically compares its performance with nine other optimization algorithms based on two standard benchmark suites, CEC2017 and CEC2022. For the CEC2017 test suite, experiments are conducted in 30-dimensional, 50-dimensional, and 100-dimensional search spaces. For the CEC2022 test suite, tests are performed under 10-dimensional and 20-dimensional conditions. To reduce the influence of randomness on the experimental results, each algorithm is independently run multiple times, and the mean and standard deviation of the optimization results are recorded to comprehensively evaluate the solution accuracy and stability of the algorithms.

To further improve the statistical reliability of the experimental conclusions, this paper introduces two nonparametric statistical methods, namely the Wilcoxon rank-sum test and the Friedman test, to analyze the performance differences and overall rankings among the algorithms. Specifically, in the Wilcoxon rank-sum test, when the (p)-value is less than 0.05, it indicates that there is a significant difference between the comparison algorithm and QCAMBHO. When the (p)-value is greater than 0.05, it indicates that there is no statistically significant difference between them. Meanwhile, “+/=/−” is used to denote that QCAMBHO performs better than, similarly to, or worse than the comparison algorithm, respectively. For the Friedman test, the average rank of each algorithm on all test functions is first calculated, and the overall performance of the algorithms is then ranked according to the average rank results, so as to determine whether there are significant statistical differences among different algorithms.

4.1. CEC2017 and CEC2022 Test Suite

This section uses the CEC 2017 test suite with dimensions of 30, 50, and 100, and the CEC 2022 test suite with dimensions of 10 and 20, to evaluate the performance of the QCAMBHO algorithm. To conduct a more comprehensive evaluation, 30 benchmark functions from the CEC 2017 test suite, namely F1–F30, and 12 benchmark functions from the CEC 2022 test suite, namely F1–F12, are selected. In total, 42 test functions are used, as shown in

Table 2. CEC 2017 benchmark functions.

Type	ID	Description	Dim	fmin
Unimodal	F1	Shifted and Rotated Bent Cigar Function	30/50/100	100
	F2	Shifted and Rotated Sum of Different Power Function	30/50/100	200
	F3	Shifted and Rotated Zakharov Function	30/50/100	300
Multimodal	F4	Shifted and Rotated Rosenbrock’s Function	30/50/100	400
	F5	Shifted and Rotated Rastrigin’s Function	30/50/100	500
	F6	Shifted and Rotated Expanded Scaffer’s F6 Function	30/50/100	600
	F7	Shifted and Rotated Lunacek Bi_Rastrigin Function	30/50/100	700
	F8	Shifted and Rotated Non-Continuous Rastrigin’s Function	30/50/100	800
	F9	Shifted and Rotated Levy Function	30/50/100	900
	F10	Shifted and Rotated Schwefel’s Function	30/50/100	1000
Hybrid	F11	Hybrid Function 1 (N = 3)	30/50/100	1100
	F12	Hybrid Function 2 (N = 3)	30/50/100	1200
	F13	Hybrid Function 3 (N = 3)	30/50/100	1300
	F14	Hybrid Function 6 (N = 4)	30/50/100	1400
	F15	Hybrid Function 6 (N = 4)	30/50/100	1500
	F16	Hybrid Function 6 (N = 4)	30/50/100	1600
	F17	Hybrid Function 6 (N = 5)	30/50/100	1700
	F18	Hybrid Function 6 (N = 5)	30/50/100	1800
	F19	Hybrid Function 6 (N = 5)	30/50/100	1900
	F20	Hybrid Function 6 (N = 6)	30/50/100	2000
Composition	F21	Composition Function 1 (N = 5)	30/50/100	2100
	F22	Composition Function 2 (N = 5)	30/50/100	2200
	F23	Composition Function 3 (N = 5)	30/50/100	2300
	F24	Composition Function 4 (N = 5)	30/50/100	2400
	F25	Composition Function 5 (N = 3)	30/50/100	2500
	F26	Composition Function 6 (N = 3)	30/50/100	2600
	F27	Composition Function 7 (N = 5)	30/50/100	2700
	F28	Composition Function 8 (N = 5)	30/50/100	2800
	F29	Composition Function 9 (N = 5)	30/50/100	2900
	F30	Composition Function 10 (N = 3)	30/50/100	3000
		Search Range: [−100, 100] in each dimension

and

Table 3. CEC 2022 benchmark functions.

Type	ID	Description	Dim	fmin
Unimodal Function	F1	Shifted and full Rotated Zakharov Function	10/20	300
Basic Functions	F2	Shifted and full Rotated Zakharov Function	10/20	400
	F3	Shifted and full Rotated Expanded Schaffer’s f6 Function	10/20	600
	F4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	10/20	800
	F5	Shifted and full Rotated Levy Function	10/20	900
Hybrid Functions	F6	Hybrid Function 1 (N = 3)	10/20	1800
	F7	Hybrid Function 2 (N = 6)	10/20	2000
	F8	Hybrid Function 3 (N = 5)	10/20	2200
Composition Functions	F9	Composition Function 1 (N = 5)	10/20	2300
	F10	Composition Function 2 (N = 4)	10/20	2400
	F11	Composition Function 3 (N = 5)	10/20	2600
	F12	Composition Function 4 (N = 6)	10/20	2700
		Search Range: [−100, 100] in each dimension

.

According to their properties and testing objectives, these benchmark functions can be divided into different types. Unimodal functions contain only one global optimum and are mainly used to evaluate the convergence speed and solution accuracy of an algorithm. Multimodal functions contain multiple local optima and are mainly used to examine the ability of an algorithm to escape from local optima and its robustness. Hybrid functions and composition functions combine the characteristics of the above two types of functions and can more effectively simulate complex optimization problems, thereby further evaluating the global search ability and overall performance of an algorithm in complex search environments. All experiments are conducted on a computer equipped with an Intel Core i5-1135G7 2.40 GHz processor and 16 GB of memory. The experimental platform is MATLAB 2021a. To ensure a fair comparison among different algorithms, the maximum number of iterations is uniformly set to 500, and the population size is set to 30 in all numerical experiments in this section.

4.2. Algorithm Parameter Settings

Table 4. Parameter settings for different algorithms.

Algorithms	Name of the Parameter	Value of the Parameter
LSHADE	$p,\left|A\right|,H$	(0.11, 6, 4)
LSHADE_cnEpSin	$p,\left|A\right|,H$	(0.11, 5, 1.4)
TACPSO	$w_{max},w_{min},c_{1i},c_{1f},c_{2i},c_{2f}$	(0.9, 0.4, 2.5, 0.5, 0.5, 2.5)
MELGWO	$P_{min},a,C_1$	10; linearly decreases from 2 to 0; 2
EWOA	$a,b$	$a \mathrm{linearly} \mathrm{decreases} \mathrm{from} 2 \mathrm{to} 0;b=1$
HPHHO	$E_0$	[−1, 1]
AOO	$c$	[0, 1]
CFOA	$r_s$	[−1, 1]
BHO	$E$	[0, 1]
QCAMBHO	$F$	[0, 1]

presents the parameter settings of QCAMBHO and the comparison algorithms. To comprehensively evaluate the performance of the proposed algorithm in numerical optimization problems, several representative and advanced metaheuristic algorithms are selected for comparison, including LSHADE [61], LSHADE_cnEpSin [62], TACPSO [63], MELGWO [64], EWOA [65], HPHHO [66], AOO [67], CFOA [68], and BHO [46]. The parameter settings of these algorithms are mainly determined according to their original references or recommended configurations. For the proposed QCAMBHO, the parameters are set based on its algorithmic design mechanisms and preliminary experimental results, so as to ensure the fairness and reproducibility of the experimental comparison.

To ensure fairness and reproducibility, the parameter settings of all comparison algorithms are determined according to their original references or recommended configurations, rather than being manually tuned for the benchmark functions used in this study. All algorithms are tested under the same experimental conditions, including the same population size, maximum number of iterations, benchmark functions, dimensions, and number of independent runs. For the proposed QCAMBHO, the parameters are set based on its algorithmic design mechanisms and the sensitivity analysis reported in Section 4.6. The detailed parameter settings are listed in Table 4.

4.3. Convergence Behavior Evaluation

To verify the effectiveness of the proposed QCAMBHO algorithm, this paper further plots its convergence graphs on different benchmark functions and analyzes its optimization process from several perspectives, including search distribution, fitness evolution, individual trajectory changes, and convergence curves. Figure 3 illustrates the convergence behavior of QCAMBHO when solving representative numerical optimization problems from the CEC benchmark suite. In these experiments, each benchmark function is treated as a minimization problem, and the explicit optimization objective is to search for the optimal decision vector $X^{*}=\left(x_1^{*},x_2^{*},\cdots ,x_D^{*}\right)$ within the predefined boundary constraints so that the objective function value $f\left(X\right)$ is minimized:

$$X^{*}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{X\in \left[L,U\right]}{\min }f\left(X\right)$$

(39)

where $X$ denotes a candidate solution, $L$ and $U$ represent the lower and upper bounds of the search space, respectively. Therefore, Figure 3 illustrates how QCAMBHO updates the population distribution, fitness values, individual trajectories, and convergence curves during the process of solving these benchmark minimization problems.

The results show that QCAMBHO exhibits good search ability and convergence characteristics on different types of functions. Its search points gradually move from a dispersed distribution to promising regions, the average fitness decreases rapidly with iterations and then becomes stable, and the individual trajectories gradually converge to potential optimal regions through dynamic adjustment. Compared with the original BHO, QCAMBHO also shows faster convergence speed and better final solution accuracy. The optimization process of QCAMBHO on complex functions shows both strong global exploration ability and later-stage local exploitation ability. In the early stage of iteration, the algorithm can maintain a wide search range to improve the effective exploration of the solution space. As the iteration proceeds, the population gradually concentrates around high-quality regions, showing good search balance and stability. This indicates that the multi-strategy mechanism introduced in this paper can effectively improve the search behavior of the original BHO, thereby enhancing its overall performance in complex numerical optimization problems.

4.4. Quantitative Analysis

Comparison with other competitive algorithms on CEC 2017:

This section evaluates the performance of the proposed QCAMBHO algorithm and other comparison algorithms on the CEC 2017 test suite under three different dimensional settings, namely 30, 50, and 100 dimensions.

Table 5. Experimental results of 10 algorithms on the CEC 2017 (Dim = 30).

Algorithm	Metric	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO	QCAMBHO
F1	Mean	4.8022 × 107	6.0805 × 106	3.4471 × 107	1.9155 × 109	1.6091 × 107	2.4203 × 109	1.0997 × 106	6.0332 × 108	5.8989 × 103	4.8457 × 103
	Std	9.7965 × 107	8.1445 × 106	1.8743 × 108	2.2546 × 109	2.2751 × 107	1.2267 × 109	5.8875 × 105	2.2917 × 108	5.3220 × 103	5.5971 × 103
F2	Mean	2.9524 × 1027	4.9061 × 1026	6.2854 × 1029	1.9238 × 1031	4.1165 × 1024	1.0000 × 1020	2.3149 × 1020	2.7467 × 1027	1.9863 × 1014	9.5753 × 1010
	Std	1.1279 × 1028	2.5908 × 1027	3.2751 × 1030	7.6875 × 1031	1.2527 × 1025	0.0000 × 100	1.0153 × 1021	1.4223 × 1028	5.7370 × 1014	3.3544 × 1011
F3	Mean	9.6977 × 104	5.9049 × 104	5.1553 × 104	4.2094 × 104	1.0311 × 105	4.6948 × 104	3.7896 × 104	5.4033 × 104	4.5648 × 103	3.6069 × 103
	Std	5.3581 × 104	3.5283 × 104	1.7264 × 104	1.0640 × 104	2.1922 × 104	8.3967 × 103	1.3405 × 104	9.9604 × 103	2.2541 × 103	1.9662 × 103
F4	Mean	5.5408 × 102	5.3857 × 102	5.3009 × 102	6.5865 × 102	5.3510 × 102	8.1201 × 102	5.1798 × 102	6.2978 × 102	4.9578 × 102	4.8149 × 102
	Std	6.0484 × 101	3.9164 × 101	5.5115 × 101	1.8043 × 102	3.9452 × 101	1.7037 × 102	2.2008 × 101	5.6390 × 101	1.8219 × 101	2.6849 × 101
F5	Mean	6.1258 × 102	6.0133 × 102	6.0276 × 102	6.7015 × 102	6.6782 × 102	7.8008 × 102	6.3763 × 102	6.6931 × 102	5.6022 × 102	5.6514 × 102
	Std	2.5796 × 101	2.2214 × 101	2.6386 × 101	3.4348 × 101	4.3927 × 101	3.0665 × 101	3.0092 × 101	2.3832 × 101	1.3392 × 101	1.5185 × 101
F6	Mean	6.0879 × 102	6.1271 × 102	6.1202 × 102	6.4400 × 102	6.3067 × 102	6.5999 × 102	6.3219 × 102	6.2942 × 102	6.0373 × 102	6.0226 × 102
	Std	6.5409 × 100	5.1840 × 100	6.0664 × 100	9.8154 × 100	1.0828 × 101	8.6766 × 100	1.1630 × 101	8.4611 × 100	1.6262 × 100	1.4748 × 100
F7	Mean	9.4564 × 102	9.1224 × 102	8.5139 × 102	1.0027 × 103	1.0057 × 103	1.2263 × 103	8.9202 × 102	9.6708 × 102	8.0888 × 102	7.9369 × 102
	Std	6.4993 × 101	4.4254 × 101	3.8823 × 101	6.9371 × 101	6.5324 × 101	9.1440 × 101	3.0547 × 101	4.3529 × 101	2.1067 × 101	2.5416 × 101
F8	Mean	9.0492 × 102	9.0511 × 102	8.9964 × 102	9.3858 × 102	9.5073 × 102	1.0115 × 103	9.0656 × 102	9.4894 × 102	8.5929 × 102	8.5993 × 102
	Std	2.2848 × 101	2.1158 × 101	2.6475 × 101	2.9309 × 101	3.6052 × 101	2.2197 × 101	3.0335 × 101	2.2613 × 101	2.3165 × 101	1.5099 × 101
F9	Mean	2.9824 × 103	2.2788 × 103	2.8435 × 103	3.6346 × 103	4.6715 × 103	6.3461 × 103	3.9938 × 103	2.6184 × 103	1.2711 × 103	1.1146 × 103
	Std	1.3500 × 103	4.5784 × 102	1.5866 × 103	6.2137 × 102	1.7759 × 103	9.5490 × 102	1.4814 × 103	8.2234 × 102	2.7136 × 102	1.1979 × 102
F10	Mean	4.8856 × 103	5.1589 × 103	5.1197 × 103	5.0617 × 103	6.3405 × 103	6.1930 × 103	4.8202 × 103	6.8864 × 103	4.6953 × 103	4.3927 × 103
	Std	4.6370 × 102	8.7415 × 102	6.9352 × 102	6.3968 × 102	1.2478 × 103	4.4510 × 102	7.2161 × 102	5.6063 × 102	4.8875 × 102	6.4551 × 102
F11	Mean	1.7826 × 103	1.3395 × 103	1.2987 × 103	1.4369 × 103	1.3746 × 103	1.6530 × 103	1.2980 × 103	1.5706 × 103	1.2129 × 103	1.1955 × 103
	Std	1.4634 × 103	7.7573 × 101	7.0218 × 101	1.9091 × 102	9.6230 × 101	1.4148 × 102	5.0164 × 101	1.5902 × 102	3.6187 × 101	3.9034 × 101
F12	Mean	2.7598 × 106	2.9897 × 106	1.4960 × 106	4.4103 × 107	3.6024 × 106	2.0892 × 108	1.8541 × 107	3.2656 × 107	2.1427 × 105	1.0313 × 105
	Std	4.3171 × 106	3.7448 × 106	2.6151 × 106	6.8810 × 107	2.4532 × 106	1.4609 × 108	1.3919 × 107	1.7685 × 107	1.7728 × 105	1.4519 × 105
F13	Mean	1.5133 × 104	3.6528 × 104	1.7096 × 105	1.2651 × 105	1.5008 × 104	4.3054 × 106	1.2650 × 105	4.2069 × 104	1.0033 × 104	2.4161 × 103
	Std	1.8746 × 104	2.0587 × 104	8.1342 × 105	1.1810 × 105	1.2257 × 104	5.9205 × 106	1.4656 × 105	1.9778 × 104	1.0943 × 104	7.9642 × 102
F14	Mean	1.1605 × 105	3.0324 × 103	3.1829 × 104	1.8148 × 105	2.2540 × 105	6.8481 × 105	1.1468 × 105	1.9619 × 104	1.5907 × 103	1.5896 × 103
	Std	5.4852 × 105	2.1365 × 103	2.9070 × 104	2.1355 × 105	2.5538 × 105	7.1608 × 105	8.8586 × 104	2.0831 × 104	6.8706 × 101	5.4482 × 101
F15	Mean	1.2477 × 104	1.0745 × 104	8.6916 × 103	2.9062 × 104	8.6703 × 103	1.1393 × 105	5.2360 × 104	2.6446 × 104	2.1362 × 103	1.7535 × 103
	Std	9.3342 × 103	1.0372 × 104	8.6563 × 103	1.8199 × 104	6.8605 × 103	1.6172 × 105	3.2223 × 104	1.4898 × 104	2.2189 × 102	1.4981 × 102
F16	Mean	2.7406 × 103	2.5670 × 103	2.6460 × 103	2.8758 × 103	2.8691 × 103	3.3267 × 103	2.8248 × 103	2.7232 × 103	2.2159 × 103	2.1486 × 103
	Std	3.1318 × 102	2.1476 × 102	2.3298 × 102	3.5703 × 102	3.4904 × 102	4.0218 × 102	3.2850 × 102	2.0900 × 102	1.9062 × 102	2.1504 × 102
F17	Mean	2.2137 × 103	2.0376 × 103	2.1884 × 103	2.2279 × 103	2.3784 × 103	2.4720 × 103	2.1993 × 103	2.0662 × 103	1.8579 × 103	1.8581 × 103
	Std	2.1439 × 102	1.5562 × 102	2.0512 × 102	2.5638 × 102	2.9463 × 102	2.1867 × 102	1.9367 × 102	1.4268 × 102	6.5249 × 101	1.0123 × 102
F18	Mean	4.1152 × 105	1.6058 × 105	2.3487 × 105	1.3946 × 106	1.5532 × 106	3.1271 × 106	1.1949 × 106	2.2572 × 105	2.6050 × 104	2.8504 × 104
	Std	1.2211 × 106	2.9007 × 105	1.6044 × 105	1.2666 × 106	1.7523 × 106	2.3537 × 106	2.0154 × 106	2.6940 × 105	1.8043 × 104	2.6019 × 104
F19	Mean	1.2146 × 104	8.5549 × 103	1.1455 × 104	8.2827 × 104	1.0087 × 104	1.5339 × 106	6.6527 × 105	2.0019 × 105	2.1326 × 103	2.0831 × 103
	Std	1.2089 × 104	1.0027 × 104	2.0272 × 104	1.1696 × 105	1.1514 × 104	1.9760 × 106	6.3862 × 105	3.2320 × 105	9.5279 × 101	6.8725 × 101
F20	Mean	2.5287 × 103	2.3903 × 103	2.5340 × 103	2.6433 × 103	2.6314 × 103	2.6374 × 103	2.5454 × 103	2.4305 × 103	2.2386 × 103	2.1918 × 103
	Std	1.6940 × 102	1.2541 × 102	2.0607 × 102	1.9636 × 102	1.6124 × 102	1.7675 × 102	1.8158 × 102	1.2854 × 102	9.5825 × 101	9.0232 × 101
F21	Mean	2.4082 × 103	2.3986 × 103	2.3862 × 103	2.4541 × 103	2.4342 × 103	2.5450 × 103	2.4252 × 103	2.4412 × 103	2.3598 × 103	2.3586 × 103
	Std	1.8896 × 101	1.8354 × 101	1.8887 × 101	3.4359 × 101	3.3797 × 101	5.4605 × 101	3.3491 × 101	2.3677 × 101	1.9108 × 101	1.5761 × 101
F22	Mean	3.4119 × 103	4.2492 × 103	3.8675 × 103	5.4897 × 103	2.3289 × 103	5.6480 × 103	5.1990 × 103	2.6277 × 103	2.3013 × 103	2.3006 × 103
	Std	1.6849 × 103	2.3886 × 103	1.8980 × 103	2.0491 × 103	1.2089 × 101	2.5051 × 103	2.1560 × 103	5.3530 × 102	1.9783 × 100	1.2187 × 100
F23	Mean	2.7930 × 103	2.7739 × 103	2.7791 × 103	2.8483 × 103	2.8096 × 103	2.9796 × 103	2.8162 × 103	2.8146 × 103	2.7261 × 103	2.7262 × 103
	Std	5.0992 × 101	3.8697 × 101	3.5745 × 101	5.4939 × 101	4.7489 × 101	9.7586 × 101	4.2735 × 101	3.0064 × 101	2.3523 × 101	2.1862 × 101
F24	Mean	2.9456 × 103	2.9585 × 103	2.9893 × 103	2.9960 × 103	2.9711 × 103	3.1919 × 103	2.9834 × 103	2.9796 × 103	2.8936 × 103	2.8834 × 103
	Std	2.9242 × 101	3.6647 × 101	6.6591 × 101	5.3390 × 101	3.5410 × 101	8.6184 × 101	5.2412 × 101	2.8511 × 101	2.4978 × 101	1.5104 × 101
F25	Mean	2.9341 × 103	2.9192 × 103	2.9100 × 103	2.9846 × 103	2.9367 × 103	3.0556 × 103	2.9194 × 103	3.0265 × 103	2.8948 × 103	2.8870 × 103
	Std	3.2644 × 101	1.9150 × 101	2.0029 × 101	6.0051 × 101	3.2226 × 101	4.0951 × 101	1.9458 × 101	4.4468 × 101	1.4854 × 101	2.9225 × 100
F26	Mean	5.2517 × 103	4.7614 × 103	4.2865 × 103	5.6212 × 103	5.5680 × 103	6.7275 × 103	5.2084 × 103	5.0772 × 103	3.9323 × 103	3.2041 × 103
	Std	7.9939 × 102	8.9254 × 102	1.2742 × 103	8.4971 × 102	7.7333 × 102	1.3155 × 103	6.0523 × 102	1.1538 × 103	6.8492 × 102	6.1943 × 102
F27	Mean	3.2580 × 103	3.2613 × 103	3.2612 × 103	3.2970 × 103	3.2658 × 103	3.3822 × 103	3.2788 × 103	3.2718 × 103	3.2302 × 103	3.2259 × 103
	Std	2.7070 × 101	3.9486 × 101	2.8774 × 101	4.0343 × 101	2.6068 × 101	8.0073 × 101	4.3464 × 101	2.5082 × 101	1.2564 × 101	1.2713 × 101
F28	Mean	3.3022 × 103	3.2924 × 103	3.2819 × 103	3.4540 × 103	3.3052 × 103	3.4867 × 103	3.2942 × 103	3.4140 × 103	3.2285 × 103	3.2113 × 103
	Std	4.0505 × 101	4.9321 × 101	6.1558 × 101	1.7327 × 102	4.7289 × 101	9.9301 × 101	3.0019 × 101	6.2880 × 101	2.4202 × 101	2.8108 × 101
F29	Mean	3.9804 × 103	3.8488 × 103	3.8508 × 103	4.4031 × 103	3.9533 × 103	4.4116 × 103	4.0244 × 103	4.1090 × 103	3.6190 × 103	3.5755 × 103
	Std	2.4074 × 102	1.7938 × 102	2.6102 × 102	2.9727 × 102	2.5215 × 102	2.4214 × 102	2.1214 × 102	1.5496 × 102	1.0276 × 102	1.4162 × 102
F30	Mean	3.3788 × 104	8.2277 × 104	4.3635 × 104	2.6680 × 106	2.7386 × 104	9.8661 × 106	4.3111 × 106	2.1553 × 106	1.8136 × 104	8.4162 × 103
	Std	2.0092 × 104	9.3325 × 104	1.5183 × 105	2.4701 × 106	2.2772 × 104	8.0010 × 106	2.5776 × 106	2.1655 × 106	8.6676 × 103	3.1274 × 103

,

Table 6. Experimental results of 10 algorithms on the CEC 2017 (Dim = 50).

Algorithm	Metric	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO	QCAMBHO
F1	Mean	3.1448 × 109	6.4168 × 108	4.2152 × 108	1.3254 × 1010	1.1602 × 109	2.2612 × 1010	1.0307 × 108	7.7332 × 109	1.3464 × 106	4.1825 × 105
	Std	1.9683 × 109	3.1281 × 108	5.2981 × 108	5.5127 × 109	7.1926 × 108	4.3978 × 109	7.6612 × 107	2.0724 × 109	1.2043 × 106	2.8299 × 105
F2	Mean	1.0000 × 1030	1.0000 × 1030	1.3608 × 1052	2.2643 × 1056	2.5029 × 1056	1.0000 × 1020	1.8776 × 1046	1.4000 × 1057	7.0209 × 1036	1.2280 × 1030
	Std	1.4314 × 1014	1.4314 × 1014	7.4481 × 1052	5.9353 × 1056	1.3146 × 1057	0.0000 × 100	5.2714 × 1046	5.2761 × 1057	3.5658 × 1037	5.2591 × 1030
F3	Mean	1.9889 × 105	1.5735 × 105	1.7919 × 105	1.2898 × 105	2.5014 × 105	1.2894 × 105	1.7313 × 105	1.5403 × 105	4.8679 × 104	4.5063 × 104
	Std	7.3682 × 104	5.4257 × 104	4.2509 × 104	2.4011 × 104	3.6166 × 104	1.8171 × 104	3.4826 × 104	2.2521 × 104	1.2161 × 104	1.1696 × 104
F4	Mean	1.0572 × 103	7.7355 × 102	7.1302 × 102	2.4146 × 103	8.0099 × 102	2.6456 × 103	6.9608 × 102	1.7288 × 103	5.9319 × 102	5.3672 × 102
	Std	3.4677 × 102	8.8532 × 101	1.1573 × 102	1.1954 × 103	9.1684 × 101	6.9550 × 102	5.1815 × 101	3.7893 × 102	4.1410 × 101	5.1054 × 101
F5	Mean	7.8835 × 102	7.8861 × 102	7.3638 × 102	8.3700 × 102	8.5366 × 102	9.7622 × 102	7.6801 × 102	8.8490 × 102	6.7517 × 102	6.8117 × 102
	Std	4.9534 × 101	4.7838 × 101	4.0070 × 101	4.5511 × 101	7.2489 × 101	3.3178 × 101	4.1600 × 101	4.4772 × 101	4.0549 × 101	5.2895 × 101
F6	Mean	6.1872 × 102	6.3293 × 102	6.2632 × 102	6.5695 × 102	6.4616 × 102	6.7674 × 102	6.4639 × 102	6.4842 × 102	6.1545 × 102	6.1179 × 102
	Std	1.0268 × 101	8.9884 × 100	7.0114 × 100	9.1182 × 100	1.0280 × 101	5.4697 × 100	9.3850 × 100	6.4951 × 100	5.6620 × 100	3.5472 × 100
F7	Mean	1.4813 × 103	1.3990 × 103	1.0828 × 103	1.4228 × 103	1.4108 × 103	1.7579 × 103	1.1571 × 103	1.3919 × 103	1.0177 × 103	9.8269 × 102
	Std	1.5796 × 102	1.0285 × 102	6.2459 × 101	1.1360 × 102	1.2180 × 102	1.1102 × 102	7.2798 × 101	7.3859 × 101	4.5995 × 101	6.3871 × 101
F8	Mean	1.1020 × 103	1.0879 × 103	1.0377 × 103	1.1386 × 103	1.1470 × 103	1.2734 × 103	1.0729 × 103	1.1889 × 103	9.6187 × 102	9.8718 × 102
	Std	4.7503 × 101	4.1458 × 101	4.4430 × 101	5.4983 × 101	5.3654 × 101	4.4228 × 101	4.1822 × 101	3.4510 × 101	3.6744 × 101	4.0799 × 101
F9	Mean	1.0753 × 104	1.2554 × 104	8.0232 × 103	1.2697 × 104	1.9179 × 104	2.3069 × 104	1.3853 × 104	1.2632 × 104	4.3536 × 103	2.6356 × 103
	Std	4.0932 × 103	4.3652 × 103	3.4851 × 103	3.3259 × 103	6.4236 × 103	4.2015 × 103	3.9801 × 103	3.1802 × 103	1.8210 × 103	5.6161 × 102
F10	Mean	8.7353 × 103	9.5650 × 103	8.6683 × 103	8.7737 × 103	1.0084 × 104	1.0864 × 104	7.9208 × 103	1.2137 × 104	7.3563 × 103	7.3509 × 103
	Std	6.7384 × 102	1.3673 × 103	1.2926 × 103	1.0651 × 103	1.3239 × 103	1.1794 × 103	9.6808 × 102	8.4818 × 102	7.9882 × 102	1.3903 × 103
F11	Mean	2.6413 × 103	2.1222 × 103	1.7040 × 103	4.1481 × 103	3.6029 × 103	3.2362 × 103	1.7335 × 103	6.6656 × 103	1.3527 × 103	1.3309 × 103
	Std	6.4735 × 102	1.0464 × 103	2.1676 × 102	1.4732 × 103	1.8585 × 103	7.5780 × 102	1.5576 × 102	2.1082 × 103	6.0794 × 101	4.8073 × 101
F12	Mean	7.5703 × 107	6.6703 × 107	4.0999 × 108	1.8516 × 109	6.1943 × 107	2.0521 × 109	1.0491 × 108	4.0151 × 108	4.4928 × 106	1.8357 × 106
	Std	5.9975 × 107	3.6794 × 107	9.1846 × 108	2.4386 × 109	3.9903 × 107	1.0740 × 109	8.6843 × 107	1.6790 × 108	3.3180 × 106	1.0302 × 106
F13	Mean	8.6749 × 104	1.0818 × 105	2.5941 × 107	6.3989 × 107	3.6146 × 104	1.2973 × 108	3.1934 × 105	7.7734 × 105	2.3113 × 104	7.8780 × 103
	Std	1.6935 × 105	6.3664 × 104	7.4853 × 107	8.6541 × 107	2.1337 × 104	1.0715 × 108	1.1127 × 106	8.4528 × 105	2.0824 × 104	5.4220 × 103
F14	Mean	1.7765 × 105	9.2730 × 104	3.3100 × 105	1.0517 × 106	1.2227 × 106	2.0912 × 106	6.6929 × 105	2.1673 × 105	5.5795 × 103	7.3947 × 103
	Std	2.3330 × 105	7.9311 × 104	5.8873 × 105	8.4969 × 105	9.8171 × 105	2.1923 × 106	5.8933 × 105	2.5082 × 105	3.6747 × 103	9.7357 × 103
F15	Mean	1.2879 × 104	2.7852 × 104	9.8952 × 103	9.5124 × 105	9.2309 × 103	7.0274 × 106	4.2083 × 104	2.3529 × 104	8.9163 × 103	5.5983 × 103
	Std	6.3947 × 103	1.5674 × 104	8.7004 × 103	4.5798 × 106	6.3006 × 103	8.9126 × 106	1.8693 × 104	1.2561 × 104	4.5506 × 103	4.0424 × 103
F16	Mean	3.6874 × 103	3.2898 × 103	3.4081 × 103	3.8251 × 103	3.6789 × 103	4.8793 × 103	3.5796 × 103	3.5021 × 103	2.9296 × 103	2.8795 × 103
	Std	4.3682 × 102	3.2537 × 102	4.2661 × 102	4.8716 × 102	5.0415 × 102	6.0577 × 102	4.7470 × 102	3.3601 × 102	4.9033 × 102	3.6456 × 102
F17	Mean	3.3354 × 103	3.0936 × 103	3.1663 × 103	3.4118 × 103	3.6114 × 103	3.7360 × 103	3.1705 × 103	3.2342 × 103	2.8184 × 103	2.7579 × 103
	Std	3.1330 × 102	2.9183 × 102	3.1326 × 102	2.8773 × 102	3.6411 × 102	3.3293 × 102	3.0317 × 102	2.7121 × 102	2.4283 × 102	2.2092 × 102
F18	Mean	9.4579 × 105	6.0130 × 105	2.0321 × 106	5.4342 × 106	6.6533 × 106	1.1337 × 107	4.3821 × 106	2.0534 × 106	1.3022 × 105	1.2224 × 105
	Std	7.4369 × 105	5.2663 × 105	2.3022 × 106	5.1159 × 106	5.0854 × 106	1.0143 × 107	3.3370 × 106	1.8059 × 106	7.2117 × 104	6.6230 × 104
F19	Mean	2.0860 × 104	3.6547 × 104	3.2813 × 104	3.8072 × 105	1.5332 × 104	2.0131 × 106	1.3304 × 106	4.4107 × 105	1.2049 × 104	6.6904 × 103
	Std	2.4014 × 104	2.3058 × 104	3.9544 × 104	3.7406 × 105	1.1901 × 104	2.2985 × 106	1.0364 × 106	5.1925 × 105	1.0378 × 104	6.4853 × 103
F20	Mean	3.5387 × 103	3.3080 × 103	3.4240 × 103	3.3132 × 103	3.6344 × 103	3.3220 × 103	3.2824 × 103	3.2463 × 103	2.8102 × 103	2.7093 × 103
	Std	2.1897 × 102	3.3119 × 102	4.0324 × 102	3.9312 × 102	4.1076 × 102	2.8749 × 102	3.6659 × 102	2.6699 × 102	2.1487 × 102	1.9479 × 102
F21	Mean	2.6011 × 103	2.5788 × 103	2.5239 × 103	2.6616 × 103	2.6246 × 103	2.8610 × 103	2.5634 × 103	2.6424 × 103	2.4778 × 103	2.4488 × 103
	Std	6.9038 × 101	4.4743 × 101	3.9195 × 101	5.6859 × 101	5.2021 × 101	8.4626 × 101	4.8104 × 101	4.5276 × 101	3.7402 × 101	4.3750 × 101
F22	Mean	1.0686 × 104	1.1584 × 104	1.0340 × 104	1.0219 × 104	1.2845 × 104	1.2644 × 104	1.0074 × 104	1.2533 × 104	7.9510 × 103	7.2157 × 103
	Std	8.4667 × 102	1.8589 × 103	1.3978 × 103	1.1362 × 103	2.1392 × 103	1.0281 × 103	9.2830 × 102	3.3648 × 103	3.4290 × 103	4.0231 × 103
F23	Mean	3.1185 × 103	3.1361 × 103	3.0698 × 103	3.1996 × 103	3.1685 × 103	3.5232 × 103	3.1019 × 103	3.1651 × 103	2.9700 × 103	2.9215 × 103
	Std	1.0026 × 102	8.6705 × 101	8.0930 × 101	1.1404 × 102	7.4938 × 101	1.0368 × 102	6.9518 × 101	7.1869 × 101	6.6768 × 101	3.9442 × 101
F24	Mean	3.2570 × 103	3.3181 × 103	3.4219 × 103	3.2772 × 103	3.2757 × 103	3.7453 × 103	3.3114 × 103	3.3408 × 103	3.1085 × 103	3.0790 × 103
	Std	8.5423 × 101	1.0564 × 102	2.0530 × 102	7.7391 × 101	8.0993 × 101	1.5676 × 102	7.3954 × 101	5.9496 × 101	6.0812 × 101	4.3959 × 101
F25	Mean	3.5132 × 103	3.2448 × 103	3.1662 × 103	4.0623 × 103	3.2856 × 103	4.5017 × 103	3.1920 × 103	4.0846 × 103	3.0836 × 103	3.0806 × 103
	Std	2.5921 × 102	8.5916 × 101	4.1045 × 101	5.2548 × 102	1.1959 × 102	4.8715 × 102	6.4643 × 101	2.7725 × 102	3.2295 × 101	2.6657 × 101
F26	Mean	7.8561 × 103	8.2948 × 103	5.9917 × 103	9.6821 × 103	8.0999 × 103	1.2206 × 104	7.9370 × 103	1.0317 × 104	5.8109 × 103	4.1526 × 103
	Std	1.0245 × 103	1.0060 × 103	2.4185 × 103	1.6978 × 103	1.0849 × 103	1.5643 × 103	1.1361 × 103	1.2896 × 103	9.6920 × 102	1.4705 × 103
F27	Mean	3.6767 × 103	3.7702 × 103	3.6376 × 103	3.8845 × 103	3.7151 × 103	4.0670 × 103	3.7459 × 103	3.7840 × 103	3.5327 × 103	3.4585 × 103
	Std	1.5586 × 102	1.7003 × 102	1.5675 × 102	2.3032 × 102	1.3939 × 102	2.0154 × 102	1.3319 × 102	1.4182 × 102	1.2415 × 102	9.1490 × 101
F28	Mean	4.1005 × 103	3.7127 × 103	3.6033 × 103	5.0559 × 103	3.6943 × 103	5.2896 × 103	3.5735 × 103	4.8299 × 103	3.3724 × 103	3.3413 × 103
	Std	4.7283 × 102	1.8808 × 102	2.4250 × 102	5.5456 × 102	1.8286 × 102	3.2188 × 102	9.6523 × 101	4.2976 × 102	4.1716 × 101	2.8925 × 101
F29	Mean	5.0911 × 103	5.0259 × 103	4.9230 × 103	6.1647 × 103	5.0420 × 103	6.9669 × 103	5.1003 × 103	5.5163 × 103	4.2966 × 103	4.0645 × 103
	Std	4.7557 × 102	3.6839 × 102	4.0724 × 102	7.0920 × 102	4.1878 × 102	8.6238 × 102	4.0118 × 102	4.3686 × 102	3.1384 × 102	2.8130 × 102
F30	Mean	4.2657 × 106	1.4456 × 107	1.6608 × 106	9.8797 × 107	3.8695 × 106	2.0416 × 108	6.4274 × 107	1.4819 × 108	2.2964 × 106	1.1036 × 106
	Std	1.4803 × 106	6.2507 × 106	7.0940 × 105	3.2635 × 107	1.6245 × 106	7.7369 × 107	2.1082 × 107	4.1882 × 107	8.7752 × 105	2.8228 × 105

and

Table 7. Experimental results of 10 algorithms on the CEC 2017 (Dim = 100).

Algorithm	Metric	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO	QCAMBHO
F1	Mean	5.1467 × 1010	2.2919 × 1010	1.4869 × 1010	7.4125 × 1010	2.2020 × 1010	1.0307 × 1011	1.0433 × 1010	6.5602 × 1010	6.2480 × 108	1.8477 × 108
	Std	1.3611 × 1010	8.9697 × 109	4.7801 × 109	1.4706 × 1010	5.0695 × 109	1.0188 × 1010	3.0879 × 109	7.8330 × 109	3.6205 × 108	1.3487 × 108
F2	Mean	1.0000 × 1030	1.0000 × 1030	3.2459 × 10128	1.1856 × 10145	8.0084 × 10145	1.0000 × 1020	2.0039 × 10131	5.1169 × 10146	3.2262 × 10111	9.2017 × 10101
	Std	1.4314 × 1014	1.4314 × 1014	1.2743 × 10129	6.4707 × 10145	4.3828 × 10146	0.0000 × 100	1.0970 × 10132	2.5301 × 10147	1.7671 × 10112	4.9715 × 10102
F3	Mean	5.0574 × 105	5.1494 × 105	5.2529 × 105	5.2645 × 105	6.4051 × 105	3.0777 × 105	6.0657 × 105	4.2817 × 105	2.1879 × 105	2.2896 × 105
	Std	1.1973 × 105	1.1418 × 105	8.0109 × 104	1.2282 × 105	7.0088 × 104	1.6859 × 104	9.8962 × 104	4.4589 × 104	1.8820 × 104	2.4588 × 104
F4	Mean	5.6025 × 103	3.3564 × 103	2.4794 × 103	9.3017 × 103	3.2569 × 103	1.2976 × 104	2.1813 × 103	9.2141 × 103	1.0562 × 103	8.7306 × 102
	Std	2.0164 × 103	9.5214 × 102	5.8381 × 102	1.9971 × 103	6.3858 × 102	1.9387 × 103	3.8348 × 102	1.6419 × 103	9.3405 × 101	5.7708 × 101
F5	Mean	1.4995 × 103	1.4040 × 103	1.2861 × 103	1.4398 × 103	1.5894 × 103	1.7810 × 103	1.3421 × 103	1.5719 × 103	1.1311 × 103	1.0584 × 103
	Std	8.4693 × 101	1.2222 × 102	1.1175 × 102	7.5342 × 101	1.2344 × 102	6.6053 × 101	7.2032 × 101	7.5837 × 101	1.3691 × 102	1.1866 × 102
F6	Mean	6.4597 × 102	6.5480 × 102	6.5362 × 102	6.7229 × 102	6.7032 × 102	6.8863 × 102	6.6655 × 102	6.7961 × 102	6.3851 × 102	6.3266 × 102
	Std	9.9505 × 100	6.4932 × 100	7.2102 × 100	4.4261 × 100	8.8826 × 100	3.6599 × 100	5.8370 × 100	5.3277 × 100	5.0326 × 100	4.6180 × 100
F7	Mean	3.9767 × 103	3.3449 × 103	2.2128 × 103	3.0009 × 103	3.3391 × 103	3.6026 × 103	2.3512 × 103	3.0429 × 103	2.0459 × 103	1.7578 × 103
	Std	5.6885 × 102	4.1227 × 102	1.9022 × 102	2.1233 × 102	2.9271 × 102	1.0478 × 102	2.2895 × 102	1.8542 × 102	1.8913 × 102	2.5552 × 102
F8	Mean	1.8432 × 103	1.7584 × 103	1.5929 × 103	1.8284 × 103	1.9057 × 103	2.1953 × 103	1.7030 × 103	1.9454 × 103	1.3831 × 103	1.3625 × 103
	Std	9.2857 × 101	1.0877 × 102	1.0271 × 102	9.7263 × 101	1.4710 × 102	7.1951 × 101	9.9643 × 101	7.0581 × 101	9.6349 × 101	1.3762 × 102
F9	Mean	4.6924 × 104	5.1090 × 104	4.4173 × 104	3.4049 × 104	7.7518 × 104	5.3084 × 104	4.1333 × 104	5.0409 × 104	1.6698 × 104	1.5014 × 104
	Std	1.1288 × 104	1.3436 × 104	2.5111 × 104	7.3712 × 103	1.9061 × 104	7.5992 × 103	8.3313 × 103	5.7083 × 103	4.3549 × 103	4.6625 × 103
F10	Mean	2.2500 × 104	2.5011 × 104	2.1112 × 104	1.9991 × 104	2.5034 × 104	2.6381 × 104	1.9501 × 104	2.8118 × 104	1.6187 × 104	1.7066 × 104
	Std	1.1448 × 103	3.9661 × 103	2.5931 × 103	1.2791 × 103	3.3802 × 103	2.1467 × 103	1.2994 × 103	1.1948 × 103	1.4737 × 103	3.4252 × 103
F11	Mean	8.6731 × 104	7.8873 × 104	5.2189 × 104	7.0569 × 104	1.7909 × 105	8.6113 × 104	5.7323 × 104	1.1237 × 105	1.1595 × 104	6.0704 × 103
	Std	4.6928 × 104	4.2895 × 104	2.5439 × 104	1.1393 × 104	3.6256 × 104	2.2697 × 104	1.2433 × 104	1.9778 × 104	4.5844 × 103	1.4387 × 103
F12	Mean	4.0121 × 109	1.8400 × 109	2.0911 × 109	1.9645 × 1010	2.0576 × 109	2.2791 × 1010	9.4645 × 108	8.7803 × 109	7.9772 × 107	3.0072 × 107
	Std	2.6141 × 109	6.6812 × 108	2.0048 × 109	1.0284 × 1010	8.0215 × 108	8.0075 × 109	3.1317 × 108	2.6451 × 109	3.8789 × 107	1.6456 × 107
F13	Mean	2.7400 × 107	3.2792 × 106	9.6013 × 107	2.1835 × 109	7.4020 × 106	2.4379 × 109	6.7452 × 105	1.8332 × 108	5.5224 × 104	1.9223 × 104
	Std	5.1434 × 107	5.2597 × 106	2.2658 × 108	2.2561 × 109	1.5655 × 107	1.3291 × 109	6.3321 × 105	8.0214 × 107	2.3405 × 104	9.1115 × 103
F14	Mean	2.6848 × 106	1.4788 × 106	2.0563 × 106	6.2370 × 106	8.6672 × 106	9.6540 × 106	7.2630 × 106	4.5136 × 106	4.0252 × 105	3.2232 × 105
	Std	1.3229 × 106	1.1014 × 106	1.1400 × 106	2.4660 × 106	4.7185 × 106	3.6224 × 106	3.9721 × 106	2.5959 × 106	3.2906 × 105	1.7806 × 105
F15	Mean	2.2201 × 105	9.2196 × 104	2.8127 × 107	2.7425 × 108	1.3644 × 105	1.5109 × 108	6.7531 × 104	2.5779 × 106	1.4991 × 104	6.5638 × 103
	Std	4.5286 × 105	4.2622 × 104	1.5396 × 108	4.0548 × 108	2.8134 × 105	1.3132 × 108	4.2061 × 104	1.5711 × 106	1.0152 × 104	4.1265 × 103
F16	Mean	8.0498 × 103	6.7035 × 103	5.8341 × 103	8.2583 × 103	6.9304 × 103	1.2396 × 104	7.1632 × 103	8.4336 × 103	5.6114 × 103	5.4264 × 103
	Std	8.7765 × 102	6.1947 × 102	7.0719 × 102	1.3681 × 103	8.5275 × 102	1.6608 × 103	8.4903 × 102	8.5189 × 102	5.7784 × 102	6.5447 × 102
F17	Mean	6.0654 × 103	5.5928 × 103	5.9641 × 103	8.1674 × 103	6.1307 × 103	9.8951 × 103	5.6619 × 103	5.7922 × 103	4.6707 × 103	4.7143 × 103
	Std	6.6668 × 102	4.2381 × 102	6.5275 × 102	4.3010 × 103	6.1637 × 102	4.8445 × 103	4.5904 × 102	7.3524 × 102	5.4601 × 102	6.2258 × 102
F18	Mean	4.8822 × 106	3.1173 × 106	3.6530 × 106	5.4731 × 106	1.4669 × 107	1.1651 × 107	7.3900 × 106	3.7107 × 106	4.9802 × 105	5.5101 × 105
	Std	3.8618 × 106	1.9691 × 106	2.1089 × 106	3.3015 × 106	7.6077 × 106	5.1441 × 106	3.3408 × 106	1.6583 × 106	2.2390 × 105	2.4211 × 105
F19	Mean	9.2303 × 105	3.3854 × 106	2.7038 × 107	1.7277 × 108	6.6320 × 105	1.4745 × 108	6.3390 × 106	9.2221 × 106	4.2980 × 104	3.3892 × 103
	Std	2.0928 × 106	1.9343 × 106	1.2730 × 108	2.6039 × 108	7.4051 × 105	1.5712 × 108	4.4240 × 106	6.2900 × 106	1.0929 × 105	1.4838 × 103
F20	Mean	6.5189 × 103	6.0874 × 103	5.8014 × 103	5.5089 × 103	6.4054 × 103	5.9235 × 103	5.4459 × 103	6.2211 × 103	5.0931 × 103	4.8425 × 103
	Std	5.0367 × 102	5.7900 × 102	6.4585 × 102	5.4177 × 102	8.1587 × 102	5.7899 × 102	6.3374 × 102	3.7652 × 102	4.6873 × 102	6.2657 × 102
F21	Mean	3.4010 × 103	3.3363 × 103	3.1327 × 103	3.3989 × 103	3.4657 × 103	3.9608 × 103	3.2149 × 103	3.4328 × 103	2.9281 × 103	2.8296 × 103
	Std	1.2771 × 102	1.2906 × 102	1.0548 × 102	1.3480 × 102	1.3451 × 102	1.6938 × 102	9.8029 × 101	7.7634 × 101	9.5839 × 101	1.1403 × 102
F22	Mean	2.4630 × 104	2.6794 × 104	2.3514 × 104	2.2600 × 104	2.6830 × 104	2.8863 × 104	2.2000 × 104	3.1032 × 104	2.0532 × 104	1.8471 × 104
	Std	1.2515 × 103	3.2600 × 103	2.0815 × 103	1.5123 × 103	2.7833 × 103	1.5656 × 103	1.2973 × 103	1.9714 × 103	1.0986 × 103	6.6611 × 103
F23	Mean	3.9493 × 103	4.0870 × 103	4.1010 × 103	4.0258 × 103	3.9630 × 103	4.7645 × 103	3.9119 × 103	4.1140 × 103	3.6176 × 103	3.5011 × 103
	Std	1.8313 × 102	1.6291 × 102	1.8445 × 102	1.6986 × 102	1.5787 × 102	2.0051 × 102	1.4834 × 102	1.2995 × 102	1.5052 × 102	1.3050 × 102
F24	Mean	4.7547 × 103	5.2375 × 103	5.7500 × 103	4.7597 × 103	4.6610 × 103	5.7402 × 103	4.8310 × 103	4.9904 × 103	4.3064 × 103	4.1049 × 103
	Std	2.0721 × 102	3.7403 × 102	6.0844 × 102	2.1445 × 102	2.2610 × 102	2.9555 × 102	2.0143 × 102	1.4525 × 102	1.7759 × 102	1.4740 × 102
F25	Mean	7.5723 × 103	5.3391 × 103	4.4595 × 103	8.3550 × 103	6.2313 × 103	9.8298 × 103	4.5390 × 103	8.6712 × 103	3.7702 × 103	3.5794 × 103
	Std	1.6276 × 103	3.8789 × 102	3.7627 × 102	1.2520 × 103	6.4777 × 102	1.2629 × 103	2.9434 × 102	7.6380 × 102	1.1593 × 102	6.3147 × 101
F26	Mean	2.2334 × 104	2.3660 × 104	2.0764 × 104	2.5906 × 104	1.9835 × 104	3.1899 × 104	2.0332 × 104	2.9715 × 104	1.5746 × 104	1.4652 × 104
	Std	3.2610 × 103	2.8145 × 103	3.2644 × 103	3.9884 × 103	2.0982 × 103	2.9656 × 103	1.3262 × 103	2.2974 × 103	1.8133 × 103	2.4463 × 103
F27	Mean	4.2257 × 103	4.4345 × 103	4.0524 × 103	4.5990 × 103	4.0275 × 103	5.2423 × 103	4.2441 × 103	4.9038 × 103	3.8319 × 103	3.7106 × 103
	Std	2.5687 × 102	3.2547 × 102	2.5461 × 102	2.7801 × 102	2.2208 × 102	3.8682 × 102	1.8626 × 102	3.0604 × 102	1.6402 × 102	1.1090 × 102
F28	Mean	1.0724 × 104	7.0884 × 103	5.8435 × 103	9.4858 × 103	7.4942 × 103	1.1954 × 104	5.3030 × 103	1.1565 × 104	3.9990 × 103	3.7260 × 103
	Std	2.7285 × 103	1.2490 × 103	1.1645 × 103	1.3150 × 103	1.3276 × 103	1.3045 × 103	6.5314 × 102	9.4483 × 102	1.8785 × 102	6.6498 × 101
F29	Mean	9.4598 × 103	9.4980 × 103	8.0489 × 103	1.1990 × 104	8.6907 × 103	1.4936 × 104	8.7731 × 103	1.1625 × 104	7.6418 × 103	7.2079 × 103
	Std	1.1328 × 103	9.0648 × 102	7.4711 × 102	1.4591 × 103	6.7442 × 102	2.4168 × 103	6.9591 × 102	1.1457 × 103	4.6556 × 102	6.3661 × 102
F30	Mean	2.0031 × 107	7.5101 × 107	5.8403 × 107	1.2659 × 109	1.7912 × 107	1.5570 × 109	1.6004 × 108	4.6372 × 108	6.3767 × 105	9.8073 × 104
	Std	1.4909 × 107	5.4665 × 107	1.5313 × 108	1.0424 × 109	1.5618 × 107	4.9610 × 108	8.9262 × 107	1.7866 × 108	4.3198 × 105	5.1761 × 104

present the mean and standard deviation of the best solutions obtained by each algorithm after 30 independent runs under different dimensional conditions. As shown in Table 4, Table 5 and Table 6, QCAMBHO generally exhibits strong optimization ability and good cross-dimensional stability in the 30-dimensional, 50-dimensional, and 100-dimensional scenarios of the CEC 2017 test suite. Under both low-dimensional and high-dimensional conditions, the mean values obtained by QCAMBHO on most test functions are clearly better than those of the original BHO and other comparison algorithms. This indicates that the proposed quantum-enhanced initialization, adaptive Levy flight mechanism, and adaptive differential operator can effectively improve the global search ability and later-stage exploitation ability of the algorithm.

A further comparison between QCAMBHO and the original BHO shows that, as the problem dimension increases from 30 to 50 and 100, the performance advantage of QCAMBHO on most functions does not weaken, but becomes more evident. For example, for functions such as F1, F2, F12, F13, and F30, the mean values of QCAMBHO are significantly lower than those of BHO. This shows that the proposed improvement strategies effectively alleviate the problems of the original BHO in complex high-dimensional search spaces, such as being easily trapped in local optima, insufficient population diversity, and limited later-stage exploitation ability. Meanwhile, QCAMBHO also maintains relatively low standard deviations on many functions, indicating that the algorithm can not only obtain better solutions but also show good robustness and result consistency in repeated runs. In particular, under the 30-dimensional and 50-dimensional conditions, QCAMBHO obtains clearly smaller standard deviations for functions such as F9, F15, F19, F26, and F30, reflecting a more stable search process.

From the overall trend across different dimensions, the solution difficulty of all algorithms increases significantly as the dimension increases, and the optimal values of most functions deteriorate to varying degrees. However, QCAMBHO can still maintain strong competitiveness in high-dimensional scenarios. Under the 100-dimensional condition, although QCAMBHO is not the best on several functions, such as F3, F10, F17, and F18, and its results are close to those of BHO or some comparison algorithms, its overall performance remains superior on most functions. In particular, it shows clear advantages on functions such as F1 and F4. This indicates that the proposed multi-strategy collaborative mechanism has good dimensional adaptability and can continuously maintain strong search efficiency and solution quality in high-dimensional complex optimization problems.

Figure 4 and Figure 5 show the convergence curves and boxplots of QCAMBHO on several representative functions, respectively. As shown in Figure 3, QCAMBHO exhibits a rapid decreasing trend in the early stages of iteration, indicating its strong initial search efficiency. As the iteration proceeds, its convergence curve continues to decrease and maintains a clearly better fitness level than the other comparison algorithms in the later stages. This suggests that the algorithm still has strong local fine search ability during the exploitation stage. Compared with the original BHO, QCAMBHO achieves faster overall convergence speed and higher final convergence accuracy, indicating that the introduced multi-strategy improvements effectively enhance the global search ability and later-stage exploitation ability of the algorithm. Figure 5 shows that, under different dimensional conditions, the boxplots of QCAMBHO are generally located at lower positions and have narrower distribution ranges. Compared with the other algorithms, QCAMBHO obtains smaller mean values and standard deviations, indicating that it can not only achieve better objective values in multiple independent runs but also has better stability and robustness. This is consistent with the numerical results in Table 7,

Table 8. Wilcoxon Rank-Sum Test results of QCAMBHO and 9 algorithms on the CEC 2017 (Dim = 30).

Algorithm	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO
F1	3.0199 × 10−11	3.0199 × 10−11	2.3768 × 10−7	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.6687 × 10−1
F2	3.0199 × 10−11	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.2118 × 10−12	3.0199 × 10−11	3.0199 × 10−11	7.0881 × 10−8
F3	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	6.3533 × 10−2
F4	7.0881 × 10−8	6.5277 × 10−8	4.9426 × 10−5	6.0658 × 10−11	9.0632 × 10−8	3.0199 × 10−11	1.7290 × 10−6	3.0199 × 10−11	4.5146 × 10−2
F5	1.5465 × 10−9	1.0666 × 10−7	1.2541 × 10−7	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.3384 × 10−11	3.0199 × 10−11	1.2967 × 10−1
F6	3.0811 × 10−8	6.0658 × 10−11	6.6955 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.2951 × 10−4
F7	6.6955 × 10−11	6.6955 × 10−11	8.3520 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.3891 × 10−11	3.0199 × 10−11	7.9590 × 10−3
F8	3.1967 × 10−9	1.1737 × 10−9	7.0881 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.0702 × 10−9	3.0199 × 10−11	5.7929 × 10−1
F9	3.6897 × 10−11	3.3384 × 10−11	3.6897 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.1877 × 10−2
F10	1.8575 × 10−3	4.6390 × 10−5	2.0058 × 10−4	5.8737 × 10−4	1.1023 × 10−8	8.9934 × 10−11	2.6077 × 10−2	3.3384 × 10−11	4.3584 × 10−2
F11	1.4643 × 10−10	4.1997 × 10−10	7.0881 × 10−8	4.0772 × 10−11	1.0937 × 10−10	3.0199 × 10−11	2.4386 × 10−9	3.0199 × 10−11	1.2967 × 10−1
F12	3.4742 × 10−10	6.0658 × 10−11	1.5581 × 10−8	3.3384 × 10−11	8.9934 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.5923 × 10−5
F13	7.3891 × 10−11	3.0199 × 10−11	1.3289 × 10−10	3.0199 × 10−11	1.2870 × 10−9	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.3715 × 10−10
F14	1.0937 × 10−10	1.0937 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	8.6499 × 10−1
F15	4.0772 × 10−11	3.0199 × 10−11	6.7220 × 10−10	3.0199 × 10−11	5.5727 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.9980 × 10−9
F16	1.5581 × 10−8	1.4294 × 10−8	5.4617 × 10−9	2.4386 × 10−9	6.7220 × 10−10	3.0199 × 10−11	2.3715 × 10−10	2.3715 × 10−10	1.7145 × 10−1
F17	1.8567 × 10−9	6.7362 × 10−6	1.0105 × 10−8	9.2603 × 10−9	4.6159 × 10−10	4.5043 × 10−11	3.1967 × 10−9	2.1959 × 10−7	4.4642 × 10−1
F18	4.1825 × 10−9	7.0430 × 10−7	1.4643 × 10−10	3.0199 × 10−11	4.0772 × 10−11	3.0199 × 10−11	7.3891 × 10−11	8.1014 × 10−10	7.7312 × 10−1
F19	2.6695 × 10−9	2.1544 × 10−10	1.6980 × 10−8	3.0199 × 10−11	3.1589 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.9428 × 10−2
F20	2.8716 × 10−10	6.0104 × 10−8	1.1023 × 10−8	2.1544 × 10−10	5.4941 × 10−11	6.0658 × 10−11	1.2870 × 10−9	2.6695 × 10−9	2.5101 × 10−2
F21	1.2057 × 10−10	1.5465 × 10−9	8.8411 × 10−7	6.0658 × 10−11	9.9186 × 10−11	3.0199 × 10−11	3.8202 × 10−10	3.0199 × 10−11	7.3940 × 10−1
F22	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	8.2357 × 10−2
F23	9.2603 × 10−9	8.1975 × 10−7	2.5721 × 10−7	2.8716 × 10−10	2.2273 × 10−9	3.0199 × 10−11	1.9568 × 10−10	4.0772 × 10−11	8.7663 × 10−1
F24	8.9934 × 10−11	1.2057 × 10−10	3.3384 × 10−11	8.1527 × 10−11	4.0772 × 10−11	3.0199 × 10−11	8.1527 × 10−11	3.0199 × 10−11	7.4827 × 10−2
F25	6.0658 × 10−11	4.5043 × 10−11	1.5964 × 10−7	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.1997 × 10−10	3.0199 × 10−11	5.0842 × 10−3
F26	1.7769 × 10−10	1.2541 × 10−7	9.2113 × 10−5	1.2057 × 10−10	5.5727 × 10−10	2.8716 × 10−10	8.9934 × 10−11	1.2023 × 10−8	9.7917 × 10−5
F27	8.8411 × 10−7	2.2780 × 10−5	2.8314 × 10−8	1.2057 × 10−10	6.5183 × 10−9	3.0199 × 10−11	3.8249 × 10−9	3.8202 × 10−10	1.0869 × 10−1
F28	1.9568 × 10−10	8.8910 × 10−10	2.6015 × 10−8	3.0199 × 10−11	1.2057 × 10−10	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	6.9125 × 10−4
F29	2.8314 × 10−8	3.2555 × 10−7	3.1573 × 10−5	3.0199 × 10−11	7.6950 × 10−8	3.0199 × 10−11	1.1737 × 10−9	3.3384 × 10−11	8.5000 × 10−2
F30	2.6099 × 10−10	3.0199 × 10−11	9.5332 × 10−7	3.0199 × 10−11	7.0881 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	8.3520 × 10−8

and

Table 9. Wilcoxon Rank-Sum Test results of QCAMBHO and 9 algorithms on the CEC 2017 (Dim = 50).

Algorithm	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO
F1	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.0907 × 10−5
F2	7.4716 × 10−10	7.4716 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.4716 × 10−10	3.0199 × 10−11	3.0199 × 10−11	4.6159 × 10−10
F3	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.5805 × 10−1
F4	3.0199 × 10−11	4.9752 × 10−11	1.4110 × 10−9	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.4941 × 10−11	3.0199 × 10−11	7.1988 × 10−5
F5	1.5581 × 10−8	9.2603 × 10−9	1.3250 × 10−4	6.0658 × 10−11	1.4643 × 10−10	3.0199 × 10−11	1.0666 × 10−7	4.5043 × 10−11	6.9522 × 10−1
F6	1.3017 × 10−3	1.2057 × 10−10	7.3891 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.5699 × 10−3
F7	3.0199 × 10−11	3.0199 × 10−11	1.2860 × 10−6	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.3803 × 10−10	3.0199 × 10−11	3.3874 × 10−2
F8	2.6099 × 10−10	1.8567 × 10−9	8.6634 × 10−5	6.6955 × 10−11	4.9752 × 10−11	3.0199 × 10−11	1.2023 × 10−8	3.0199 × 10−11	7.9590 × 10−3
F9	3.0199 × 10−11	3.0199 × 10−11	1.9568 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.0907 × 10−5
F10	6.2828 × 10−6	3.2555 × 10−7	1.7836 × 10−4	4.0840 × 10−5	3.0811 × 10−8	1.5465 × 10−9	9.0688 × 10−3	4.9752 × 10−11	2.9727 × 10−1
F11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	9.0490 × 10−2
F12	3.0199 × 10−11	3.0199 × 10−11	8.1014 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.6973 × 10−4
F13	8.1527 × 10−11	3.6897 × 10−11	7.7725 × 10−9	3.0199 × 10−11	1.9568 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	7.2208 × 10−6
F14	8.9934 × 10−11	3.1589 × 10−10	4.0772 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.6897 × 10−11	9.9186 × 10−11	4.9178 × 10−1
F15	9.5332 × 10−7	6.1210 × 10−10	2.2360 × 10−2	3.0199 × 10−11	3.8481 × 10−3	3.0199 × 10−11	4.0772 × 10−11	3.1967 × 10−9	2.1265 × 10−4
F16	3.3520 × 10−8	4.6390 × 10−5	4.7445 × 10−6	7.7725 × 10−9	9.8329 × 10−8	3.3384 × 10−11	2.0283 × 10−7	1.1567 × 10−7	4.8252 × 10−1
F17	1.6980 × 10−8	1.0907 × 10−5	2.1540 × 10−6	2.8716 × 10−10	6.7220 × 10−10	3.6897 × 10−11	8.8411 × 10−7	3.0811 × 10−8	3.2553 × 10−1
F18	1.3289 × 10−10	8.1014 × 10−10	4.5043 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	9.9186 × 10−11	4.1191 × 10−1
F19	2.5306 × 10−4	2.6695 × 10−9	2.3768 × 10−7	3.3384 × 10−11	2.1566 × 10−3	3.0199 × 10−11	3.0199 × 10−11	6.6955 × 10−11	6.9724 × 10−3
F20	3.3384 × 10−11	2.6099 × 10−10	1.5581 × 10−8	6.5277 × 10−8	4.9752 × 10−11	1.8567 × 10−9	5.0922 × 10−8	2.2273 × 10−9	7.7272 × 10−2
F21	1.3289 × 10−10	1.6132 × 10−10	1.5964 × 10−7	3.0199 × 10−11	3.6897 × 10−11	3.0199 × 10−11	8.8910 × 10−10	3.3384 × 10−11	2.0681 × 10−2
F22	1.2362 × 10−3	2.1327 × 10−5	3.6709 × 10−3	6.6689 × 10−3	1.1567 × 10−7	5.9673 × 10−9	8.6844 × 10−3	2.0283 × 10−7	2.2257 × 10−1
F23	4.5043 × 10−11	3.0199 × 10−11	1.2057 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	6.6955 × 10−11	3.0199 × 10−11	1.3017 × 10−3
F24	1.6132 × 10−10	4.5043 × 10−11	6.6955 × 10−11	8.1527 × 10−11	8.1527 × 10−11	3.0199 × 10−11	3.3384 × 10−11	3.0199 × 10−11	5.1877 × 10−2
F25	3.0199 × 10−11	3.0199 × 10−11	3.8202 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	6.1210 × 10−10	3.0199 × 10−11	6.2040 × 10−1
F26	5.4941 × 10−11	6.0658 × 10−11	9.7917 × 10−5	6.6955 × 10−11	1.6132 × 10−10	3.0199 × 10−11	2.3715 × 10−10	3.0199 × 10−11	1.1058 × 10−4
F27	1.1567 × 10−7	3.8202 × 10−10	4.1178 × 10−6	4.5043 × 10−11	7.3803 × 10−10	3.0199 × 10−11	1.7769 × 10−10	7.3891 × 10−11	1.2732 × 10−2
F28	3.0199 × 10−11	3.0199 × 10−11	3.6897 × 10−11	3.0199 × 10−11	3.6897 × 10−11	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	6.3772 × 10−3
F29	2.1544 × 10−10	2.1544 × 10−10	1.5465 × 10−9	3.6897 × 10−11	4.1997 × 10−10	3.0199 × 10−11	1.7769 × 10−10	4.0772 × 10−11	1.3017 × 10−3
F30	3.0199 × 10−11	3.0199 × 10−11	7.6588 × 10−5	3.0199 × 10−11	4.0772 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.8249 × 10−9

.

Table 8, Table 9 and

Table 10. Wilcoxon Rank-Sum Test results of QCAMBHO and 9 algorithms on the CEC 2017 (Dim = 100).

Algorithm	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO
F1	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.4617 × 10−9
F2	1.2118 × 10−12	1.2118 × 10−12	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.2118 × 10−12	3.0199 × 10−11	3.0199 × 10−11	9.0632 × 10−8
F3	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.3345 × 10−1
F4	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.0338 × 10−9
F5	3.0199 × 10−11	3.1589 × 10−10	2.0152 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.4941 × 10−11	3.0199 × 10−11	5.5546 × 10−2
F6	6.0104 × 10−8	5.4941 × 10−11	9.9186 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.2650 × 10−5
F7	3.0199 × 10−11	3.0199 × 10−11	5.0922 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.1825 × 10−9	3.0199 × 10−11	1.8682 × 10−5
F8	3.6897 × 10−11	9.9186 × 10−11	7.6950 × 10−8	4.5043 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.8716 × 10−10	3.0199 × 10−11	3.1830 × 10−1
F9	3.3384 × 10−11	3.6897 × 10−11	1.6132 × 10−10	2.1544 × 10−10	3.0199 × 10−11	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	1.4945 × 10−1
F10	1.7290 × 10−6	3.6459 × 10−8	5.0912 × 10−6	1.4298 × 10−5	2.1947 × 10−8	3.4742 × 10−10	2.1327 × 10−5	3.0199 × 10−11	9.5873 × 10−1
F11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.5329 × 10−8
F12	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.9980 × 10−9
F13	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.4617 × 10−9
F14	3.3384 × 10−11	2.4386 × 10−9	3.6897 × 10−11	3.0199 × 10−11	4.9752 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	8.1875 × 10−1
F15	3.0199 × 10−11	3.0199 × 10−11	2.0152 × 10−8	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.2780 × 10−5
F16	4.9752 × 10−11	1.6980 × 10−8	3.6439 × 10−2	1.0937 × 10−10	5.4617 × 10−9	3.0199 × 10−11	1.0702 × 10−9	3.0199 × 10−11	2.4581 × 10−1
F17	1.5581 × 10−8	9.5332 × 10−7	3.6459 × 10−8	1.4643 × 10−10	2.6695 × 10−9	3.0199 × 10−11	2.5721 × 10−7	8.8411 × 10−7	7.3940 × 10−1
F18	6.0658 × 10−11	6.7220 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.0772 × 10−11	5.7929 × 10−1
F19	3.0199 × 10−11	3.0199 × 10−11	1.7769 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.1825 × 10−9
F20	2.3715 × 10−10	3.1967 × 10−9	2.1540 × 10−6	1.8916 × 10−4	3.4971 × 10−9	1.1567 × 10−7	5.2640 × 10−4	1.7769 × 10−10	1.4128 × 10−1
F21	3.0199 × 10−11	4.0772 × 10−11	5.0723 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.5043 × 10−11	3.0199 × 10−11	7.6973 × 10−4
F22	1.0105 × 10−8	8.1014 × 10−10	6.0459 × 10−7	2.9590 × 10−5	8.8910 × 10−10	3.6897 × 10−11	6.2027 × 10−4	3.6897 × 10−11	5.8945 × 10−1
F23	9.9186 × 10−11	3.0199 × 10−11	4.0772 × 10−11	5.4941 × 10−11	4.0772 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.2259 × 10−3
F24	4.5043 × 10−11	3.3384 × 10−11	6.0658 × 10−11	3.6897 × 10−11	1.9568 × 10−10	3.0199 × 10−11	3.6897 × 10−11	3.0199 × 10−11	1.9963 × 10−5
F25	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.9980 × 10−9
F26	1.0937 × 10−10	3.0199 × 10−11	1.1023 × 10−8	3.0199 × 10−11	1.0702 × 10−9	3.0199 × 10−11	7.3891 × 10−11	3.0199 × 10−11	5.9428 × 10−2
F27	1.4643 × 10−10	4.9752 × 10−11	1.0277 × 10−6	3.3384 × 10−11	3.4971 × 10−9	3.0199 × 10−11	6.6955 × 10−11	3.0199 × 10−11	3.1821 × 10−4
F28	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.4386 × 10−9
F29	1.3289 × 10−10	6.0658 × 10−11	4.9426 × 10−5	3.0199 × 10−11	3.1967 × 10−9	3.0199 × 10−11	2.0338 × 10−9	3.0199 × 10−11	5.0842 × 10−3
F30	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	1.0937 × 10−10

present the Wilcoxon rank-sum test results between QCAMBHO and the nine comparison algorithms on the CEC 2017 test suite under different dimensional conditions. Specifically, these tables report the results under 30-dimensional, 50-dimensional, and 100-dimensional settings. At the significance level of 0.05, QCAMBHO shows statistically significant advantages over the comparison algorithms on most test functions. This indicates that its performance improvement does not come from random fluctuations, but has strong statistical reliability. In particular, when compared with LSHADE, LSHADE_cnEpSin, MELGWO, EWOA, HPHHO, AOO, and CFOA, the (p)-values on most functions are far less than 0.05, which further verifies the superiority of QCAMBHO on different types of optimization problems.

From the comparison with the original BHO, QCAMBHO also achieves significant advantages on most functions under the three-dimensional settings. This shows that the proposed improvement strategies effectively enhance the overall search performance of the original algorithm. Although the differences between the two algorithms do not reach a significant level on a few functions, this also indicates that QCAMBHO can at least maintain comparable performance to BHO on these functions, while achieving significant improvements on most other functions.

Table 11. Friedman Mean Rank Test Results.

Suites	CEC2017
Dimensions	30		50		100
Algorithm	AvgRank	Overall Rank	AvgRank	Overall Rank	AvgRank	Overall Rank
LSHADE	5.67	5	5.53	6	6.10	6
LSHADE_cnEpSin	4.47	3	5.23	4	5.53	5
TACPSO	4.77	4	4.53	3	4.93	4
MELGWO	8.23	9	7.80	9	7.13	8
EWOA	6.57	7	6.70	7	6.57	7
HPHHO	9.47	10	9.20	10	8.97	10
AOO	6.17	6	5.33	5	4.67	3
CFOA	6.67	8	7.43	8	7.90	9
BHO	1.83	2	2.03	2	1.97	2
QCAMBHO	1.17	1	1.20	1	1.23	1

presents the Friedman mean rank test results on the CEC2017 test suite under 30-dimensional, 50-dimensional, and 100-dimensional conditions. It can be seen that QCAMBHO obtains average ranks of 1.17, 1.20, and 1.23 in the three-dimensional scenarios, respectively, and ranks first overall in all cases. This indicates that QCAMBHO has the best comprehensive performance among all comparison algorithms. Meanwhile, the original BHO ranks second under all three-dimensional settings, suggesting that BHO itself has certain competitiveness, while QCAMBHO achieves further stable and significant performance improvement on this basis.

Figure 6a–c show the radar plots of the Friedman test results under 30-dimensional, 50-dimensional, and 100-dimensional conditions, respectively. Consistent with the results in Table 11, these visualizations clearly show that QCAMBHO is always located in the innermost position, further verifying its highest overall ranking.

From a practical perspective, the numerical optimization results indicate that QCAMBHO has good applicability to complex real-world optimization scenarios. The superior performance on unimodal functions suggests that the proposed algorithm has strong convergence accuracy and can efficiently refine promising solutions. Its competitive results on multimodal functions indicate that the adaptive Lévy flight strategy and quantum-enhanced initialization help the population explore different regions of the search space and reduce the risk of premature convergence. Moreover, the stable performance on hybrid and composition functions shows that QCAMBHO can maintain a better balance between global exploration and local exploitation in complex landscapes with multiple interacting structures. These characteristics are important for practical optimization tasks, where objective functions are often nonlinear, high-dimensional, and difficult to solve by deterministic methods. Therefore, the numerical experiments not only verify the benchmark performance of QCAMBHO, but also provide methodological support for its subsequent application to KELM parameter optimization in bankruptcy prediction.

4.5. Ablation Experiment

To further clarify the purpose of the ablation experiment, the meanings of the three single-strategy variants are first explained. QCAMBHO1 denotes the variant of BHO that only incorporates the quantum-computing-enhanced initialization strategy, while the adaptive Lévy flight strategy and the adaptive differential operator are not used. QCAMBHO2 denotes the variant that only introduces the adaptive Lévy flight strategy into the Rough Search stage of BHO, without using quantum-enhanced initialization or the adaptive differential operator. QCAMBHO3 denotes the variant that only applies the adaptive differential operator in the later stage of the algorithm, while the other two improvement strategies are removed. The complete QCAMBHO integrates all three strategies simultaneously. Therefore, the ablation experiment is designed to evaluate the independent contribution of each proposed strategy and to examine whether their integration can produce a synergistic effect. By comparing BHO, the three single-strategy variants, and the complete QCAMBHO, the role of each mechanism in improving population diversity, global exploration, and later-stage exploitation can be more clearly identified. The ablation results on the CEC2017 30-dimensional test suite are reported in

Table 12. Ablation experiment of QCAMBHO on the CEC 2017 (Dim = 30).

Algorithm	Metric	BHO	QCAMBHO1	QCAMBHO2	QCAMBHO3	QCAMBHO
F1	Mean	7.6348 × 103	5.4822 × 103	5.4501 × 103	6.4736 × 103	4.8646 × 103
	Std	7.5962 × 103	5.6937 × 103	4.8830 × 103	6.2062 × 103	5.7222 × 103
F2	Mean	2.0539 × 1015	2.8135 × 1014	7.4541 × 1013	1.1069 × 1012	2.0121 × 1011
	Std	7.7533 × 1015	1.0036 × 1015	3.8692 × 1014	3.4118 × 1012	1.0744 × 1012
F3	Mean	4.6310 × 103	4.5265 × 103	4.7761 × 103	2.8237 × 103	4.1645 × 103
	Std	2.2138 × 103	2.0234 × 103	2.8186 × 103	1.6366 × 103	2.9074 × 103
F4	Mean	4.9162 × 102	4.9397 × 102	4.8092 × 102	4.8231 × 102	4.8804 × 102
	Std	2.6872 × 101	2.5718 × 101	2.8038 × 101	2.3146 × 101	2.7127 × 101
F5	Mean	5.6519 × 102	5.6344 × 102	5.7448 × 102	5.6979 × 102	5.6692 × 102
	Std	2.2350 × 101	2.0638 × 101	1.6632 × 101	2.1228 × 101	1.7539 × 101
F6	Mean	6.0339 × 102	6.0349 × 102	6.0264 × 102	6.0316 × 102	6.0202 × 102
	Std	2.1395 × 100	1.8167 × 100	1.5654 × 100	1.3739 × 100	1.1367 × 100
F7	Mean	8.0749 × 102	8.1341 × 102	8.0545 × 102	8.0229 × 102	8.1107 × 102
	Std	2.1529 × 101	2.7220 × 101	2.9585 × 101	3.1260 × 101	2.9440 × 101
F8	Mean	8.5661 × 102	8.5664 × 102	8.6240 × 102	8.6843 × 102	8.6655 × 102
	Std	1.8339 × 101	1.7170 × 101	2.1205 × 101	2.0717 × 101	1.8833 × 101
F9	Mean	1.2715 × 103	1.2869 × 103	1.1053 × 103	1.1406 × 103	1.0803 × 103
	Std	3.4489 × 102	2.9938 × 102	1.7477 × 102	2.0241 × 102	1.6786 × 102
F10	Mean	4.5643 × 103	4.9200 × 103	4.4613 × 103	4.4436 × 103	4.3556 × 103
	Std	6.9110 × 102	5.8632 × 102	7.6933 × 102	5.8475 × 102	6.2754 × 102
F11	Mean	1.2127 × 103	1.2244 × 103	1.2124 × 103	1.2039 × 103	1.1950 × 103
	Std	4.6598 × 101	4.2231 × 101	4.1840 × 101	3.6176 × 101	3.8829 × 101
F12	Mean	2.0800 × 105	1.6261 × 105	1.2796 × 105	1.8466 × 105	9.2413 × 104
	Std	2.6518 × 105	2.0110 × 105	8.7683 × 104	1.7557 × 105	7.8754 × 104
F13	Mean	8.7228 × 103	8.1419 × 103	7.7870 × 103	5.0828 × 103	2.5494 × 103
	Std	3.8288 × 103	2.6356 × 103	1.1252 × 104	2.1448 × 103	1.2663 × 103
F14	Mean	1.6053 × 103	1.6031 × 103	1.6278 × 103	1.5761 × 103	1.6043 × 103
	Std	8.0173 × 101	7.3150 × 101	8.7727 × 101	6.1681 × 101	6.8031 × 101
F15	Mean	2.3735 × 103	2.0667 × 103	2.0412 × 103	1.8204 × 103	1.7914 × 103
	Std	1.2428 × 103	3.1635 × 102	2.4795 × 102	1.2949 × 102	2.0750 × 102
F16	Mean	2.2473 × 103	2.1873 × 103	2.2058 × 103	2.2056 × 103	2.1275 × 103
	Std	2.6033 × 102	2.3868 × 102	2.3201 × 102	2.3190 × 102	2.0553 × 102
F17	Mean	1.8976 × 103	1.8647 × 103	1.9093 × 103	1.8505 × 103	1.8612 × 103
	Std	9.9400 × 101	6.8903 × 101	8.4865 × 101	7.6994 × 101	7.4567 × 101
F18	Mean	2.5432 × 104	1.9375 × 104	2.0539 × 104	2.3952 × 104	2.6082 × 104
	Std	1.6056 × 104	1.3300 × 104	1.1718 × 104	1.8154 × 104	1.5421 × 104
F19	Mean	2.0940 × 103	2.1002 × 103	2.1262 × 103	2.0303 × 103	2.0811 × 103
	Std	7.3477 × 101	9.5667 × 101	1.1744 × 102	5.8367 × 101	1.3181 × 102
F20	Mean	2.2481 × 103	2.2455 × 103	2.2379 × 103	2.2372 × 103	2.1552 × 103
	Std	8.5785 × 101	8.9133 × 101	1.0096 × 102	8.1093 × 101	5.1190 × 101
F21	Mean	2.3524 × 103	2.3632 × 103	2.3613 × 103	2.3638 × 103	2.3640 × 103
	Std	4.6129 × 101	3.6232 × 101	3.6979 × 101	2.1269 × 101	1.5727 × 101
F22	Mean	2.3012 × 103	2.3016 × 103	2.3009 × 103	2.3010 × 103	2.3010 × 103
	Std	1.3342 × 100	2.4689 × 100	1.2308 × 100	1.4758 × 100	1.8069 × 100
F23	Mean	2.7235 × 103	2.7216 × 103	2.7329 × 103	2.7258 × 103	2.7295 × 103
	Std	2.4440 × 101	2.1220 × 101	2.1611 × 101	2.0482 × 101	2.3552 × 101
F24	Mean	2.9070 × 103	2.8993 × 103	2.8907 × 103	2.9004 × 103	2.8879 × 103
	Std	2.3066 × 101	2.1220 × 101	2.1516 × 101	2.5599 × 101	1.9803 × 101
F25	Mean	2.8913 × 103	2.8922 × 103	2.8891 × 103	2.8913 × 103	2.8893 × 103
	Std	1.0722 × 101	1.3399 × 101	4.7530 × 100	7.1125 × 100	6.8421 × 100
F26	Mean	3.9245 × 103	4.0298 × 103	3.6537 × 103	3.6180 × 103	3.2192 × 103
	Std	8.3588 × 102	6.2959 × 102	8.1713 × 102	7.7259 × 102	6.6277 × 102
F27	Mean	3.2309 × 103	3.2262 × 103	3.2270 × 103	3.2244 × 103	3.2247 × 103
	Std	2.1362 × 101	1.5164 × 101	1.1750 × 101	1.2759 × 101	1.0763 × 101
F28	Mean	3.2223 × 103	3.2252 × 103	3.2242 × 103	3.2156 × 103	3.2148 × 103
	Std	2.8317 × 101	2.0697 × 101	2.0279 × 101	1.5814 × 101	1.5597 × 101
F29	Mean	3.6123 × 103	3.6293 × 103	3.6454 × 103	3.6166 × 103	3.6274 × 103
	Std	1.1437 × 102	1.4898 × 102	1.2871 × 102	1.3195 × 102	1.4036 × 102
F30	Mean	1.5601 × 104	1.5116 × 104	9.1246 × 103	1.3159 × 104	7.9897 × 103
	Std	5.9555 × 103	5.3207 × 103	3.2200 × 103	4.5429 × 103	2.2887 × 103
AvgRank		3.77	3.60	3.07	2.50	2.07
Overall Rank		5	4	3	2	1

.

This indicates that all three improvement strategies can improve the search behavior of the original algorithm in different ways. Among them, QCAMBHO2 and QCAMBHO3 show more obvious improvements on several functions, while the complete QCAMBHO algorithm integrating all three strategies achieves the best overall performance. This suggests that the proposed improvement mechanisms have a good synergistic effect.

QCAMBHO obtains the best mean values for several functions, such as F1 and F2, showing the strongest overall optimization ability. In particular, for functions such as F2 and F12, QCAMBHO shows a more significant performance advantage over BHO and the single-strategy variants. This indicates that multi-strategy integration can effectively improve global exploration ability, local exploitation ability, and ability to escape from local optima. Meanwhile, QCAMBHO also maintains relatively low standard deviations on most functions, suggesting that it has good stability and robustness while obtaining better solutions.

From the results of the single-strategy variants, different strategies contribute differently to different types of functions. For example, QCAMBHO3 performs better on functions such as F3 and F14, indicating that this strategy has a positive effect on later-stage exploitation or search balance for some problems. QCAMBHO2 shows strong competitiveness for functions such as F4 and F6, suggesting that it has a certain effect on improving population quality and search accuracy. QCAMBHO1 also outperforms the original BHO on some functions, indicating that this strategy can effectively enhance the basic search performance.

Comparison with other competitive algorithms on CEC 2022:

This section evaluates the performance of the proposed QCAMBHO algorithm and other comparison algorithms on the CEC 2022 test suite under two-dimensional settings, namely 10 and 20 dimensions. The tables present the mean and standard deviation of the best solutions obtained by each algorithm on 12 test functions after 30 independent runs.

Table 13. Experimental results of 10 algorithms on the CEC 2022 (Dim = 10).

Algorithm	Metric	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO	QCAMBHO
F1	Mean	6.4086 × 102	3.0006 × 102	3.0000 × 102	3.0215 × 102	1.3372 × 103	4.9872 × 102	3.0044 × 102	5.7188 × 102	3.0000 × 102	3.0000 × 102
	Std	1.7256 × 103	1.7495 × 10−1	6.1712 × 10−6	5.0761 × 100	6.5673 × 102	2.1225 × 102	4.6132 × 10−1	4.1619 × 102	5.1711 × 10−14	6.8408 × 10−14
F2	Mean	4.0943 × 102	4.1022 × 102	4.0955 × 102	4.1050 × 102	4.1276 × 102	4.4145 × 102	4.0924 × 102	4.0389 × 102	4.0461 × 102	4.0386 × 102
	Std	1.4906 × 101	1.8156 × 101	1.8389 × 101	1.8521 × 101	2.5252 × 101	3.2416 × 101	1.2619 × 101	3.9397 × 100	2.5476 × 100	3.3519 × 100
F3	Mean	6.0036 × 102	6.0018 × 102	6.0018 × 102	6.0604 × 102	6.0553 × 102	6.2367 × 102	6.0440 × 102	6.0228 × 102	6.0000 × 102	6.0000 × 102
	Std	1.0709 × 100	3.2424 × 10−1	4.0170 × 10−1	5.9899 × 100	6.4811 × 100	1.4012 × 101	3.5245 × 100	1.8517 × 100	1.2492 × 10−3	4.6918 × 10−4
F4	Mean	8.1034 × 102	8.0796 × 102	8.1821 × 102	8.1920 × 102	8.2103 × 102	8.2698 × 102	8.2145 × 102	8.1043 × 102	8.0517 × 102	8.0571 × 102
	Std	3.7412 × 100	3.1740 × 100	1.1583 × 101	8.3373 × 100	7.3527 × 100	8.7261 × 100	1.0112 × 101	2.8022 × 100	2.2309 × 100	2.8585 × 100
F5	Mean	9.2218 × 102	9.0514 × 102	9.0207 × 102	9.8075 × 102	9.6368 × 102	1.1910 × 103	9.0264 × 102	9.0045 × 102	9.0009 × 102	9.0004 × 102
	Std	3.7580 × 101	8.4598 × 100	3.1464 × 100	1.0871 × 102	7.1420 × 101	1.5901 × 102	2.9064 × 100	5.3888 × 10−1	1.8684 × 10−1	8.8056 × 10−2
F6	Mean	1.9152 × 103	1.8602 × 103	3.3170 × 103	3.6678 × 103	3.0581 × 103	4.0223 × 103	5.1964 × 103	2.9471 × 103	1.8257 × 103	1.8212 × 103
	Std	1.8809 × 102	3.7190 × 101	1.7910 × 103	1.7439 × 103	1.0103 × 103	1.9562 × 103	2.3553 × 103	1.2209 × 103	1.6472 × 101	1.7822 × 101
F7	Mean	2.0160 × 103	2.0177 × 103	2.0214 × 103	2.0328 × 103	2.0287 × 103	2.0433 × 103	2.0322 × 103	2.0314 × 103	2.0087 × 103	2.0072 × 103
	Std	7.4461 × 100	7.5333 × 100	4.9945 × 100	1.0916 × 101	1.5020 × 101	1.7383 × 101	8.6149 × 100	5.0336 × 100	8.6589 × 100	8.9923 × 100
F8	Mean	2.2197 × 103	2.2235 × 103	2.2242 × 103	2.2235 × 103	2.2216 × 103	2.2292 × 103	2.2253 × 103	2.2234 × 103	2.2139 × 103	2.2057 × 103
	Std	5.0163 × 100	2.2681 × 101	2.2853 × 101	4.3064 × 100	3.9337 × 100	9.0780 × 100	2.5999 × 100	4.8222 × 100	8.7673 × 100	8.3489 × 100
F9	Mean	2.5342 × 103	2.5238 × 103	2.5369 × 103	2.5397 × 103	2.5293 × 103	2.5916 × 103	2.5324 × 103	2.5305 × 103	2.5202 × 103	2.5191 × 103
	Std	2.6826 × 101	5.8821 × 100	2.9714 × 101	3.7248 × 101	2.0682 × 10−3	5.8085 × 101	4.6871 × 100	3.3057 × 100	5.6346 × 100	6.8053 × 100
F10	Mean	2.5426 × 103	2.5404 × 103	2.5096 × 103	2.5645 × 103	2.5007 × 103	2.5212 × 103	2.5883 × 103	2.5333 × 103	2.5077 × 103	2.5112 × 103
	Std	6.2110 × 101	5.3304 × 101	3.2606 × 101	9.0513 × 101	2.2470 × 10−1	4.6224 × 101	1.5439 × 102	5.1156 × 101	2.7892 × 101	3.3104 × 101
F11	Mean	2.7487 × 103	2.7427 × 103	2.7472 × 103	2.6985 × 103	2.6819 × 103	2.6994 × 103	2.6845 × 103	2.6530 × 103	2.6451 × 103	2.6133 × 103
	Std	1.4300 × 102	1.3878 × 102	1.5052 × 102	1.7047 × 102	1.1270 × 102	1.2657 × 102	1.6155 × 102	9.6499 × 101	8.9486 × 101	7.3029 × 101
F12	Mean	2.8645 × 103	2.8594 × 103	2.8667 × 103	2.8680 × 103	2.8673 × 103	2.8721 × 103	2.8648 × 103	2.8648 × 103	2.8546 × 103	2.8554 × 103
	Std	1.6994 × 100	4.4705 × 100	2.8239 × 100	1.1858 × 101	2.5524 × 100	1.3050 × 101	2.6190 × 100	2.2587 × 100	2.7676 × 100	2.8552 × 100

and

Table 14. Experimental results of 10 algorithms on the CEC 2022 (Dim = 20).

Algorithm	Metric	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO	QCAMBHO
F1	Mean	1.9673 × 104	3.1564 × 104	3.0020 × 103	6.0981 × 103	2.2351 × 104	9.4875 × 103	2.4637 × 103	1.3494 × 104	3.0630 × 102	3.0010 × 102
	Std	1.8818 × 104	1.8695 × 104	1.9164 × 103	3.2031 × 103	4.5042 × 103	4.0638 × 103	1.5182 × 103	3.8238 × 103	1.3027 × 101	2.2531 × 10−1
F2	Mean	4.6707 × 102	4.8388 × 102	4.5809 × 102	5.1727 × 102	4.6375 × 102	5.2719 × 102	4.6517 × 102	5.1381 × 102	4.5129 × 102	4.4720 × 102
	Std	2.9672 × 101	4.1563 × 101	2.2452 × 101	4.2183 × 101	1.3827 × 101	4.8287 × 101	1.8875 × 101	4.6386 × 101	1.2560 × 101	1.0182 × 101
F3	Mean	6.0154 × 102	6.1822 × 102	6.0428 × 102	6.2943 × 102	6.1669 × 102	6.5234 × 102	6.2267 × 102	6.1461 × 102	6.0088 × 102	6.0032 × 102
	Std	2.2225 × 100	8.8193 × 100	2.5298 × 100	8.3716 × 100	7.3413 × 100	1.0694 × 101	6.3292 × 100	6.3844 × 100	9.0038 × 10−1	3.1163 × 10−1
F4	Mean	8.4886 × 102	8.7447 × 102	8.5545 × 102	8.6891 × 102	8.7529 × 102	8.9491 × 102	8.5961 × 102	8.6051 × 102	8.3160 × 102	8.2897 × 102
	Std	1.8024 × 101	1.3774 × 101	1.7086 × 101	1.5005 × 101	2.2458 × 101	1.1630 × 101	2.2454 × 101	1.1631 × 101	1.5225 × 101	1.0362 × 101
F5	Mean	1.4470 × 103	1.9497 × 103	1.1681 × 103	1.6194 × 103	1.8060 × 103	2.5485 × 103	1.6045 × 103	1.0471 × 103	9.2425 × 102	9.1966 × 102
	Std	3.6083 × 102	6.7940 × 102	2.7065 × 102	3.0079 × 102	5.3373 × 102	3.7349 × 102	6.0687 × 102	1.3170 × 102	2.6164 × 101	3.0208 × 101
F6	Mean	6.0581 × 103	6.5053 × 103	5.1122 × 103	8.7368 × 103	6.0699 × 103	1.2689 × 106	5.1829 × 103	3.7078 × 103	2.0347 × 103	2.1081 × 103
	Std	5.0717 × 103	5.2169 × 103	3.9097 × 103	1.0345 × 104	4.2855 × 103	2.7577 × 106	4.5230 × 103	2.1391 × 103	1.7822 × 102	5.0326 × 102
F7	Mean	2.0694 × 103	2.1088 × 103	2.0628 × 103	2.1259 × 103	2.0920 × 103	2.1306 × 103	2.0954 × 103	2.0888 × 103	2.0352 × 103	2.0356 × 103
	Std	4.6509 × 101	3.5736 × 101	2.0624 × 101	4.9214 × 101	3.3198 × 101	4.0338 × 101	3.6068 × 101	1.9046 × 101	9.0316 × 100	9.6408 × 100
F8	Mean	2.2547 × 103	2.2545 × 103	2.2478 × 103	2.2841 × 103	2.2600 × 103	2.2460 × 103	2.2581 × 103	2.2345 × 103	2.2232 × 103	2.2221 × 103
	Std	5.0449 × 101	4.6303 × 101	4.5563 × 101	6.8891 × 101	5.9719 × 101	2.4593 × 101	4.9060 × 101	2.2300 × 101	1.7666 × 100	1.6492 × 100
F9	Mean	2.4809 × 103	2.4883 × 103	2.4952 × 103	2.5040 × 103	2.4817 × 103	2.5179 × 103	2.4864 × 103	2.5019 × 103	2.4807 × 103	2.4808 × 103
	Std	1.0706 × 10−1	9.6559 × 100	2.4033 × 101	1.2883 × 101	8.8033 × 10−1	2.1991 × 101	5.6868 × 100	1.2049 × 101	2.9470 × 10−1	1.2278 × 10−2
F10	Mean	2.5298 × 103	3.2572 × 103	2.8018 × 103	3.8224 × 103	2.5126 × 103	2.5013 × 103	3.6307 × 103	3.0804 × 103	2.5369 × 103	2.5105 × 103
	Std	1.0540 × 102	5.2551 × 102	5.4835 × 102	8.8888 × 102	4.3723 × 101	3.1320 × 10−1	8.6122 × 102	1.0127 × 103	8.2165 × 101	3.8362 × 101
F11	Mean	2.9682 × 103	3.0963 × 103	2.9421 × 103	3.1181 × 103	2.9484 × 103	3.4424 × 103	2.9713 × 103	3.1060 × 103	2.9520 × 103	2.8633 × 103
	Std	1.7135 × 102	1.1862 × 102	2.1117 × 102	3.3004 × 102	1.1980 × 102	2.0591 × 102	2.1972 × 102	1.8019 × 102	9.1136 × 101	1.2452 × 102
F12	Mean	2.9746 × 103	2.9171 × 103	2.9708 × 103	2.9894 × 103	2.9958 × 103	3.0492 × 103	2.9863 × 103	2.9726 × 103	2.9355 × 103	2.9390 × 103
	Std	2.9359 × 101	1.6864 × 101	3.2584 × 101	3.8822 × 101	3.2219 × 101	6.1553 × 101	2.7567 × 101	2.1033 × 101	1.4589 × 101	1.1690 × 101

report the experimental results of QCAMBHO and the nine comparison algorithms on the CEC 2022 test suite under the 10-dimensional and 20-dimensional conditions.

Overall, QCAMBHO shows strong comprehensive optimization performance in both dimensional scenarios. Under the 10-dimensional condition, QCAMBHO obtains the best results for functions such as F2 and F5, and achieves results that are almost consistent with the best values for F1 and F3. This indicates that QCAMBHO already has high solution accuracy and stability in low-dimensional scenarios. Under the 20-dimensional condition, QCAMBHO further obtains the best or near-best results for functions such as F1 and F2, suggesting that the proposed algorithm can still maintain strong competitiveness as the problem complexity increases.

Compared with the original BHO, QCAMBHO obtains better mean values on most test functions. The improvement is especially clear for functions such as F1 and F2 in the 20-dimensional scenario, indicating that the proposed multi-strategy collaborative mechanism effectively improves the global search ability and solution quality of the algorithm. Meanwhile, QCAMBHO also maintains relatively low standard deviations for several functions, such as F1, F3, and F5 under the 10-dimensional condition, and F1, F3, F8, and F9 under the 20-dimensional condition. This shows that it has good robustness and result consistency in multiple independent runs.

From the overall trend across different dimensions, when the dimension increases from 10 to 20, the solution results of most algorithms deteriorate to varying degrees. However, QCAMBHO can still maintain good performance for most functions, reflecting its favorable dimensional adaptability. Nevertheless, QCAMBHO does not achieve the absolute best results for a few functions, such as F4, F10, and F12 under the 10-dimensional condition, and F6, F7, F9, and F12 under the 20-dimensional condition, where its results are close to those of BHO or some comparison algorithms. This indicates that some functions in the CEC 2022 test suite still place high requirements on local exploitation ability and search balance. However, based on the overall results, QCAMBHO still shows the best or highly competitive comprehensive performance.

Figure 7 and Figure 8 show the convergence curves and boxplots of QCAMBHO on several representative functions, respectively. As shown in Figure 7, QCAMBHO can rapidly reduce the fitness value in the early stages of iteration, showing strong early global search ability. As the iteration proceeds, its convergence curve continues to decrease and remains stable at a better level, indicating that the algorithm also has good local search performance in the later exploitation stages. Compared with the original BHO and other comparison algorithms, QCAMBHO generally achieves faster convergence speed and better final convergence accuracy.

As shown in Figure 8, the boxes corresponding to QCAMBHO are generally lower and more compact, and the median values are also located at better levels. This indicates that QCAMBHO can obtain better solution quality in multiple independent runs and has high stability. In contrast, the other comparison algorithms generally show wider boxes, longer whiskers, or more outliers, suggesting greater result fluctuations and relatively weaker robustness.

Table 15. Wilcoxon Rank-Sum Test results of QCAMBHO and 9 algorithms on the CEC 2022 (Dim = 10).

Algorithm	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO
F1	7.6870 × 10−12	7.6870 × 10−12	8.5486 × 10−12	7.6870 × 10−12	7.6870 × 10−12	7.6870 × 10−12	7.6870 × 10−12	7.6870 × 10−12	5.9090 × 10−1
F2	1.2362 × 10−3	1.7649 × 10−2	1.6235 × 10−1	1.1711 × 10−2	2.8378 × 10−1	1.8731 × 10−7	8.2919 × 10−6	2.5188 × 10−1	4.2889 × 10−1
F3	3.1589 × 10−10	3.6897 × 10−11	8.9934 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.7460 × 10−2
F4	1.6035 × 10−6	2.2642 × 10−3	4.7573 × 10−9	1.2837 × 10−9	9.8892 × 10−11	4.4905 × 10−11	8.9666 × 10−11	3.7982 × 10−7	6.8973 × 10−1
F5	1.7525 × 10−11	2.0916 × 10−9	1.4425 × 10−7	1.7525 × 10−11	1.5808 × 10−11	1.5808 × 10−11	3.1659 × 10−10	7.7774 × 10−7	8.0361 × 10−1
F6	1.1937 × 10−6	1.0277 × 10−6	2.3715 × 10−10	3.0199 × 10−11	7.3891 × 10−11	3.3384 × 10−11	3.0199 × 10−11	3.0199 × 10−11	9.9258 × 10−2
F7	2.5974 × 10−5	3.5708 × 10−6	3.4971 × 10−9	3.3384 × 10−11	6.5183 × 10−9	1.0937 × 10−10	4.5043 × 10−11	3.0199 × 10−11	1.5798 × 10−1
F8	1.5581 × 10−8	1.0702 × 10−9	1.0105 × 10−8	3.8202 × 10−10	1.7769 × 10−10	6.0658 × 10−11	3.3384 × 10−11	4.6159 × 10−10	3.0059 × 10−4
F9	3.0047 × 10−11	5.0842 × 10−3	9.2707 × 10−12	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.4933 × 10−1
F10	1.6062 × 10−6	7.2208 × 10−6	1.0277 × 10−6	1.1937 × 10−6	1.7294 × 10−7	3.3520 × 10−8	1.2023 × 10−8	5.9706 × 10−5	2.9047 × 10−1
F11	9.9355 × 10−10	1.4456 × 10−9	1.9387 × 10−7	2.6006 × 10−10	1.9107 × 10−9	2.3593 × 10−10	2.3593 × 10−10	2.8660 × 10−10	7.4805 × 10−2
F12	3.6897 × 10−11	2.3885 × 10−4	3.3342 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	6.0658 × 10−11	3.3384 × 10−11	6.6273 × 10−1

and

Table 16. Wilcoxon Rank-Sum Test results of QCAMBHO and 9 algorithms on the CEC 2022 (Dim = 20).

Algorithm	LSHADE	LSHADE_cnEpSin	TACPSO	MELGWO	EWOA	HPHHO	AOO	CFOA	BHO
F1	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	2.8716 × 10−10
F2	7.1186 × 10−9	1.1567 × 10−7	3.3386 × 10−3	7.3803 × 10−10	1.0105 × 10−8	3.6897 × 10−11	5.4620 × 10−6	4.0772 × 10−11	5.7460 × 10−2
F3	2.0058 × 10−4	3.0199 × 10−11	3.1589 × 10−10	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	4.2259 × 10−3
F4	2.6784 × 10−6	5.4941 × 10−11	6.0104 × 10−8	6.6955 × 10−11	6.6955 × 10−11	3.0199 × 10−11	6.5183 × 10−9	1.7769 × 10−10	9.2344 × 10−1
F5	5.4941 × 10−11	3.3384 × 10−11	6.5183 × 10−9	3.0199 × 10−11	4.0772 × 10−11	3.0199 × 10−11	3.6897 × 10−11	1.2023 × 10−8	5.3951 × 10−1
F6	1.4294 × 10−8	3.8249 × 10−9	1.3111 × 10−8	1.6947 × 10−9	2.3897 × 10−8	3.0199 × 10−11	2.0152 × 10−8	3.0103 × 10−7	3.7108 × 10−1
F7	1.8608 × 10−6	3.0199 × 10−11	2.3897 × 10−8	3.0199 × 10−11	2.1544 × 10−10	3.0199 × 10−11	8.9934 × 10−11	5.4941 × 10−11	9.8231 × 10−1
F8	6.1210 × 10−10	4.0772 × 10−11	1.3594 × 10−7	4.5043 × 10−11	7.0881 × 10−8	3.0199 × 10−11	4.0772 × 10−11	4.0772 × 10−11	1.7649 × 10−2
F9	3.0199 × 10−11	5.5727 × 10−10	3.0161 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	3.0199 × 10−11	5.2978 × 10−1
F10	1.0763 × 10−2	9.9186 × 10−11	1.8567 × 10−9	4.6159 × 10−10	7.7725 × 10−9	8.4848 × 10−9	4.1997 × 10−10	1.3111 × 10−8	6.3772 × 10−3
F11	5.8587 × 10−6	2.2273 × 10−9	1.5798 × 10−1	2.3897 × 10−8	3.5201 × 10−7	3.0199 × 10−11	7.0430 × 10−7	1.4294 × 10−8	1.4067 × 10−4
F12	3.0811 × 10−8	1.8608 × 10−6	7.6950 × 10−8	2.3715 × 10−10	3.6897 × 10−11	3.3384 × 10−11	1.3289 × 10−10	4.1997 × 10−10	1.4945 × 10−1

present the Wilcoxon rank-sum test results comparing QCAMBHO and the nine comparison algorithms on the CEC2022 test suite under 10-dimensional and 20-dimensional conditions. At the significance level of 0.05, QCAMBHO shows statistically significant advantages over the comparison algorithms for most test functions. This indicates that its performance improvement has strong statistical reliability and is not caused by random fluctuations. In particular, when compared with LSHADE, LSHADE_cnEpSin, MELGWO, EWOA, HPHHO, AOO, and CFOA, the (p)-values for most functions are significantly less than 0.05, further verifying the superiority of QCAMBHO on the CEC2022 test suite.

From the comparison with the original BHO, QCAMBHO also achieves significant advantages in several functions in the 10-dimensional and 20-dimensional scenarios. For example, under the 20-dimensional condition, the differences between QCAMBHO and BHO for F1, F3, F8, F10, and F11 all reach the significance level. This indicates that, as the problem dimension increases, the proposed improvement strategies can further enhance the search performance of the original algorithm. However, for some functions, the differences between the two algorithms do not reach a significant level, suggesting that QCAMBHO performs similarly to BHO for these functions, but this does not weaken its overall advantage.

Table 17. Friedman Mean Rank Test Results on CEC2022.

Suites	CEC2022
Dimensions	10		20
Algorithm	AvgRank	Overall Rank	AvgRank	Overall Rank
LSHADE	5.75	5	5.00	4
LSHADE_cnEpSin	4.83	3	7.08	8
TACPSO	5.75	5	4.17	3
MELGWO	8.00	9	8.42	9
EWOA	6.33	7	6.42	7
HPHHO	9.00	10	8.42	9
AOO	6.92	8	6.17	6
CFOA	5.08	4	5.67	5
BHO	1.83	2	2.17	2
QCAMBHO	1.50	1	1.50	1

presents the Friedman mean rank test results. It can be seen that QCAMBHO obtains an average rank of 1.50 in both dimensional scenarios and ranks first overall, indicating that it has the best comprehensive optimization performance among all comparison algorithms. The original BHO ranks second with average ranks of 1.83 and 2.17, respectively, suggesting that BHO itself has strong competitiveness, while QCAMBHO achieves further stable improvement on this basis. As shown in Figure 9a,b, QCAMBHO is always located in the innermost position, further verifying its highest overall ranking.

4.6. Sensitivity Analysis

To analyze the sensitivity of the adaptive Lévy flight parameter, five values of ${\beta }_{base}$, namely 0.9, 1.2, 1.5, 1.8, and 2.1, were tested on the CEC2017 30-dimensional benchmark functions. The five corresponding variants are denoted as QCAMBHO_S1 to QCAMBHO_S5. As reported in

Table 18. Results of sensitivity analysis.

Algorithm	QCAMBHO_S1	QCAMBHO_S2	QCAMBHO_S3	QCAMBHO_S4	QCAMBHO_S5
F1	5	2	3	1	4
F2	2	1	4	3	5
F3	5	4	1	3	2
F4	4	5	2	1	3
F5	5	2	3	1	4
F6	3	1	2	5	4
F7	4	3	1	2	5
F8	3	4	2	1	5
F9	1	3	2	5	4
F10	3	1	4	5	2
F11	4	3	2	5	1
F12	4	2	1	5	3
F13	3	2	1	4	5
F14	5	1	2	4	3
F15	2	3	1	5	4
F16	4	1	3	2	5
F17	4	5	2	3	1
F18	2	3	4	5	1
F19	4	3	1	5	2
F20	5	1	4	2	3
F21	5	1	4	2	3
F22	1	3	4	5	2
F23	5	3	1	2	4
F24	5	2	3	1	4
F25	4	5	1	2	3
F26	5	3	1	2	4
F27	2	1	3	5	4
F28	5	2	3	4	1
F29	2	4	3	1	5
F30	3	4	1	5	2
${\beta }_{base}$	0.9	1.2	1.5	1.8	2.1
Avg Rank	3.633333333	2.6	2.3	3.2	3.266666667
Overall Rank	5	2	1	3	4

, QCAMBHO_S3 obtains the best average rank of 2.30, followed by QCAMBHO_S2 with an average rank of 2.60. This indicates that moderate values of ${\beta }_{base}$ are more suitable for the proposed adaptive Lévy flight mechanism. When ${\beta }_{base}$ is too small, the search process may be overly disturbed by large jumps, resulting in reduced convergence stability. Conversely, when ${\beta }_{base}$ is too large, the heavy-tailed characteristic of Lévy flight may be weakened, which limits the ability of the algorithm to escape from local optima. Overall, ${\beta }_{base}=1.5$ achieves the best comprehensive performance and is therefore used as the default setting in this study.

5. Bankruptcy Prediction Problem

This section focuses on bankruptcy prediction using the constructed QCAMBHO-KELM model. It covers the experimental design, evaluation metrics, experimental results, and related analysis.

5.1. QCAMBHO-KELM

Kernel Extreme Learning Machine (KELM) is an improved form of Extreme Learning Machine (ELM), and its core idea is to introduce the kernel mapping mechanism into the learning framework. Compared with standard ELM, KELM avoids the uncertainty caused by random initialization of hidden-layer parameters and can implicitly map input samples into a high-dimensional feature space through kernel functions. Therefore, it is more suitable for describing nonlinear relationships in financial distress and bankruptcy prediction tasks.

KELM has been increasingly applied to bankruptcy prediction because of its nonlinear learning ability, fast training speed, and good generalization performance. Zhao et al. [69] proposed an effective bankruptcy prediction model based on KELM and adopted a two-step grid search strategy to tune the model parameters, showing that KELM can outperform several traditional statistical and machine learning models in terms of classification accuracy, Type I error, Type II error, and AUC. Wang et al. [70] further introduced a grey wolf optimizer-based KELM model for bankruptcy prediction, demonstrating that metaheuristic optimization can effectively improve the parameter selection of KELM and enhance prediction performance. More recently, several studies [71,72] have continued this research stream by combining improved swarm intelligence algorithms with KELM for bankruptcy prediction, such as EEOOA-KELM and EAZOA-KELM. These studies indicate that optimizing the key parameters of KELM, especially the regularization coefficient and kernel parameter, is crucial for improving its predictive accuracy and robustness in financial distress and bankruptcy prediction tasks. Motivated by this line of research, this study constructs a QCAMBHO-KELM model, in which the proposed QCAMBHO is used to search for the optimal KELM parameters and improve bankruptcy prediction performance. The structure of the KELM model is shown in Figure 10.

Based on KELM, this paper further constructs a QCAMBHO-KELM bankruptcy prediction model. In this framework, the proposed QCAMBHO algorithm is used to optimize the key parameters of KELM, such as the regularization coefficient and kernel parameter, thereby improving the classification performance and stability of the model. By combining the nonlinear learning ability of KELM with the global search ability of QCAMBHO, the proposed model can more effectively capture complex patterns in corporate financial data and improve bankruptcy prediction accuracy under high-dimensional and imbalanced data conditions. The workflow of the proposed QCAMBHO-KELM model is illustrated in Figure 11.

5.2. Experimental Setup

This study uses the Wieslaw dataset to verify the effectiveness of the QCAMBHO-KELM model in bankruptcy prediction. The dataset contains 240 samples, and each sample includes 30 feature indicators, mainly covering financial indicators and basic firm information. Typical features include the asset-liability ratio, net profit margin, current ratio, cash flow, and operating revenue. These attributes comprehensively reflect the financial status and operating performance of enterprises and serve as an important data source for bankruptcy prediction research. By analyzing these feature indicators, potential financial distress and bankruptcy risk can be effectively identified. The dataset can be obtained from Kaggle.

Table 19. List of the features based on the Wieslaw dataset and their definition.

NO	Features	NO	Features
C1	Cash/current liabilities	C16	Sales/receivables
C2	Cash/total assets	C17	Sales/total assets
C3	Current assets/current liabilities	C18	Sales/current assets
C4	Current assets/total assets	C19	365$\times$ receivables/sales
C5	Working capital/total assets	C20	Sales/total assets
C6	Working capital/sales	C21	Liabilities/total income
C7	Sales/inventory	C22	Current liabilities/total income
C8	Sales/receivables	C23	Receivables/liabilities
C9	Net profit/total assets	C24	Net profit/sales
C10	Net profit/current assets	C25	Liabilities/total assets
C11	Net profit/sales	C26	Liabilities/equity
C12	Gross profit/sales	C27	Long-term liabilities/equity
C13	Net profit/liabilities	C28	Current liabilities/equity
C14	Net profit/equity	C29	EBIT/total assets
C15	Net profit/(equity + long term liabilities)	C30	Current assets/sales

summarizes the detailed information of the Wieslaw dataset used in this study.

To evaluate the performance of the constructed QCAMBHO-KELM model in bankruptcy prediction, this paper compares it with several existing optimized models, including EWOA-KELM, HPHHO-KELM, MELGWO-KELM, TACPSO-KELM, CFOA-KELM, ALA-KELM, AOO-KELM, RIME-KELM, and BHO-KELM. For all these optimized models, the population size is uniformly set to 30, the problem dimension is determined by the number of features in the dataset, and the maximum number of iterations is set to 50, so as to ensure fairness and consistency in the comparative experiments. To ensure the objectivity and reliability of the experimental results, tenfold cross-validation is used to evaluate the classification performance of each model. Specifically, the dataset is first divided into 10 subsets of similar size. In each experiment, nine subsets are selected as the training set, while the remaining subset is used as the test set. This process is repeated 10 times, so that each subset is used exactly once as the test set. Finally, the average value of 30 experimental results is taken as the comprehensive evaluation result of each model.

5.3. Data Preprocessing and Experimental Settings

Before constructing the bankruptcy prediction model, the Wieslaw dataset was preprocessed to ensure that all input variables were suitable for KELM-based classification. The original dataset was imported from an Excel file, where the first several columns were labeled as financial feature variables, and the last column was used as the class label. The feature matrix and label vector were then separated. To meet the input requirement of the classification model, the class labels were converted into one-hot encoded vectors. In addition, since the original financial indicators may have different value ranges and scales, min–max normalization was applied to all feature variables, and each feature was linearly mapped into the interval [0, 1].

In the proposed QCAMBHO-KELM framework, QCAMBHO was used to optimize two key hyperparameters of KELM, namely the regularization parameter and the RBF kernel parameter. Therefore, the optimization dimension was set to 2. The lower and upper bounds of the two parameters were set to [1, 1] and [20, 20], respectively, and the radial basis function was adopted as the kernel function. The population size of each optimizer was set to 30, and the maximum number of iterations was set to 100. To ensure the fairness of comparison, all competing algorithms were configured with the same population size, maximum iteration number, search dimension, parameter boundaries, and fitness evaluation function. For comparative analysis, QCAMBHO-KELM was compared with nine optimizer-assisted KELM models, including EWOA-KELM, HPHHO-KELM, MELGWO-KELM, TACPSO-KELM, CFOA-KELM, ALA-KELM, AOO-KELM, RIME-KELM, and BHO-KELM. Each algorithm was independently executed 30 times to reduce the influence of randomness. In each run, the optimizer first searched for the optimal KELM hyperparameter combination, and then the obtained optimal parameters were used to evaluate the classification performance of KELM. The fitness function was defined based on the classification performance of KELM, and cross-validation was adopted during model evaluation to improve the reliability of the results.

5.4. Measures for Performance Evaluation

To comprehensively evaluate the classification performance of the proposed model in bankruptcy prediction, this paper selects accuracy (ACC), Matthews correlation coefficient (MCC), sensitivity, specificity, precision, recall, and F1-score as evaluation metrics. These metrics can systematically measure model performance from multiple perspectives, including overall classification correctness, positive and negative sample identification ability, and the balance of prediction results. Therefore, they can more comprehensively reflect the effectiveness and robustness of the model in bankruptcy prediction.

5.5. Experimental Results and Analysis

According to the results in the table, QCAMBHO achieves the best results on all evaluation metrics, indicating that it has the best overall classification performance in the bankruptcy prediction task. Specifically, the ACC, MCC, and F1-score of QCAMBHO reach 81.872%, 0.6463, and 81.120%, respectively, all of which are higher than those of the original BHO and the other comparison algorithms. This shows that the proposed model can not only effectively improve overall classification accuracy but also achieve better performance in comprehensive discrimination between positive and negative samples and the balance of class identification.

QCAMBHO achieves 83.553% in both Sensitivity and Recall, and 80.427% in Specificity, indicating that the model maintains a good balance in identifying both bankrupt and non-bankrupt firms. The higher Sensitivity and Recall values show that the model has a stronger ability to capture bankrupt samples, which helps reduce the possibility of missing high-risk firms. Meanwhile, the higher Specificity indicates that it can also distinguish normal firms effectively and avoid excessive false alarms. In addition, the Precision of QCAMBHO reaches 79.882%, and its F1-score also remains at the highest level, further showing that the model achieves a better trade-off between precision and recall.

Compared with the original BHO, QCAMBHO achieves further improvements in key metrics such as ACC, MCC, Sensitivity, Precision, Recall, and F1-score. This indicates that the proposed improvement strategies can effectively enhance the ability of the basic optimization algorithm to search for the optimal KELM parameters, thereby improving the classification performance of the model. Overall, the experimental results verify the effectiveness and superiority of the QCAMBHO-KELM model in enterprise bankruptcy prediction, and also show that the proposed optimization mechanism has good application potential in practical classification problems.

Figure 12 presents the boxplot results of each algorithm in the bankruptcy prediction task. Overall, QCAMBHO shows better distribution characteristics on the main evaluation metrics, including ACC, MCC, Precision, Recall, Sensitivity, and F1-score. Its boxes are generally located at higher positions, the median values are better, and the box widths are relatively smaller. This indicates that the model not only achieves better classification performance, but also maintains high stability and robustness in multiple independent runs. In contrast, the other comparison algorithms generally show lower box positions, larger fluctuation ranges, or more outliers on most metrics, suggesting relatively weaker consistency in classification results. The bankruptcy prediction results of QCAMBHO-KELM are reported in

Table 20. QCAMBHO-KELM Bankruptcy Prediction Results.

Algorithm	ACC Mean	MCC Mean	Sensitivity Mean	Specificity Mean	Precision Mean	Recall Mean	F1 Mean
LSHADE	7.5274 × 101	5.1451 × 10−1	7.6164 × 101	7.4476 × 101	7.3334 × 101	7.6164 × 101	7.3964 × 101
LSHADE_cnEpSin	7.1780 × 101	4.4395 × 10−1	7.3556 × 101	7.0199 × 101	6.9110 × 101	7.3556 × 101	7.0618 × 101
TACPSO	7.6310 × 101	5.3452 × 10−1	7.7497 × 101	7.5316 × 101	7.4154 × 101	7.7497 × 101	7.5147 × 101
MELGWO	7.6130 × 101	5.3226 × 10−1	7.6899 × 101	7.5487 × 101	7.4348 × 101	7.6899 × 101	7.4829 × 101
EWOA	7.5545 × 101	5.1968 × 10−1	7.6490 × 101	7.4729 × 101	7.3686 × 101	7.6490 × 101	7.4367 × 101
HPHHO	7.6310 × 101	5.3481 × 10−1	7.7227 × 101	7.5509 × 101	7.4510 × 101	7.7227 × 101	7.5166 × 101
AOO	7.5717 × 101	5.2344 × 10−1	7.6556 × 101	7.4985 × 101	7.3859 × 101	7.6556 × 101	7.4403 × 101
CFOA	7.5974 × 101	5.2863 × 10−1	7.6586 × 101	7.5442 × 101	7.4198 × 101	7.6586 × 101	7.4613 × 101
BHO	8.1645 × 101	6.4106 × 10−1	8.3088 × 101	8.0368 × 101	7.9563 × 101	8.3088 × 101	8.0743 × 101
QCAMBHO	8.1872 × 101	6.4629 × 10−1	8.3553 × 101	8.0427 × 101	7.9882 × 101	8.3553 × 101	8.1120 × 101

.

6. Conclusions and Future Works

This paper proposes an improved BHO algorithm named QCAMBHO to address the limitations of the original BHO in complex optimization problems, such as insufficient population diversity, weak balance between global exploration and local exploitation, and limited later-stage search ability. QCAMBHO integrates a quantum-computing-enhanced initialization strategy, an adaptive Levy flight strategy, and an adaptive differential operator to improve initial solution quality, enhance global exploration, strengthen local exploitation, and reduce the risk of falling into local optima. Numerical experiments on the CEC2017 and CEC2022 test suites show that QCAMBHO achieves better or competitive performance compared with the comparison algorithms on most benchmark functions. The results of mean value, standard deviation, Wilcoxon rank-sum test, Friedman mean rank test, convergence curves, and boxplots further verify its strong optimization accuracy, stability, robustness, and cross-dimensional adaptability. The ablation experiments also confirm that the three improvement strategies contribute positively to the algorithm and that their integration produces a good synergistic effect. In addition, QCAMBHO is used to optimize the parameters of KELM, and the constructed QCAMBHO-KELM model achieves superior classification performance in enterprise bankruptcy prediction, further demonstrating the practical application value of the proposed algorithm.

Although QCAMBHO achieves promising results in numerical optimization and bankruptcy prediction, this study still has several limitations that should be acknowledged. First, the effectiveness of QCAMBHO is mainly verified through empirical experiments on benchmark functions and bankruptcy prediction datasets, while its convergence mechanism and dynamic search behavior have not yet been fully supported by rigorous theoretical analysis. Future work can further investigate the convergence properties, parameter-action mechanisms, and search stability of QCAMBHO from a mathematical perspective. Second, although the proposed adaptive mechanisms improve the balance between exploration and exploitation, their parameter settings and triggering conditions are still designed based on the current algorithmic framework. For problems with different structures, constraints, or search scales, more flexible feedback control strategies may be needed to further enhance the self-adaptive ability of the algorithm.

Third, the computational cost of QCAMBHO is higher than that of the original BHO because the proposed quantum-computing-enhanced initialization, adaptive Lévy flight, and adaptive differential operator introduce additional operations. Although the improvement in optimization accuracy and prediction performance can partly justify this extra cost, future studies should further examine the trade-off between performance gain and computational efficiency under equal function evaluations or equal running time. Fourth, the bankruptcy prediction experiment in this paper mainly focuses on improving classification performance through KELM parameter optimization. The interpretability of QCAMBHO-KELM and its robustness across broader financial datasets still require further investigation. In future research, QCAMBHO can be extended to constrained optimization, multi-objective optimization, high-dimensional feature selection, and complex engineering optimization problems. For bankruptcy prediction, future studies can further combine QCAMBHO with more machine learning or deep learning models, introduce multi-source information such as textual disclosures, corporate governance variables, and market indicators, and incorporate feature importance or SHAP-based explanation methods to build more accurate, generalizable, and interpretable enterprise risk early-warning models.