A Multi-Strategy Harris Hawks Optimization and Its Application in Feature Selection

Abstract

Feature selection (FS) is a pivotal preprocessing task in data mining aimed at identifying optimal feature subsets to improve model generalization and reduce computational overhead. However, its NP-hard nature poses significant challenges for traditional optimizers in terms of search efficiency and solution quality. The Harris Hawks Optimization (HHO) algorithm is a state-of-the-art population-based metaheuristic method that demonstrates powerful capabilities in various optimization challenges. Despite its advantages, HHO encounters problems such as early stagnation and reduced accuracy. To mitigate these problems, we introduce an advanced algorithm called the Hybrid Strategy Harris Hawks Optimization (HSHHO). The HSHHO combines three key enhancements to support global search diversity and local refinement: (1) an exploration mechanism that utilizes the Self-Parameterized Map (SPM) alongside a dynamic logarithmic spiral to expand search breadth; (2) a nonlinear adjustment to the escape energy parameter for improved phase equilibrium; and (3) an elite perturbation approach that uses Cauchy–Gaussian mutation to strengthen local optimization and solution quality. We assessed HSHHO against eight well-known algorithms on 30 benchmark functions, where it exhibited superior results in the majority of cases. Finally, HSHHO is applied to address 18 feature selection tasks. The results demonstrated that HSHHO achieved highly competitive outcomes in terms of objective values, feature subset size, and classification performance in most datasets, reaching an average accuracy of 94.47%.

1. Introduction

Nowadays, various industries generate a large amount of data, and finding valuable insights from multi-source data is a daunting challenge for data processing [1]. Model performance often requires clean, non-redundant input data. However, the collected raw data often has redundant, irrelevant, and noisy features [2]. High-dimensional data can lead to a curse of dimensionality, severely reducing the model’s generalization ability.

To address these issues, researchers widely use feature selection (FS) methods to identify key attributes in the data. The goal of feature selection is to find the optimal subset of features to eliminate irrelevant or redundant features, thereby improving model accuracy and reducing computational complexity [3]. Essentially, FS is a combinatorial optimization problem, where given a dataset with $n$ features, the objective is to select the optimal feature subset from $2^n$ candidate feature combinations [4]. Due to its NP hard nature and inherent multi-objective nature, this task has attracted in-depth research in recent years [5]. At present, the FS method has been applied in classification and regression tasks [6,7], engineering optimization [8,9], text classification [10,11], and many other fields. Especially in complex engineering applications, such as railway transportation emergency response scenarios, due to the involvement of multi-disciplinary collaboration, the emergency response process generates massive amounts of multi feature data. Researchers need to identify key influencing factors from these redundant and noisy raw features. FS, as a key preprocessing task, can effectively eliminate irrelevant variables in railway emergency data, thereby improving emergency decision-making capabilities in the event of sudden accidents.

FS approaches fall into three categories: filter, wrapper, and embedded methods [12,13,14,15,16]. Filter methods are computationally lightweight and classifier-independent, which explains their widespread popularity. However, they often produce suboptimal feature subsets and are prone to overfitting [17]. Wrapper methods, by contrast, evaluate subsets directly through a specific classifier; although they are more accurate and less susceptible to overfitting than filters, they can still select biased features when used in isolation and are computationally more demanding. Embedded methods integrate feature selection into the training process of the classifier itself and have therefore drawn considerable interest; numerous hybrid strategies have been proposed to further enhance their effectiveness [18,19]. Regardless of category, these techniques are typically paired with well-known classifiers such as decision trees [20,21], support vector machines [22], Bayesian networks [23,24], k-nearest neighbors [25], and artificial neural networks [26,27].

In recent years, metaheuristic optimization algorithms have become the main method for solving FS problems due to their powerful global search and local optimization capabilities [28,29,30,31,32,33]. The landscape optimized by metaheuristics can be divided into two main schools of thought. The first genre includes mature classic algorithms that have been rigorously tested for decades, such as genetic algorithms (GA), differential evolution (DE), and simulated annealing (SA). The second school represents recent biologically inspired swarm intelligence methods that utilize collective behavior in nature to solve complex engineering problems. For example, Particle Swarm Optimization (PSO), Gray Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), etc. These optimization algorithms can generate a set of candidate solutions. Their search process consists of two basic stages, the exploration stage and the exploitation stage [34]. However, the imbalance between the two stages is the main reason why these algorithms fall into local optima when processing high-dimensional data.

To overcome these limitations, researchers are increasingly turning to hybrid algorithms that combine complementary advantages. For example, chaotic cosine firefly (CSCF) algorithm [35], PSO-GWO hybrid algorithm [36], GSA-GA fusion algorithm [37], quantum enhanced bat algorithm (CCQBA) [38], and opponent based social spider optimizer (OBSSO) [39]. These hybrid algorithms demonstrate excellent solving capabilities in feature selection problems.

Among the popular swarm intelligence algorithms, the HHO algorithm has received widespread attention due to its simple mathematical structure and effectiveness in solving various engineering problems [40]. However, HHO performs poorly in dealing with high-dimensional and complex multimodal problems. This is because HHO has limitations of premature convergence and low solution accuracy. Therefore, it is urgent to propose improvement strategies to optimize HHO in order to enhance the algorithm’s fast gaming performance. At present, several enhancement methods have been proposed to alleviate these weaknesses: neighborhood perturbation [41], sine escape energy adjustment [42], logarithmic spiral exploration [43], and tent graph plus crossover mutation strategy [44]. Based on these insights, the present work introduces a novel algorithm called Hybrid Strategy Harris Hawks Optimization (HSHHO). As an improved feature selection wrapper method, HSHHO aims to obtain the optimal feature subset while improving classification accuracy.

The core contribution lies in three targeted improvement measures aimed at strengthening the exploration, transition, and mining stages of HHO:

(1)

SPM chaotic mapping and variable logarithmic spiral strategy were proposed to enhance burst diversity and global search.

(2)

A nonlinear escape energy function has been proposed, achieving a smoother and more adaptive balance between exploration and exploitation.

(3)

The Cauchy–Gaussian elite perturbation mechanism has been proposed, which can promote local refinement and help algorithms overcome local optima.

The rest of this article is organized as follows. Section 2 reviewed the standard HHO algorithm and proposed three improvement strategies for constructing HSHHO. Section 3 reports a comprehensive experiment on 30 benchmark functions and 18 real-world feature selection datasets, comparing HSHHO with eight well-known methods. Finally, Section 4 summarizes the research findings and outlines future research directions.

2. Methods

2.1. Harris Hawks Optimization

The Harris Hawks Optimization (HHO) is a population-based metaheuristic, inspired by the collaborative hunting tactics of Harris hawks in pursuing prey. This algorithm has shown robust performance in addressing various optimization challenges. To cope with the variability in prey behavior and environmental conditions, hawks adapt their strategies dynamically. Empirical evaluations indicate that HHO often surpasses other metaheuristics in terms of solution quality. The algorithm operates in two primary phases—exploration and exploitation—driven by the prey’s escape energy, as illustrated in Figure 1. Exploration employs two position-update mechanisms, while exploitation features four distinct strategies. The detail of the HHO is as follows.

Figure 1 shows the framework of the HHO algorithm, demonstrating the dynamic behavior between different hunting stages. The optimization process is mainly determined by the escape energy $E$ of the prey. When $\left|E\right|\ge 1$, the algorithm is in the exploration stage. When $\left|E\right|<1$, the algorithm will enter the exploitation stage. In these stages, the specific strategy is determined by two variables: $q$ determines the habitat mechanism during exploration, while $r$ represents the probability of prey successfully escaping during exploitation. The arrow indicates the behavioral change in the eagle’s position from global search to local capture.

2.1.1. Exploration Phase

In HHO, each hawk represents a candidate solution within the population. Through iterative updates, hawks adjust their positions relative to the current best location. During exploration, two strategies are applied based on a random threshold $q$. If $q \ge 0.5$, hawks update positions using their own location and those of nearby individuals. If $q < 0.5$, they select a random perch within the search space for observation. The mathematical formulations are:

$$X\left(t+1\right)=\left\{\begin{array}{c}{X}_{rant}\left(t\right)-r_1\left|X_{rant}\left(t\right)-2r_{2}X\left(t\right)\right| q\ge 0.5\\ \left(X_{rabbit}\left(t\right)-X_m\left(t\right)\right)-r_3\left(LB+r_4\left(UB-LB\right)\right) q<0.5\end{array}\right.$$

(1)

where $X\left(t+1\right)$ is the expressed hawk position at iteration $t+1$. $X_{rant}\left(t\right)$ is a randomly chosen hawk position at iteration $t$. $X_{rabbit}\left(t\right)$ is the prey’s position (best solution) at iteration $t$. $r_1$, $r_2$, $r_3$, $r_4$ are random values in (0, 1). $UB$ and $LB$ denote the upper and lower bounds of the search space. The average population position $X_m\left(t\right)$ is computed as:

$$X_m\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{X}_i\left(t\right)$$

(2)

2.1.2. Transition Phase

The transition from exploration to exploitation hinges on the prey’s escape energy $E$, which dictates the hawks’ pursuit tactics. Initially, hawks scan broadly for prey; upon detection, they shift to targeted attacks. This phase ensures an equilibrium between global and local search, enhancing overall efficiency. The escape energy is modeled as:

$$E={2E}_0\left(1-{\displaystyle \frac{t}{T}}\right)$$

(3)

where $E_0$ represents is the initial escape energy, which is a random value with [−1, 1]. $t$ is the current iteration, and $T$ is the maximum iterations. $E$ decreases linearly from 2 to 0 over iterations, as depicted in the light green area of Figure 1.

2.1.3. Exploitation Phase

Exploitation encompasses four strategies, selected based on $E$ and an escape probability $r$. If $r\ge 0.5$, the prey escapes successfully; otherwise, it fails. Figure 1 outlines the sequential application of these strategies.

• Soft besiege strategy

When $\left|E\right|\ge 0.5$ and $r\ge 0.5$, hawks gradually encircle the prey with sufficient energy remaining:

$$X\left(t+1\right)=∆X\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X\left(t\right)\right|$$

(4)

$$∆X\left(t\right)=X_{rabbit}\left(t\right)-X\left(t\right)$$

(5)

where $J$ is a random number with [0, 2] denoting the strength of the rabbit’s jumps during its escape. $X_{rabbit}\left(t\right)$ denotes the best solution in iteration $t$.

2.

Hard besiege strategy

When $\left|E\right|<0.5$ and $r\ge 0.5$, the prey is fatigued, prompting a direct assault:

$$X\left(t+1\right)=X_{rabbit}\left(t\right)-E\left|∆X\left(t\right)\right|$$

(6)

3.

Soft besiege with progressive rapid dives

When $\left|E\right|\ge 0.5$ and $r<0.5$, the prey evades energetically, leading to Lévy-flight-based dives:

$$Y=X_{rabbit}\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X\left(t\right)\right|$$

(7)

$$Z=Y+S\times LF\left(D\right)$$

(8)

$$X\left(t+1\right)=\left\{\begin{array}{c}Y if F\left(Y\right)<F\left(X\left(t\right)\right)\\ Z if F\left(Z\right)<F\left(X\left(t\right)\right)\end{array}\right.$$

(9)

where $S$ is a random vector in [0, 1]. Y and Z denote the two strategies, respectively. $LF$ denotes the Lévy flight function [45]:

$$LF\left(x\right)=0.01\times {\displaystyle \frac{\mu \times \sigma }{{\left|v\right|}^{\frac{1}{p}}}}$$

(10)

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

(11)

where $\mu$ and $v$ normally distributed in (0, 1). $\beta$ is 1.5 acquiescently.

4.

Hard besiege with progressive rapid dives

When $\left|E\right|<0.5$ and $r<0.5$, a similar dive strategy is applied but with reduced energy:

$$Y^{\prime }=X_{rabbit}\left(t\right)-E\left|J{X}_{rabbit}\left(t\right)-X_m\left(t\right)\right|$$

(12)

$$Z^{\prime }=Y^{\prime }+S\times LF\left(D\right)$$

(13)

$$X\left(t+1\right)=\left\{\begin{array}{c}{Y}^{\prime } if F\left(Y^{\prime }\right)<F\left(X\left(t\right)\right)\\ Z^{\prime } if F\left(Z^{\prime }\right)<F\left(X\left(t\right)\right)\end{array}\right.$$

(14)

2.2. Hybrid Strategy Harris Hawks Optimization

While HHO excels in dynamic search, it shares common pitfalls with other swarm methods, such as limited accuracy and local optima entrapment when facing complex problems. These arise from random initialization and rigid exploration/exploitation mechanics. To remedy this, we propose HSHHO, incorporating: (1) SPM chaotic map and variable logarithmic spiral (SPM-VLS) for exploration; (2) nonlinear escape energy for transition; and (3) Cauchy–Gaussian elite perturbation for exploitation. The three proposed improvement strategies effectively address the inherent weaknesses of HHO. Although SPM-VLS expands the initial search range, the nonlinear energy method maintains a balance between the exploration stage and the mining stage during the mid-term iteration, and the dual mutation mechanism in the mining stage escapes the local optimal solution. This synergy ensures that each stage of the optimization process is particularly strengthened.

2.2.1. Population Initialization Based on a Chaotic Map

The population initialization of optimization algorithms is one of the key steps. The distribution of the population affects the search performance of the algorithm in the early stage. In the HHO algorithm, the initial population position is randomly generated, which may lead to an uneven distribution of individuals. The diversity of the initial population is the key to algorithm optimization, and it is necessary to enhance population diversity to improve overall search capability. Introducing chaotic mapping into the algorithm initialization phase can alleviate this problem. Chaotic mapping can provide traversal and non-repetitive sequences, which are superior to random initialization in improving population diversity [46]. This study chooses the SPM mapping method to replace the traditional random generation method. Figure 2 and Figure 3 show the distribution histograms and scatter plots of SPM, respectively. It is not difficult to see that the population generated using SPM mapping exhibits a uniform distribution. Therefore, SPM effectively enhances the diversity of the initial population, which is beneficial for the global search of the algorithm.

Therefore, we introduce the SPM to improve the population initialization of HHO:

$$x\left(t+1\right)=\left\{\begin{array}{c}mod\left({\displaystyle \frac{x\left(t\right)}{\mu }}+\delta \mathit{sin}\left(\pi x\left(t\right)\right)+r,1\right) 0\le x\left(t\right)\le \mu \\ mod\left({\displaystyle \frac{\raisebox{1ex}{$x\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.}{0.5-\mu }}+\delta \mathit{sin}\left(\pi x\left(t\right)\right)+r,1\right) \mu \le x\left(t\right)\le 0.5\\ mod\left({\displaystyle \frac{1-\raisebox{1ex}{$x\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.}{0.5-\mu }}+\delta \mathit{sin}\left(\pi \left(1-x\left(t\right)\right)\right)+r,1\right) 0.5\le x\left(t\right)\le 1-\mu \\ mod\left({\displaystyle \frac{1-\raisebox{1ex}{$x\left(t\right)$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.}{\mu }}+\delta \mathit{sin}\left(\pi \left(1-x\left(t\right)\right)\right)+r,1\right) 1-\mu \le x\left(t\right)\le 1\end{array}\right.$$

(15)

where if $\mu \in \left(0,1\right), \delta \in (0,1)$, the system is in a chaotic state. $r\in (0, 1)$ is a random number.

Therefore, we can obtain the position of $x_{position}$:

$$x_{position}=lb+x\left(t\right)\times \left(ub-lb\right)$$

(16)

2.2.2. Variable Logarithmic Spiral Strategy

HHO’s exploration relies heavily on random factors, which can lead to premature convergence. We integrate a variable logarithmic spiral, inspired by whale and moth algorithms [47], to refine position updates:

$$X_{t+1}=\left|X_{best}-X_t\right|\times e^{bl}\times cos\left(2\pi l\right)+X_{best}$$

(17)

where $X_{best}$ represents the best solution. $l$ represents a random number in [−1, 1]. $b$ dynamically adjusts:

$$b=e^{\pi \left(1-{\displaystyle \frac{t}{T}}\right)}$$

(18)

This enables larger spirals early on and finer adjustments later.

2.2.3. Improvement Strategies for the Transition Phase

The transition phase of HHO is based on the escape energy $E$. The main purpose of the transition phase is to maintain a balance between exploration phase and exploitation phase. We propose a nonlinear method to address the limitation of linear function.

$$E\left(t\right)=\left\{\begin{array}{ll}{E}_{min}+\left(E_{max}-E_{min}\right)\times {\left(cos{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{t}{T}}\right)}^2& q>0.5\\ E_{min}+\left(E_{max}-E_{min}\right)\times \left(1-{\displaystyle \frac{‖X_t-X_{best}‖}{‖{X_m-X}_{best}‖}}\right)& q<0.5\end{array}\right.$$

(19)

where $X_m$ represents the average position of the population. $E_{min}$ is the minimum value, $E_{max}$ is the maximum value. $\rho \in \left[0,1\right]$ of random numbers.

Equation (19) shows that $E\left(t\right)$ varies nonlinearly with the number of iterations. Based on Figure 1, when $q>0.5$, $E\left(t\right)$ decreases nonlinearly from the exploration to the exploitation. According to the variation curve of the cos(x) function, with $E\left(t\right)$ in the middle of the algorithm iteration, the rate of change accelerates, which enables the HHO to quickly switch to the exploitation phase. When $q<0.5$ and $E\left(t\right)$ depends on the distance between the position of the Harris hawks and the position of the optimal solution, the closer to the optimal solution, the larger $E\left(t\right)$ is, and the algorithm will shift from the exploitation stage to the exploration phase, which can relieve local population accumulation and avoid getting into a local optimum in the later stage of the algorithm.

2.2.4. Improvement Strategies for the Exploitation Phase

To overcome limitations of HHO, we introduce an elite-guided perturbation strategy [48]. The disturbance probability $P_{dis}$ is introduced to determine whether the current optimal individual $X_{best}$ performs the elite perturbation operation:

$$P_{dis}=\left(D-1\right)\times {\displaystyle \frac{e^{\left(\frac{t-1}{T}\right)}}{4D}}$$

(20)

where $P_{dis}$ is the disturbance probability of an elite individual and $D$ is the problem dimension.

To improve the capability of running away from local optima, an improved method is proposed by integrating the Cauchy–Gaussian mutation mechanism into the traditional elite perturbation strategy. The specific calculation formula of the Cauchy–Gaussian mutation is expressed as follows:

$${Positon}_{best}\left(t\right)=X\left(t\right)\times \left[1+{\lambda }_{1}Cauchy\left(0,1\right)+{\lambda }_{2}Guass\left(0,1\right)\right]$$

(21)

where $Cauchy\left(0,1\right)$ represents a Cauchy random variable, and $Guass\left(0,1\right)$ represents a Gaussian variable. ${\lambda }_1=1-{\displaystyle \raisebox{1ex}{$t^2$}\!\left/ \!\raisebox{-1ex}{$T^2$}\right.}$, whose value gradually decreases with the iterations, ${\lambda }_2={\displaystyle \raisebox{1ex}{$t^2$}\!\left/ \!\raisebox{-1ex}{$T^2$}\right.}$, and the value gradually increases with the iterations.

Therefore, the best individual position is as shown:

$$X_{best}^{\prime }=\left\{\begin{array}{c}{Positon}_{best} \alpha >P_{dis}\cup F\left(X_{best}^{\prime }\right)>F\left(X_{bset}\right)\\ X_{bset} otherwise\end{array}\right.$$

(22)

In summary, the proposed HSHHO algorithm is shown in Figure 4.

2.3. Complexity of the HSHHO Algorithm

The computational complexity of the HSHHO algorithm is mainly composed of the following three parts:

• Initialization of the population. The computational complexity of this part mainly depends on population size $N$ and the problem dimension $D$, and the calculated complexity is $\mathrm{O}(N\times D)$.
• Evaluating the fitness of the initial population. The computational complexity of this part is mainly determined by the population size $N$. The computational complexity is $\mathrm{O}\left(N\right)$.
• The population position is cyclically updated. The computational complexity of this part mainly depends by the population size $N$, the problem dimension $D$ and the number of iterations $T$. It is composed of individual position updates and fitness calculations. The position of the individual is updated through different search strategies. The execution of two different stages is determined by whether the algorithm satisfies the judgment condition. The computational complexity of this part is calculated as $\mathrm{O}(N\times D\times T)$.

Based on the above analysis, we can deduce that the time complexity of our proposed HSHHO algorithm follows a similar structure to that of the standard HHO algorithm. Specifically, the time complexity primarily includes population position initialization, population position update, and population fitness calculation. Consequently, the computational complexity of these components should remain unchanged. Similarly, the different strategy in this paper is to improve the method of updating the population position without increasing the number of populations and the iteration time, which does not change the time complexity of this component.

Therefore, $\mathrm{O}\left(HSHHO\right)$ = $\mathrm{O}\left(\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{p}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)$ + $\mathrm{O}(\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})$ + $\mathrm{O}(\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{u}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e})$. where $\mathrm{O}\left(\mathrm{I}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{p}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{O}(N\times D)$, $\mathrm{O}\left(\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)=\mathrm{O}\left(N\right)$, $\mathrm{O}\left(\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{s} \mathrm{H}\mathrm{a}\mathrm{w}\mathrm{k}\mathrm{s} \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{u}\mathrm{p}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{e}\right)=\mathrm{O}(N\times D\times T)$. Total complexity of the HSHHO is $\mathrm{O}\left(HSHHO\right)=\mathrm{O}(N\times D\times T)$.

3. Results

To assess the efficacy and optimization potential of HSHHO, we conduct a series of simulation experiments. Initially, the algorithm is tested on 23 standard benchmark functions to evaluate the impact of its three enhancement strategies relative to the baseline HHO. Subsequently, HSHHO is benchmarked against eight established metaheuristic algorithms using IEEE CEC 2017 problems, with statistical analyses to highlight its advantages. Finally, HSHHO is deployed on 18 feature selection datasets to demonstrate its practical utility. All simulations are executed on a Windows 10 system equipped with an Intel Core i7-7700, CPU @ 3.60 GHz and 32 G of RAM, using MATLAB R2022b for implementation.

The common parameters for all algorithms are set such that the population size $N$ is 30 and the maximum iteration $T$ is 500. In this study, to ensure fairness in computational workload between algorithms, the total number of function evaluations $FE$ is set to $N\times T=15000$. In addition, we define one iteration of the algorithm as a complete loop. In this loop, the positions of all individuals in the group are updated, followed by a round of fitness assessment.

3.1. Comparison of the Effects of Different Improvement Strategies

To bolster HHO’s optimization capabilities, we introduce three targeted enhancements: a hybrid SPM chaotic map with variable logarithmic spiral (SPM-VLS), a nonlinear escape energy mechanism for smoother phase transitions, and a Cauchy–Gaussian elite perturbation approach. These are integrated to form the comprehensive HSHHO algorithm. This section aims to empirically verify the independent contribution of each strategy to HHO and demonstrate the excellent performance achieved through its fusion effect.

To conduct this analysis, we decomposed HSHHO into three variants, including HSHHO1, HSHHO2, and HSHHO3. Each variant contains only one improvement strategy. HSHHO1 only enhances HHO through SPM-VLS and ignores other functions. HSHHO2 only applies nonlinear escape energy updates. HSHHO3 is only available for the police Cauchy Gauss Elite. These variants, as well as the proposed HSHHO and HHO, are evaluated and compared on 23 standard benchmark functions, as shown in

Table 1. Descriptions of the 23 standard benchmark functions.

Function	Dim	Range	Optimum	Class
$F_1(x)={\displaystyle \sum _{i=1}^n}{x}_i^2$	30	[−100, 100]	0	Unimodal benchmark functions
$F_2(x)={\displaystyle \sum _{i=1}^n}\left|x_i\right|+{\displaystyle \prod _{i=1}^n}\left|x_i\right|$	30	[−10, 10]	0
$F_3(x)={\displaystyle \sum _{i=1}^n}{\left({\displaystyle \sum _{j-1}^i}{x}_j\right)}^2$	30	[−100, 100]	0
$F_4(x)={max}_i\left\{\left|x_i\right|,1\le i\le n\right\}$	30	[−100, 100]	0
$F_5(x)={\displaystyle \sum _{i=1}^{n-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	30	[−30, 30]	0
$F_6(x)={\displaystyle \sum _{i=1}^n}{\left(\left|x_i+0.5\right|\right)}^2$	30	[−100, 100]	0
$F_7(x)={\displaystyle \sum _{i=1}^n}i{x}_i^4+random[0,1]$	30	[−1.28, 1.28]	0
$F_8(x)={\displaystyle \sum _{i=1}^n}-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	$-418.98\times Dim$	Multimodal benchmark functions
$F_9(x)={\displaystyle \sum _{i=1}^n}\left[x_i^2-10cos\left(2\pi x_i\right)+10\right]$	30	[−5.12, 5.12]	0
$F_{10}\left(x\right)=-20exp\left(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{x}_i^2}\right)-exp\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\cos \left(2\pi x_i\right)\right)+20+e$	30	[−32, 32]	0
$F_{11}\left(x\right)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\displaystyle \prod _{i=1}^n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0
$F_{12}\left(x\right)={\displaystyle \frac{\pi }{n}}\left\{10sin\left(\pi y_i\right)+{\displaystyle \sum _{i=1}^{n-1}}{\left(y_i-1\right)}^2\left[1+10{sin}^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\right\}+{\displaystyle \sum _{i=1}^n}u(x_i,10,100,4)$ $y_i=1+{\displaystyle \frac{x_i+1}{4}}$ $u\left(x_i,a,k,m\right)=\left\{\begin{array}{c}k{(x_i-a)}^m x_i>a\\ 0 -a<x_i<a\\ k{({-x}_i-a)}^m x_i<-a\end{array}\right.$	30	[−50, 50]	0
$F_{13}\left(x\right)=0.1\left\{{sin}^2\left(3\pi x_1\right)+\left({\displaystyle \sum _{i=1}^n}{\left(x_i-1\right)}^2\right)\left[1+{sin}^2\left(3\pi x_1+1\right)\right]+{\left(x_n-1\right)}^2\left[1+{sin}^2\left(2\pi x_n\right)\right]\right\}+{\displaystyle \sum _{i=1}^n}u\left(x_i,5,100,4\right)$	30	[−50, 50]	0
$F_{14}\left(x\right)={\left({\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}}{\displaystyle \frac{1}{j+{\displaystyle {\sum }_{i=1}^2}{\left(x_i-a_{ij}\right)}^6}}\right)}^{-1}$	2	[−65, 65]	1	Fixed-dimension multimodal benchmark functions
$F_{15}\left(x\right)={\displaystyle \sum _{i=1}^{11}}{\left[a_i-{\displaystyle \frac{x_1(b_i^2+b_1{x}_2)}{b_i^2+b_i{x}_3+x_4}}\right]}^2$	4	[−5, 5]	0.00030
$F_{16}\left(x\right)=4x_1^2-2.1x_1^2+{\displaystyle \frac{1}{3}}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[−5, 5]	−1.0316
$F_{17}\left(x\right)={\left(x_2-{\displaystyle \frac{5.1}{{4\pi }^2}}{x}_1^2+{\displaystyle \frac{5}{\pi }}{x}_1-6\right)}^2+10\left(1-{\displaystyle \frac{1}{8\pi }}\right)cos{x}_1+10$	2	[−5, 5]	0.398
$F_{18}\left(x\right)=\left[1+{(x_1+x_2+1)}^2\left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)\right]\times \left[30+{\left(2x_1-3x_2\right)}^2\left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]$	2	[−2, 2]	3
$F_{19}\left(x\right)=-{\displaystyle \sum _{i=1}^4}{c}_{i}exp\left(-{\displaystyle \sum _{j=1}^3}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	3	[1, 3]	−3.86
$F_{20}\left(x\right)=-{\displaystyle \sum _{i=1}^4}{c}_{i}exp\left(-{\displaystyle \sum _{j=1}^6}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	6	[0, 1]	−3.32
$F_{21}\left(x\right)=-{\displaystyle \sum _{i=1}^5}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.1532
$F_{22}\left(x\right)=-{\displaystyle \sum _{i=1}^7}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.4028
$F_{23}\left(x\right)=-{\displaystyle \sum _{i=1}^{10}}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.5363

.

Table 2. Experimental results.

Function	Index	HHO	HSHHO1	HSHHO2	HSHHO3	HSHHO
F1	AVG	9.2202 × 10−94	1.9297 × 10−109	1.6754 × 10−142	2.3264 × 10−284	0.0000
	STD	5.0487 × 10−93	1.0565 × 10−108	9.1533 × 10−142	0.0000	0.0000
F2	AVG	3.0247 × 10−50	1.2447 × 10−56	5.5461 × 10−76	3.0287 × 10−140	0.0000
	STD	1.0607 × 10−49	6.1266 × 10−56	2.1252 × 10−75	1.6589 × 10−139	0.0000
F3	AVG	8.1637 × 10−78	1.7914 × 10−94	3.7932 × 10−100	3.0046 × 10−286	0.0000
	STD	4.0442 × 10−77	9.5497 × 10−94	2.0373 × 10−99	0.0000	0.0000
F4	AVG	6.1139 × 10−48	6.6739 × 10−55	1.0344 × 10−71	2.7746 × 10−144	0.0000
	STD	3.3448 × 10−47	2.5437 × 10−54	5.6653 × 10−71	1.5197 × 10−143	0.0000
F5	AVG	7.4754 × 10−3	5.1007 × 10−3	2.7718	1.2976 × 10−2	3.1645 × 10−3
	STD	9.9405 × 10−3	7.0617 × 10−3	8.2381	2.2918 × 10−2	5.4109 × 10−3
F6	AVG	1.1492 × 10−4	5.4432 × 10−5	7.8722 × 10−4	2.4963 × 10−5	1.4019 × 10−5
	STD	1.1033 × 10−4	1.3313 × 10−4	3.9139 × 10−4	4.8599 × 10−5	1.8865 × 10−5
F7	AVG	1.4853 × 10−4	1.1531 × 10−4	2.9132 × 10−4	1.0458 × 10−4	6.9998 × 10−5
	STD	1.5813 × 10−4	9.0161 × 10−5	7.5267 × 10−4	1.2357 × 10−4	6.4866 × 10−5
F8	AVG	−1.2568 × 104	−1.2568 × 104	−1.1717 × 104	−1.2541 × 104	−1.2507 × 104
	STD	8.5821 × 10−1	1.8305	2.1650 × 103	9.7629 × 101	3.1601 × 102
F9	AVG	0.0000	0.0000	0.0000	0.0000	0.0000
	STD	0.0000	0.0000	0.0000	0.0000	0.0000
F10	AVG	4.4409 × 10−16	4.4409 × 10−16	4.4409 × 10−16	4.4409 × 10−16	4.4409 × 10−16
	STD	0.0000	0.0000	0.0000	0.0000	0.0000
F11	AVG	0.0000	0.0000	0.0000	0.0000	0.0000
	STD	0.0000	0.0000	0.0000	0.0000	0.0000
F12	AVG	5.0048 × 10−6	3.0985 × 10−6	1.5908 × 10−5	2.9053 × 10−6	2.6737 × 10−6
	STD	7.5440 × 10−6	5.2977 × 10−6	8.1661 × 10−6	8.7679 × 10−6	6.2248 × 10−6
F13	AVG	7.3854 × 10−5	4.1608 × 10−5	6.1878 × 10−4	6.4770 × 10−5	1.4577 × 10−5
	STD	1.0268 × 10−4	5.3595 × 10−5	2.0259 × 10−3	1.6530 × 10−4	2.6671 × 10−5
F14	AVG	1.7230	1.2626	2.5698	9.9800 × 10−1	9.9800 × 10−1
	STD	1.4866	6.8599 × 10−1	2.9206	1.9366 × 10−6	1.5070 × 10−7
F15	AVG	3.7550 × 10−4	3.1184 × 10−4	3.9282 × 10−4	4.7645 × 10−4	3.3515 × 10−4
	STD	3.4535 × 10−4	1.0682 × 10−5	3.1044 × 10−4	3.4231 × 10−4	2.6490 × 10−5
F16	AVG	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	STD	9.9897 × 10−10	3.6767 × 10−10	3.0363 × 10−10	1.8941 × 10−5	6.0994 × 10−6
F17	AVG	3.9789 × 10−1	3.9789 × 10−1	3.9789 × 10−1	3.9789 × 10−1	3.9789 × 10−1
	STD	7.8303 × 10−7	5.5480 × 10−7	3.2770 × 10−4	4.9781 × 10−4	1.1796 × 10−7
F18	AVG	9.3000	8.4000	9.3000	3.9876	3.0000
	STD	1.1615 × 101	1.0984 × 101	1.1615 × 101	5.4051	3.7568 × 10−8
F19	AVG	−3.8112	−3.0897	−3.0898	−3.6037	−3.8615
	STD	1.9611 × 10−1	3.1446 × 10−1	2.3587 × 10−1	9.3659 × 10−2	1.8628 × 10−3
F20	AVG	−3.2858	−3.2929	−3.2584	−2.9789	−3.2823
	STD	5.5976 × 10−1	5.3445 × 10−1	6.0406 × 10−1	1.5164 × 10−1	5.7081 × 10−2
F21	AVG	−5.395	−5.0548	−5.395	−1.0129 × 101	−1.0152 × 101
	STD	1.2933	8.1976 × 10−5	1.2934	3.9687 × 10−2	3.1446 × 10−3
F22	AVG	−4.9338	−5.0876	−4.9313	−1.0383 × 101	−1.0402 × 101
	STD	6.3550 × 10−1	4.9794 × 10−5	1.8883	3.3322 × 10−2	3.7548 × 10−4
F23	AVG	−5.1282	−5.1284	−4.2499	−1.0510 × 101	−1.0535 × 101
	STD	1.2303	5.4647 × 10−5	1.8040	4.1428 × 10−2	7.4586 × 10−4

Bold represents the optimal value.

shows the experimental results of HHO, HSHHO, and these function variants, with the optimal values displayed in bold. Figure 5 shows the convergence curves of these algorithms. Overall, the ARV in Table 2 indicates that in most tests, HSHHO and its variants perform better than HHO. This confirms the effectiveness of SPM-VLS, nonlinear escape energy, and Cauchy–Gaussian perturbation strategies in improving the quality of HHO solutions.

The Wilcoxon rank sum test in

Table 3. The p-values of the Wilcoxon rank-sum test with 5% significance of experimental results.

Function	HHO	HSHHO1	HSHHO2	HSHHO3	HSHHO
F1	1.2118 × 10−12	1.9457 × 10−9	1.2118 × 10−12	1.6572 × 10−11	NaN
F2	1.2118 × 10−12	1.2118 × 10−12	1.2118 × 10−12	1.2118 × 10−12	NaN
F3	1.2118 × 10−12	1.9346 × 10−10	1.2118 × 10−12	1.2118 × 10−12	NaN
F4	1.2118 × 10−12	1.2118 × 10−12	1.2118 × 10−12	1.2118 × 10−12	NaN
F5	3.1830 × 10−3	1.4945 × 10−1	3.0059 × 10−4	4.8413 × 10−2	1
F6	1.8608 × 10−6	6.9522 × 10−1	3.0199 × 10−11	5.6922 × 10−1	1
F7	2.0681 × 10−2	1.4412 × 10−2	9.4683 × 10−3	2.9727 × 10−1	1
F8	7.6171 × 10−3	1.2212 × 10−2	1.2023 × 10−8	2.2257 × 10−1	1
F9	NaN	NaN	NaN	NaN	NaN
F10	NaN	NaN	NaN	NaN	NaN
F11	NaN	NaN	NaN	NaN	NaN
F12	4.6371 × 10−3	9.3519 × 10−1	1.5964 × 10−7	6.3088 × 10−1	1
F13	9.5207 × 10−4	5.1877 × 10−2	4.5043 × 10−11	2.8129 × 10−2	1
F14	9.7052 × 10−1	5.9706 × 10−5	4.2896 × 10−1	4.5146 × 10−2	1
F15	2.8790 × 10−6	6.0104 × 10−8	1.8682 × 10−5	6.5486 × 10−4	1
F16	3.3384 × 10−11	3.0199 × 10−11	3.0180 × 10−11	7.2208 × 10−6	1
F17	3.0180 × 10−11	3.0180 × 10−11	3.0199 × 10−11	5.8737 × 10−4	1
F18	2.6784 × 10−6	1.6062 × 10−6	2.0620 × 10−1	3.0199 × 10−11	1
F19	2.1959 × 10−7	8.1200 × 10−4	1.0666 × 10−7	1.8731 × 10−7	1
F20	4.2259 × 10−3	1.0315 × 10−2	4.6756 × 10−2	3.0199 × 10−11	1
F21	1.6132 × 10−10	3.0199 × 10−11	3.8202 × 10−10	1.3250 × 10−4	1
F22	3.0199 × 10−11	3.0199 × 10−11	5.0723 × 10−10	1.2212 × 10−2	1
F23	9.9186 × 10−11	3.0199 × 10−11	5.5727 × 10−10	3.5012 × 10−3	1

shows significant differences between HSHHO1 and HHO in most functions. HSHHO2 performs well on eight functions, but performs poorly on F5–F8, F12–F14, and F23. This indicates that nonlinear escape energy faces difficulties in maintaining phase balance in unimodal and multimodal scenarios. However, the p-values in Table 3 indicate that all optimization strategies except F14 can improve the performance of HHO. Meanwhile, HSHHO3 ensures an optimal solution for all problems except F5, emphasizing the promoting effect of the perturbation strategy on local refinement. The p-value indicates that all functions except F6 and F12 show significance.

3.2. Comparison Between HSHHO and Other Optimization Algorithms

This section evaluates the performance comparison between HSHHO and eight popular optimization algorithms based on 30 benchmark functions of CEC 2017.

Table 4. Description of the CEC 2017 benchmark function.

Function	Class	Name	Optimum
F1	Unimodal	Shifted and rotated bent cigar function	100
F2		Shifted and rotated sum of different power function	200
F3		Shifted and rotated Zakharov function	300
F4	Multimodal	Shifted and rotated Rosenbrock’s function	400
F5		Shifted and rotated Rastrigin’s function	500
F6		Shifted and rotated expanded Scaffer’s F6 function	600
F7		Shifted and rotated Lunacek Bi-Rastrigin function	700
F8		Shifted and rotated non-continuous Rastrigin’s function	800
F9		Shifted and rotated Lévy function	900
F10		Shifted and rotated Schwefel’s function	1000
F11	Hybrid	Hybrid function 1 (N = 3)	1100
F12		Hybrid function 2 (N = 3)	1200
F13		Hybrid function 3 (N = 3)	1300
F14		Hybrid function 4 (N = 4)	1400
F15		Hybrid function 5 (N = 4)	1500
F16		Hybrid function 6 (N = 4)	1600
F17		Hybrid function 6 (N = 5)	1700
F18		Hybrid function 6 (N = 5)	1800
F19		Hybrid function 6 (N = 5)	1900
F20		Hybrid function 6 (N = 6)	2000
F21	Composition	Composition function 1 (N = 3)	2100
F22		Composition function 2 (N = 3)	2200
F23		Composition function 3 (N = 4)	2300
F24		Composition function 4 (N = 4)	2400
F25		Composition function 5 (N = 5)	2500
F26		Composition function 6 (N = 5)	2600
F27		Composition function 7 (N = 6)	2700
F28		Composition function 8 (N = 6)	2800
F29		Composition function 9 (N = 3)	2900
F30		Composition function 10 (N = 3)	3000

shows the description of the CEC 2017. Especially, the F2 of the CEC 2017 test function has been officially proven to have issues. However, in order to maintain the integrity and continuity of the testing, F2 is retained but not used in the experiment in this study.

Eight popular optimization algorithms include WOA, SSA, MFO, GOA, SCA, PSO, and GWO.

Table 5. Experimental results on CEC 2017.

Function	Index	HSHHO	HHO	PSO	GWO	WOA	SSA	MFO	GOA	SCA
F1	AVG	9.7554 × 108	1.4424 × 109	5.2538 × 109	2.4450 × 109	2.5804 × 109	2.3372 × 109	2.8575 × 109	8.0766 × 1010	5.8199 × 1010
	STD	5.4143 × 109	6.5741 × 109	5.5493 × 109	3.4282 × 109	4.3683 × 109	8.1278 × 109	2.6039 × 109	2.7315 × 109	1.3062 × 109
F2	AVG	/	/	/	/	/	/	/	/	/
	STD	/	/	/	/	/	/	/	/	/
F3	AVG	7.7346 × 103	2.8155 × 104	1.0714 × 105	1.9994 × 104	3.7978 × 104	6.2690 × 104	7.7053 × 104	1.9390 × 105	8.1068 × 104
	STD	2.0756 × 104	2.1616 × 104	5.5314 × 104	1.5863 × 104	2.7229 × 104	2.7250 × 104	3.3063 × 104	7.7895 × 104	1.5926 × 104
F4	AVG	2.4112 × 102	5.3278 × 102	1.2129 × 103	2.2951 × 102	4.0682 × 102	6.7315 × 102	9.9462 × 103	1.9333 × 104	1.1779 × 104
	STD	9.5843 × 102	1.3907 × 103	1.8819 × 103	6.1623 × 102	9.8022 × 102	2.7435 × 103	2.0508 × 103	1.5372 × 104	5.2096 × 102
F5	AVG	5.0963 × 102	6.9076 × 102	6.5948 × 102	6.5181 × 102	6.2562 × 102	7.0892 × 102	8.1565 × 102	8.2432 × 102	8.4106 × 102
	STD	5.5005 × 101	3.8815 × 102	3.2668 × 102	3.0402 × 101	3.5242 × 101	3.2511 × 101	1.6248 × 101	1.3062 × 102	1.6025 × 102
F6	AVG	6.2109 × 102	6.7078 × 102	6.6349 × 102	6.2159 × 102	6.5872 × 102	6.4165 × 102	6.9733 × 102	7.1644 × 102	6.9524 × 102
	STD	1.9088 × 101	6.3082	1.4431 × 102	1.1583 × 101	4.4150	9.5642	4.2081	1.3201 × 101	3.2228
F7	AVG	1.0692 × 103	1.1702 × 103	1.2815 × 103	1.0671 × 103	1.4792 × 103	1.1979 × 103	1.4280 × 103	2.5667 × 103	1.3778 × 103
	STD	1.7055 × 102	8.3467 × 101	1.4583 × 102	1.1888 × 102	5.8544 × 101	1.5932 × 102	3.3147 × 101	1.0580 × 103	8.9837 × 101
F8	AVG	9.6935 × 102	1.0152 × 103	1.0497 × 103	9.9726 × 102	1.0949 × 103	1.0276 × 103	1.1746 × 103	1.3347 × 103	1.1614 × 103
	STD	4.5695 × 101	4.5533 × 101	7.7940 × 101	5.9885 × 101	2.7073 × 101	7.1590 × 101	1.0623 × 101	1.1185 × 102	2.5866 × 101
F9	AVG	1.1732 × 103	3.4357 × 103	1.1845 × 104	4.2391 × 103	3.9085 × 103	6.9409 × 103	9.8570 × 103	2.1962 × 104	1.0510 × 104
	STD	1.7086 × 103	1.2991 × 103	2.6032 × 103	2.3797 × 103	1.9680 × 103	2.1619 × 103	1.8012 × 103	7.3233 × 103	1.5287 × 103
F10	AVG	2.6175 × 103	5.3673 × 103	5.0912 × 103	4.6130 × 103	6.0938 × 103	5.3090 × 103	7.9793 × 103	8.3024 × 103	7.5284 × 103
	STD	1.2863 × 103	7.3927 × 102	8.2120 × 102	1.7038 × 103	3.3632 × 102	1.5889 × 103	2.5281 × 102	8.7545 × 103	3.1626 × 103
F11	AVG	1.5683 × 103	2.3048 × 103	4.3565 × 103	1.6463 × 103	1.6190 × 103	2.8943 × 103	3.2019 × 104	1.0292 × 104	9.0029 × 103
	STD	1.2944 × 103	2.4938 × 103	2.0074 × 103	1.2138 × 103	7.3213 × 102	2.5054 × 103	2.2547 × 103	8.5421 × 103	5.4543 × 102
F12	AVG	4.2102 × 107	5.9459 × 108	7.0906 × 108	4.0726 × 108	3.2593 × 108	3.9936 × 108	1.0424 × 108	2.1557 × 108	1.4221 × 108
	STD	2.1806 × 108	1.7850 × 109	1.1717 × 109	1.0358 × 109	1.0214 × 109	1.8102 × 108	1.3041 × 109	6.8755 × 109	9.9834 × 108
F13	AVG	8.1013 × 107	7.9847 × 108	6.6440 × 108	2.7493 × 108	2.2803 × 108	2.1827 × 108	1.5359 × 108	2.1256 × 109	1.2800 × 108
	STD	6.9252 × 108	2.1941 × 109	1.5194 × 108	9.7112 × 108	2.2425 × 109	1.2221 × 109	5.6837 × 108	1.2771 × 109	1.5157 × 109
F14	AVG	1.3977 × 104	3.6344 × 106	1.7095 × 105	6.8557 × 105	1.0412 × 105	5.4506 × 105	4.8018 × 107	1.2024 × 107	1.5302 × 107
	STD	1.3511 × 105	2.2885 × 105	9.3932 × 105	2.7528 × 105	7.1384 × 105	18,003 × 106	3.7231 × 106	1.5338 × 107	6.9547 × 105
F15	AVG	1.9451 × 105	5.7895 × 106	1.4988 × 108	1.7983 × 107	1.3562 × 107	4.1858 × 107	8.8706 × 108	7.2240 × 109	1.2949 × 109
	STD	1.8413 × 108	3.4082 × 107	3.4754 × 108	1.1703 × 108	1.6151 × 108	4.9335 × 108	1.4493 × 108	5.5912 × 109	5.0938 × 107
F16	AVG	2.9309 × 103	4.1662 × 103	3.4670 × 103	3.3016 × 103	3.6725 × 103	3.8787 × 103	6.3544 × 103	8.2050 × 103	5.9973 × 103
	STD	4.6591 × 102	1.6355 × 102	3.0824 × 102	5.0062 × 102	3.3962 × 102	6.1309 × 102	5.8768 × 102	1.2939 × 103	1.1485 × 102
F17	AVG	2.2657 × 103	2.9546 × 103	2.6320 × 103	2.1743 × 103	2.7152 × 103	2.6852 × 103	7.9980 × 103	1.4139 × 104	1.4521 × 104
	STD	9.3955 × 101	1.8629 × 102	1.2796 × 102	2.5942 × 102	1.7518 × 102	4.0460 × 103	7.7827 × 102	1.4003 × 104	4.4589 × 103
F18	AVG	8.6175 × 104	2.8386 × 106	7.2322 × 106	3.9432 × 105	5.2915 × 105	8.5918 × 106	1.3414 × 108	7.9891 × 108	1.8390 × 108
	STD	1.0155 × 106	7.8252 × 106	1.5028 × 107	9.1560 × 106	2.7621 × 106	6.5267 × 107	7.3483 × 107	1.1184 × 109	2.8442 × 107
F19	AVG	1.6928 × 107	6.8774 × 107	1.1339 × 108	3.0044 × 107	2.5613 × 107	5.2711 × 107	3.7551 × 108	6.9040 × 109	5.6175 × 108
	STD	1.9357 × 108	4.6015 × 108	5.7385 × 108	1.2692 × 107	2.9562 × 108	6.0870 × 108	4.4435 × 108	5.0033 × 109	2.8577 × 108
F20	AVG	2.5585 × 103	2.8746 × 103	3.0259 × 103	2.7561 × 103	2.6048 × 103	2.9083 × 103	3.0248 × 103	3.5063 × 103	3.0591 × 103
	STD	1.1734 × 102	1.2163 × 102	1.1190 × 102	1.8765 × 102	1.1117 × 102	1.7356 × 102	1.6338 × 102	4.2353 × 102	1.7163 × 102
F21	AVG	2.4670 × 103	2.5092 × 103	2.6093 × 103	2.5084 × 103	2.5626 × 103	2.5388 × 103	2.8623 × 103	2.8970 × 103	2.7270 × 103
	STD	4.2120 × 101	5.3443 × 101	4.2037 × 101	3.1562 × 101	3.2519 × 101	7.4717 × 101	1.4568 × 101	1.1938 × 102	4.4329 × 101
F22	AVG	2.8911 × 103	9.7133 × 103	5.0588 × 103	9.5586 × 103	4.8736 × 103	2.7969 × 103	1.0536 × 104	1.0944 × 104	1.0621 × 104
	STD	1.2239 × 103	5.0372 × 102	2.8100 × 103	9.5533 × 102	1.0690 × 103	1.2912 × 103	18,570 × 102	1.0526 × 103	2.6537 × 102
F23	AVG	2.8797 × 103	2.9789 × 103	3.0721 × 103	2.8891 × 103	3.0603 × 103	2.9458 × 103	3.5757 × 103	3.7814 × 1031	3.7595 × 103
	STD	7.5436 × 101	6.7255 × 101	7.9489 × 101	6.0165 × 101	4.0440 × 101	7.4876 × 101	1.4033 × 101	2.2267 × 102	4.5520 × 10−12
F24	AVG	3.0174 × 103	3.2164 × 103	3.1769 × 103	3.0648 × 103	3.0504 × 103	3.0993 × 103	3.6709 × 103	4.3821 × 103	3.9736 × 103
	STD	6.1227 × 101	4.6854 × 101	1.0267 × 102	3.8740 × 101	4.1633 × 101	6.5872 × 101	1.1944 × 101	9.6448 × 101	2.3097 × 102
F25	AVG	3.0424 × 103	3.2084 × 103	3.7651 × 103	3.1629 × 103	3.2049 × 103	3.1182 × 103	6.1033 × 103	1.4818 × 104	5.7999 × 103
	STD	4.5556 × 102	6.1504 × 102	9.2433 × 102	4.7527 × 102	5.9302 × 102	9.3944 × 102	1.4116 × 102	1.1568 × 104	2.5444 × 102
F26	AVG	3.1925 × 103	4.6352 × 103	4.6003 × 103	3.3619 × 103	3.6146 × 103	4.7397 × 103	8.8722 × 103	1.0881 × 104	7.6574 × 103
	STD	1.1725 × 103	9.2160 × 102	7.1127 × 102	6.3521 × 102	6.4589 × 102	4.9364 × 102	1.5270 × 102	2.2137 × 103	1.0669 × 103
F37	AVG	3.2358 × 103	3.4859 × 103	3.4342 × 103	3.3020 × 103	3.3448 × 103	3.2209 × 103	5.1244 × 103	5.5392 × 103	4.4582 × 103
	STD	6.9045 × 101	9.3751 × 101	1.9107 × 102	6.4110 × 101	6.8558 × 101	1.3339 × 102	2.3661 × 101	3.1244 × 102	1.5417 × 102
F28	AVG	3.3520 × 103	3.4450 × 103	3.7245 × 103	3.4335 × 103	3.5450 × 103	3.5965 × 103	6.6752 × 103	1.1238 × 104	8.6292 × 103
	STD	6.2793 × 102	3.9747 × 102	3.6686 × 102	3.2141 × 102	5.7435 × 102	8.6539 × 102	1.8916 × 102	3.8618 × 103	1.4005 × 102
F29	AVG	1.5134 × 103	2.1430 × 103	1.6622 × 103	1.5884 × 103	2.4300 × 103	1.7066 × 103	3.0916 × 103	7.5102 × 104	4.7342 × 103
	STD	6.2592 × 102	4.9253 × 102	2.5559 × 102	3.4425 × 102	3.4094 × 102	1.1404 × 102	3.7210 × 101	1.0100 × 105	8.0851 × 102
F30	AVG	1.3070 × 107	3.0329 × 107	7.5994 × 107	5.2071 × 107	1.9112 × 107	4.4620 × 107	1.9878 × 109	3.8709 × 109	3.1083 × 109
	STD	1.1167 × 107	1.5930 × 108	1.1917 × 108	1.3604 × 108	1.2049 × 108	3.0214 × 108	1.6568 × 108	2.9225 × 109	8.7558 × 104

Bold represents the optimal value.

reports the AVG and STD metrics, with the best performers highlighted in bold. To emphasize performance differences, we applied the non-parametric Wilcoxon rank sum test, with p-values detailed in

Table 6. The p-values of the Wilcoxon rank-sum test.

Function	HSHHO	HHO	PSO	GWO	WOA	SSA	MFO	GOA	SCA
F1	1	2.4505 × 10−52	1.3578 × 10−101	2.1937 × 10−91	7.1341 × 10−96	1.6461 × 10−50	3.2687 × 10−170	1.6826 × 10−166	1.2052 × 10−184
F2	/	/	/	/	/	/	/	/	/
F3	1	2.7364 × 10−106	1.0698 × 10−150	1.2169 × 10−102	4.0594 × 10−119	7.5041 × 10−146	2.1346 × 10−175	1.7823 × 10−162	2.2654 × 10−174
F4	1	1.5807 × 10−86	5.9609 × 10−136	8.3792 × 10−84	1.5402 × 10−116	1.0254 × 10−57	1.9029 × 10−171	1.0770 × 10−165	9.0280 × 10−180
F5	1	5.1281 × 10−123	1.0307 × 10−65	3.9458 × 10−45	2.3397 × 10−20	4.2295 × 10−156	2.9920 × 10−171	8.6114 × 10−168	1.5884 × 10−172
F6	1	2.1907 × 10−138	9.3983 × 10−134	2.3177 × 10−8	1.8893 × 10−133	3.3498 × 10−103	6.3278 × 10−160	8.8139 × 10−165	2.6404 × 10−158
F7	1	1.2578 × 10−78	6.5618 × 10−105	1.7044 × 10−18	1.3242 × 10−144	1.1097 × 10−80	3.6721 × 10−142	3.2632 × 10−160	7.0028 × 10−144
F8	1	7.1216 × 10−97	1.6341 × 10−101	1.0671 × 10−27	7.5621 × 10−148	4.2674 × 10−25	5.3393 × 10−172	3.4864 × 10−163	9.0326 × 10−158
F9	1	8.5290 × 10−134	9.6441 × 10−162	1.9577 × 10−140	2.0053 × 10−139	3.4686 × 10−158	2.7326 × 10−164	1.0824 × 10−166	9.3955 × 10−163
F10	1	1.2651 × 10−143	1.1409 × 10−142	1.1424 × 10−78	4.3445 × 10−149	1.7836 × 10−116	6.3156 × 10−166	2.1529 × 10−163	5.9644 × 10−162
F11	1	3.3557 × 10−105	2.7481 × 10−148	5.1738 × 10−94	2.3019 × 10−106	2.3703 × 10−129	1.0447 × 10−180	6.5783 × 10−157	1.8588 × 10−162
F12	1	5.6236 × 10−101	3.9015 × 10−119	7.4343 × 10−110	2.5835 × 10−101	4.5002 × 10−64	3.2022 × 10−166	3.8472 × 10−167	8.8867 × 10−169
F13	1	1.8683 × 10−135	1.3924 × 10−145	5.3217 × 10−142	2.3769 × 10−121	3.9456 × 10−143	1.4514 × 10−183	1.8803 × 10−166	3.8959 × 10−177
F14	1	8.6576 × 10−159	1.1420 × 10−158	8.6996 × 10−158	5.3128 × 10−136	8.8309 × 10−146	3.7732 × 10−171	3.8842 × 10−165	8.5869 × 10−169
F15	1	4.3809 × 10−132	5.9466 × 10−153	8.3282 × 10−143	1.4954 × 10−119	1.1148 × 10−105	1.6725 × 10−167	6.6941 × 10−165	2.5733 × 10−181
F16	1	4.9359 × 10−149	3.4985 × 10−116	1.3640 × 10−63	6.7324 × 10−141	3.9267 × 10−143	1.2781 × 10−164	1.9035 × 10−167	7.8800 × 10−167
F17	1	8.5216 × 10−161	2.3512 × 10−155	3.3784 × 10−5	1.3423 × 10−155	1.3702 × 10−42	1.2047 × 10−178	8.9322 × 10−169	1.6800 × 10−185
F18	1	1.2741 × 10−147	2.2843 × 10−141	6.6136 × 10−149	6.2608 × 10−101	2.4879 × 10−145	8.0513 × 10−175	2.6433 × 10−166	7.2446 × 10−168
F19	1	1.1257 × 10−137	9.6772 × 10−146	5.8565 × 10−143	1.9589 × 10−109	5.3174 × 10−59	7.1361 × 10−174	5.4396 × 10−167	1.2369 × 10−167
F20	1	2.0556 × 10−153	1.2510 × 10−163	3.8778 × 10−114	1.3509 × 10−112	3.9971 × 10−156	7.8589 × 10−160	3.9635 × 10−163	7.8907 × 10−164
F21	1	1.4426 × 10−71	5.1796 × 10−151	8.3808 × 10−84	1.7256 × 10−140	8.4876 × 10−95	4.5765 × 10−170	1.3651 × 10−165	6.1059 × 10−173
F22	1	6.6759 × 10−161	1.3576 × 10−103	6.5272 × 10−161	4.7120 × 10−132	4.9235 × 10−7	3.3084 × 10−173	2.2345 × 10−165	2.1766 × 10−163
F23	1	1.4315 × 10−135	6.9279 × 10−148	1.3726 × 10−16	1.9428 × 10−146	2.3292 × 10−118	1.7869 × 10−177	4.4065 × 10−169	3.1364 × 10−188
F24	1	4.5083 × 10−155	8.2203 × 10−151	2.7798 × 10−130	6.2504 × 10−111	7.7082 × 10−144	5.7836 × 10−178	1.1461 × 10−173	3.3953 × 10−165
F25	1	1.3838 × 10−57	7.4399 × 10−136	1.1967 × 10−98	1.0154 × 10−99	3.5981 × 10−8	1.1203 × 10−178	1.2006 × 10−166	3.8691 × 10−164
F26	1	1.1804 × 10−106	2.9114 × 10−103	1.3490 × 10−9	1.9441 × 10−14	6.5831 × 10−123	8.2492 × 10−164	8.6851 × 10−168	3.0531 × 10−157
F27	1	1.0250 × 10−157	2.0206 × 10−155	3.6442 × 10−147	3.6280 × 10−153	9.4130 × 10−97	1.9708 × 10−170	2.0139 × 10−169	2.3166 × 10−180
F28	1	1.3695 × 10−81	3.4568 × 10−130	1.6581 × 10−88	4.3739 × 10−102	2.4493 × 10−73	1.5066 × 10−179	4.0790 × 10−164	5.2728 × 10−177
F29	1	3.0822 × 10−109	1.0659 × 10−79	5.9170 × 10−64	2.0918 × 10−125	1.2884 × 10−3	5.8818 × 10−167	1.6202 × 10−164	3.4176 × 10−182
F30	1	1.4871 × 10−124	4.3231 × 10−140	1.4709 × 10−133	2.5568 × 10−122	1.3894 × 10−16	5.2403 × 10−182	6.6758 × 10−167	1.2636 × 10−178

.

The unimodal benchmark has a single global optimum, making it an ideal choice for convergence speed and accuracy. According to Table 4, F1–F3 belong to this category. As shown in Table 5, HSHHO ensures the lowest AVG and STD. The p-values in Table 6 show significant differences compared to competitors on all unimodal functions. These results highlight HSHHO’s enhanced handling of unimodal challenges, thanks to the robust global exploration of the SPM-VLS strategy during the search phase.

In contrast, multimodal benchmarks contain multiple local optima, which test the algorithm’s ability to avoid getting stuck and optimize locally. The CEC 2017 test ensured comprehensive validation, improving the reliability and breadth of the experiment. Table 5 shows that HSHHO dominates all multimodal functions except for F4 and F7. However, for most functions, HSHHO exhibits significant advantages. This highlights its powerful utilization ability, which is driven by Cauchy–Gaussian elite perturbations and helps escape local minima and broader global pursuits.

To increase rigor, we studied the mixed (F11–F20) and combined (F21–F30) functions of CEC 2017, which are more complex than single or multi peak types. Table 5 confirms the highest AVG and STD of HSHHO on all functions except F17 and F27. On F17 and F27, HSHHO ranks second in value. The p-values in Table 6 confirm that they are significantly better than other algorithms on each mixing and combining function. These results stem from the nonlinear escape energy mechanism, which promotes optimal phase balance in HSHHO and produces excellent performance on complex problems.

In summary, according to the results in Table 5, HSHHO exhibits excellent performance in 26 out of 30 benchmark functions. However, according to the “no free lunch” theorem, it can be observed that HSHHO is not the optimal executor in all cases. For example, it ranks second on functions F4, F7, F17, and F27 or is comparable to GWO and SSA. This indicates that although HSHHO has strong robustness to mixing and combining problems, specific landscape features in certain multimodal functions may still be more effectively navigated through other metaheuristic methods.

The convergence graph in Figure 6, Figure 7 and Figure 8 visually compares HSHHO with different optimization algorithms. Figure 6 depicts the rapid decline and sustained accuracy advantage of HSHHO. Figure 7 shows that HSHHO quickly reaches its optimal value, surpassing other algorithms. Figure 8 highlights the leading speed of HSHHO on all functions except F22, ultimately achieving unparalleled accuracy.

In addition, we compare the average time. According to

Table 7. Average running time on CEC 2017.

Function	HSHHO	HHO	PSO	GWO	WOA	SSA	MFO	GOA	SCA
F1	2.485 × 10−1	2.036 × 10−1	9.21 × 10−2	1.629 × 10−1	9.57 × 10−2	8.8799 × 10−2	1.242 × 10−1	3.046	1.391 × 10−1
F2	/	/	/	/	/	/	/	/	/
F3	3.121 × 10−1	2.266 × 10−1	9.77 × 10−2	1.769 × 10−1	1.089 × 10−1	1.791 × 10−1	1.517 × 10−1	3.492	1.594 × 10−1
F4	2.715 × 10−1	2.305 × 10−1	1.044 × 10−1	1.770 × 10−1	9.94 × 10−2	1.702 × 10−1	1.418 × 10−1	3.349	1.487 × 10−1
F5	4.047 × 10−1	3.387 × 10−1	1.58 × 10−1	2.532 × 10−1	1.572 × 10−1	2.260 × 10−1	2.072 × 10−1	4.255	2.188 × 10−1
F6	4.433 × 10−1	3.408 × 10−1	1.679 × 10−1	2.337 × 10−1	1.725 × 10−1	2.203 × 10−1	2.045 × 10−1	2.961	2.153 × 10−1
F7	4.236 × 10−1	3.461 × 10−1	1.423 × 10−1	2.293 × 10−1	1.551 × 10−1	2.211 × 10−1	1.910 × 10−1	3.962	2.113 × 10−1
F8	3.115 × 10−1	2.498 × 10−1	1.125 × 10−1	1.816 × 10−1	1.19 × 10−1	1.722 × 10−1	1.527 × 10−1	3.006	1.635 × 10−1
F9	3.370 × 10−1	2.801 × 10−1	1.234 × 10−1	1.947 × 10−1	1.328 × 10−1	1.805 × 10−1	1.561 × 10−1	3.22	1.805 × 10−1
F10	4.253 × 10−1	3.330 × 10−1	1.478 × 10−1	2.288 × 10−1	1.490 × 10−1	2.115 × 10−1	1.966 × 10−1	3.349	2.003 × 10−1
F11	5.082 × 10−1	3.991 × 10−1	1.885 × 10−1	3.307 × 10−1	2.006 × 10−1	2.737 × 10−1	2.446 × 10−1	5.158	2.811 × 10−1
F12	5.915 × 10−1	4.641 × 10−1	2.102 × 10−1	3.366 × 10−1	2.032 × 10−1	3.243 × 10−1	2.734 × 10−1	5.353	2.793 × 10−1
F13	4.737 × 10−1	3.690 × 10−1	1.731 × 10−1	3.058 × 10−1	1.717 × 10−1	2.789 × 10−1	2.349 × 10−1	4.855	2.504 × 10−1
F14	6.786 × 10−1	5.008 × 10−1	2.255 × 10−1	3.751 × 10−1	2.616 × 10−1	3.436 × 10−1	3.331 × 10−1	5.867	3.392 × 10−1
F15	7.823 × 10−1	6.416 × 10−1	2.617 × 10−1	4.748 × 10−1	2.544 × 10−1	4.730 × 10−1	4.501 × 10−1	8.403	3.738 × 10−1
F16	5.396 × 10−1	4.156 × 10−1	1.845 × 10−1	3.035 × 10−1	2.036 × 10−1	2.923 × 10−1	2.630 × 10−1	5.260	2.745 × 10−1
F17	7.151 × 10−1	4.757 × 10−1	2.347 × 10−1	3.404 × 10−1	2.293 × 10−1	3.210 × 10−1	3.084 × 10−1	4.285	2.966 × 10−1
F18	5.645 × 10−1	4.416 × 10−1	1.871 × 10−1	3.406 × 10−1	2.095 × 10−1	3.059 × 10−1	2.758 × 10−1	6.132	2.911 × 10−1
F19	1.827	1.359	8.538 × 10−1	8.775 × 10−1	7.597 × 10−1	7.929 × 10−1	7.979 × 10−1	5.924	8.388 × 10−1
F20	6.111 × 10−1	4.527 × 10−1	2.240 × 10−1	3.207 × 10−1	2.261 × 10−1	2.960 × 10−1	2.744 × 10−1	3.775	2.873 × 10−1
F21	9.612 × 10−1	6.025 × 10−1	3.063 × 10−1	4.262 × 10−1	3.222 × 10−1	3.883 × 10−1	3.937 × 10−1	4.888	3.816 × 10−1
F22	1.157	8.730 × 10−1	4.302 × 10−1	5.959 × 10−1	4.446 × 10−1	5.615 × 10−1	5.699 × 10−1	6.115	5.346 × 10−1
F23	1.010	7.321 × 10−1	3.884 × 10−1	4.746 × 10−1	3.717 × 10−1	4.985 × 10−1	4.544 × 10−1	4.474	4.431 × 10−1
F24	9.775 × 10−1	7.216 × 10−1	3.804 × 10−1	4.922 × 10−1	3.915 × 10−1	4.350 × 10−1	4.245 × 10−1	4.310	4.541 × 10−1
F25	8.633 × 10−1	6.273 × 10−1	3.277 × 10−1	4.363 × 10−1	3.263 × 10−1	4.049 × 10−1	3.855 × 10−1	4.186	4.000 × 10−1
F26	1.471	1.077	5.276 × 10−1	6.695 × 10−1	5.275 × 10−1	6.465 × 10−1	7.273 × 10−1	5.821	6.262 × 10−1
F27	1.332	9.058 × 10−1	4.765 × 10−1	5.886 × 10−1	4.996 × 10−1	5.604 × 10−1	6.032 × 10−1	4.947	5.460 × 10−1
F28	1.158	7.613 × 10−1	4.246 × 10−1	5.236 × 10−1	4.289 × 10−1	5.074 × 10−1	4.869 × 10−1	4.630	4.995 × 10−1
F29	9.254 × 10−1	6.879 × 10−1	3.503 × 10−1	4.640 × 10−1	3.457 × 10−1	4.326 × 10−1	3.994 × 10−1	4.534	4.142 × 10−1
F30	2.202	1.487	8.179 × 10−1	9.519 × 10−1	8.427 × 10−1	9.272 × 10−1	8.962 × 10−1	5.431	9.000 × 10−1

Bold represents the optimal value.

, PSO and WOA achieved good results on the benchmark test function. Compared with HHO, the average time of HSHHO increased due to the influence of three improvement strategies. However, achieving better computational results at a smaller time cost is often acceptable.

3.3. HSHHO for Feature Selection

To demonstrate the practical value of HSHHO, we apply it to feature selection tasks spanning datasets with feature counts ranging from a few to several hundred. Its performance is benchmarked against several established metaheuristic algorithms.

3.3.1. Datasets and Parameter Settings

We selected 18 widely used UCI datasets, detailed in

Table 8. Detailed features of the 18 feature selection datasets.

Number	Datasets	Number of Features	Number of Samples
1	Exactly	13	1000
2	Exactly2	13	1000
3	Breastcancer	9	699
4	Lymphography	18	148
5	Vote	16	300
6	Zoo	16	101
7	BreastEW	30	569
8	CongressEW	16	435
9	HeartEW	13	270
10	IonosphereEW	34	351
11	KrvskpEW	36	3196
12	WaveformEW	40	5000
13	WineEW	13	178
14	SonarEW	60	208
15	SpectEW	22	267
16	Tic-tac-toe	9	958
17	M-of-n	13	100
18	PenglungEW	325	73

, that have been extensively validated in prior feature selection research, ensuring both reliability and relevance.

Using the wrapper approach, the optimizer is combined with the classifier to identify the optimal feature subset. In order to evaluate the accuracy of the selected feature subset, a classifier must be used for classification after feature selection. The KNN classifier is a popular classification method in the field of feature selection. The selection of K value and distance measurement in KNN affect the effectiveness of KNN. In this study, the selection of K value refers to the existing literature, with a value of 5 [49]. For each dataset, the samples are divided into 80% training set and 20% testing set, and reliability is improved through 10-fold cross validation. All experiments were repeated 10 times to eliminate the influence of random fluctuations.

For comparative analysis, HSHHO is tested alongside HHO, PSO, GWO, WOA, SSA, and GOA. Common parameters are fixed as follows: population size = 10, maximum iterations = 100, and search dimension equal to the number of features in each dataset. The parameters of each algorithm are summarized in

Table 9. Parameter setting.

Algorithm	Parameter	Value
HSHHO/HHO	$E_0, r$	$E_0$: Random in [−1, 1] $r$: Random in [0, 1]
PSO	${\omega , c}_1, c_2$	$\omega =0.7$; $c_1=c_2=2$
GWO	$a$	$a=2$
WOA	$b,p$	$b=1$; $p=0.5$
SSA	$c_2, c_3$	Random in [0, 1]
GOA	$f,l$	$f=0.5$; $l=1.5$

. All implementations are developed and executed in MATLAB R2022b.

3.3.2. Performance Indicators

To evaluate the performance of the HSHHO, three statistical indicators are introduced, including average fitness, average accuracy, and average feature selection size.

• Average fitness (AvgFit): Average value of the fitness obtained.

  $$AvgFit={\displaystyle \frac{1}{T}}{\displaystyle \sum _{i=1}^T}{Fit}_{best}^i$$

  (23)

  where ${Fit}_{best}^i$ denotes the best fitness obtained in the $i$-th iteration.
• Average accuracy (AvgAcc): Accuracy of the classifier in selecting the best feature subset.

  $$AvgAcc={\displaystyle \frac{1}{T}}{\displaystyle \sum _{i=1}^T}{Acc}_i$$

  (24)

  where ${Acc}^i$ is the accuracy achieved by the optimal subset in the $i$-th iteration.
• Average number of feature selections (AvgSel): Average value of the ratio of the number of features selected.

  $$AvgSel={\displaystyle \frac{1}{T}}{\displaystyle \sum _{i=1}^T}{\displaystyle \frac{{FSnumber}_i}{D}}$$

  (25)

  where ${FSnumber}_i$ the number of features selected in the $i$-th iteration, and $D$ is the total number of original features.
• Average run-time (AvgRt): Average value of the algorithm’s run time.

  $$AvgRt={\displaystyle \frac{1}{T}}{\displaystyle \sum _{i=1}^T}{Time}_i$$

  (26)

  where ${Time}_i$ denotes the run time in the $i$-th iteration.

3.3.3. Comparison Results on the 18 Feature Datasets

For average fitness, the results are reported in

Table 10. Comparison of the results for average fitness.

Datasets	Index	HSHHO	HHO	PSO	GWO	WOA	SSA	GOA
Exactly	AVG	0.0186	0.1782	0.1395	0.2788	0.1643	0.0445	0.2876
	STD	0.0533	0.1036	0.1667	0.1833	0.0935	0.0684	0.1308
Exactly2	AVG	0.2154	0.2637	0.2915	0.2669	0.2743	0.2557	0.2648
	STD	0.0742	0.0235	0.0364	0.0280	0.0176	0.0113	0.0383
Breastcancer	AVG	0.0237	0.0285	0.0364	0.0338	0.0378	0.0356	0.0493
	STD	0.0082	0.0066	0.0137	0.0105	0.0133	0.0075	0.0063
Lymphography	AVG	0.0417	0.1523	0.1983	0.1397	0.1176	0.1034	0.1451
	STD	0.0062	0.0153	0.0271	0.0196	0.0133	0.0127	0.0184
Vote	AVG	0.0173	0.0295	0.0558	0.0623	0.0573	0.0472	0.0086
	STD	0.0129	0.0273	0.0235	0.0288	0.0254	0.0176	0.0067
Zoo	AVG	0.0136	0.0422	0.1065	0.0603	0.0255	0.0157	0.0336
	STD	0.0253	0.0353	0.0377	0.0286	0.0147	0.0136	0.0241
BreastEW	AVG	0.0239	0.0327	0.0388	0.0436	0.0385	0.0314	0.0539
	STD	0.0068	0.0071	0.0073	0.0076	0.0073	0.0068	0.0084
CongressEW	AVG	0.0217	0.0326	0.0469	0.0435	0.0389	0.0264	0.0477
	STD	0.0056	0.0077	0.0083	0.0085	0.0075	0.0067	0.0085
HeartEW	AVG	0.0427	0.1635	0.2451	0.1443	0.1532	0.1206	0.2135
	STD	0.0165	0.0238	0.0295	0.0226	0.0255	0.0184	0.0209
IonosphereEW	AVG	0.0621	0.0757	0.1425	0.1086	0.0853	0.1140	0.0743
	STD	0.0133	0.0268	0.0236	0.0173	0.0212	0.0158	0.0254
KrvskpEW	AVG	0.0203	0.0364	0.0536	0.1934	0.0336	0.0259	0.0557
	STD	0.0047	0.0055	0.0067	0.1337	0.0053	0.0047	0.0079
WaveformEW	AVG	0.1608	0.1873	0.2294	0.1720	0.1835	0.1565	0.2380
	STD	0.0055	0.0135	0.0178	0.0164	0.0127	0.0073	0.0215
WineEW	AVG	0.0066	0.0184	0.0213	0.0322	0.0076	0.0058	0.0368
	STD	0.0027	0.0133	0.0162	0.0186	0.0060	0.0053	0.0155
SonarEW	AVG	0.0116	0.0463	0.1376	0.0855	0.0361	0.0378	0.0996
	STD	0.0074	0.0136	0.0746	0.0355	0.0228	0.0238	0.0330
SpectEW	AVG	0.1283	0.1772	0.1457	0.1970	0.1738	0.1653	0.2371
	STD	0.0137	0.0163	0.0177	0.0166	0.0173	0.0176	0.0236
Tic-tac-toe	AVG	0.1385	0.1596	0.2433	0.2123	0.1825	0.1672	0.2677
	STD	0.0154	0.0254	0.0269	0.0187	0.0352	0.0246	0.0371
M-of-n	AVG	0.0053	0.0238	0.1174	0.0363	0.0331	0.0163	0.0578
	STD	0.0011	0.0174	0.0566	0.0165	0.0183	0.0138	0.0221
PenglungEW	AVG	0.0048	0.0267	0.0319	0.0476	0.0524	0.0184	0.0553
	STD	0.0035	0.0158	0.0194	0.0283	0.0266	0.0139	0.0258

Bold represents the optimal value.

. According to Table 10, it can be observed that HSHHO obtains the best value on the 16-feature datasets. Compared to HHO, PSO, GWO and WOA, HSHHO achieves the best results on all datasets. SSA achieves the optimal fitness value on the WaveformEW dataset, while GOA achieves better results on the Vote dataset.

For average accuracy, the results are reported in

Table 11. Comparison results for accuracy.

Datasets	Index	HSHHO	HHO	PSO	GWO	WOA	SSA	GOA
Exactly	AVG	0.9931	0.8527	0.8753	0.8215	0.8674	0.9664	0.7681
	STD	0.0734	0.1147	0.1094	0.1487	0.1073	0.0916	0.1628
Exactly2	AVG	0.7833	0.7254	0.7336	0.7547	0.7323	0.7734	0.7496
	STD	0.0141	0.0176	0.0182	0.0122	0.0188	0.0106	0.0337
Breastcancer	AVG	0.9847	0.9736	0.9471	0.9674	0.9730	0.9648	0.9534
	STD	0.0034	0.0076	0.0184	0.0164	0.0086	0.0137	0.0155
Lymphography	AVG	0.9290	0.9116	0.7653	0.8782	0.9046	0.9273	0.8692
	STD	0.0053	0.0087	0.0364	0.0284	0.0126	0.0076	0.0314
Vote	AVG	0.9873	0.9581	0.8966	0.9582	0.9633	0.9671	0.9347
	STD	0.0094	0.0148	0.0237	0.0137	0.0125	0.0078	0.0183
Zoo	AVG	0.9826	0.9682	0.8893	0.9527	0.9643	0.9714	0.9580
	STD	0.0076	0.0126	0.0257	0.0121	0.0116	0.0096	0.0175
BreastEW	AVG	0.9862	0.9613	0.9457	0.9566	0.9873	0.9734	0.9407
	STD	0.0051	0.0103	0.0176	0.0124	0.0061	0.0082	0.0137
CongressEW	AVG	0.9682	0.9534	0.9456	0.9637	0.9786	0.9743	0.9581
	STD	0.0135	0.0103	0.0257	0.0128	0.0057	0.0066	0.0216
HeartEW	AVG	0.8843	0.8379	0.7833	0.8742	0.8307	0.8874	0.8173
	STD	0.0145	0.0195	0.0266	0.0151	0.0182	0.0173	0.0214
IonosphereEW	AVG	0.9635	0.9362	0.9078	0.9133	0.9157	0.9722	0.9353
	STD	0.0083	0.0178	0.3842	0.2457	0.2239	0.0264	0.0133
KrvskpEW	AVG	0.9839	0.9746	0.9384	0.9375	0.9661	0.9703	0.9564
	STD	0.0033	0.0108	0.0296	0.0173	0.0184	0.0165	0.0192
WaveformEW	AVG	0.8503	0.8057	0.7693	0.8375	0.7832	0.8572	0.8174
	STD	0.0043	0.0195	0.0286	0.0163	0.0271	0.0115	0.0188
WineEW	AVG	0.9978	0.9733	0.9572	0.9766	0.9834	0.9945	0.9763
	STD	0.0054	0.0162	0.0197	0.0133	0.0099	0.0086	0.0145
SonarEW	AVG	0.9683	0.9248	0.7830	0.8677	0.9186	0.9635	0.9178
	STD	0.0143	0.0272	0.0464	0.0238	0.0306	0.0157	0.0349
SpectEW	AVG	0.9033	0.8672	0.7921	0.8493	0.9096	0.8776	0.8530
	STD	0.0076	0.0166	0.0265	0.0193	0.0131	0.0154	0.0175
Tic-tac-toe	AVG	0.8397	0.7918	0.8134	0.7785	0.8073	0.7806	0.7849
	STD	0.0128	0.0156	0.0087	0.0217	0.0109	0.0132	0.0174
M-of-n	AVG	1.0000	0.9577	0.8770	0.9538	0.9364	0.9769	0.9137
	STD	0.0178	0.0261	0.0589	0.0352	0.0387	0.0143	0.0225
PenglungEW	AVG	1.0000	0.9751	0.7885	0.8873	0.9682	1.0000	0.9744
	STD	0.0024	0.0139	0.0288	0.0183	0.0122	0.0081	0.0163

Bold represents the optimal value.

. It is evident that HSHHO demonstrates superior performance. Compared to SSA, HSHHO also achieves the best results on other datasets except the HeartEW, IonosphereEW, and WaveformEW datasets. For PSO, GWO, and GOA, HSHHO achieves the best results on all datasets. Additionally, SSA ranks second for accuracy.

In terms of the number of selected features, based on the average number of selected features by HSHHO and other optimization algorithms reported in

Table 12. Comparison results for number of selected features.

Datasets	Index	HSHHO	HHO	PSO	GWO	WOA	SSA	GOA
Exactly	AVG	4.2333	6.6000	9.8000	6.1333	8.5000	6.7333	6.9000
	STD	0.8657	2.7312	2.5146	2.8940	1.6845	0.7735	2.2875
Exactly2	AVG	2.1333	6.8333	6.2000	4.1333	4.3333	6.2333	6.2000
	STD	1.4352	2.6824	2.0467	2.3647	2.7821	1.8403	2.5473
Breastcancer	AVG	3.4333	5.1000	5.7000	6.3000	5.0000	4.3333	5.6667
	STD	1.5376	2.7422	2.2457	2.4725	1.4973	1.8478	2.1447
Lymphography	AVG	3.6000	10.3667	9.0000	8.4667	7.4333	8.5000	8.5667
	STD	1.3862	3.1457	2.7549	2.2967	2.4369	1.3645	1.5783
Vote	AVG	3.6333	8.7667	7.3000	8.9667	7.9667	8.1000	8.5000
	STD	1.3465	1.6743	1.3345	2.4724	2.6948	1.9064	2.1445
Zoo	AVG	4.2333	9.6333	18.7333	8.5667	6.6667	5.3667	6.7000
	STD	1.6754	2.6137	3.6476	2.4336	1.8544	1.7648	1.8532
BreastEW	AVG	7.6333	15.3333	16.6000	17.5000	15.8000	10.6333	12.5000
	STD	2.6843	3.9227	3.9853	3.5872	3.6954	2.8947	3.3427
CongressEW	AVG	2.3667	6.6333	4.0000	6.0000	7.6333	5.1333	8.1000
	STD	1.3047	2.4183	1.6737	2.1641	2.7349	1.8452	2.3457
HeartEW	AVG	4.4333	7.6667	6.2000	8.4333	8.6000	4.6333	6.4667
	STD	2.3584	1.6427	1.4576	2.0458	1.8674	1.7343	1.8174
IonosphereEW	AVG	4.3000	9.7333	8.6000	7.6333	10.1667	8.2666	10.2000
	STD	1.3542	2.3768	2.1975	2.6841	1.6743	2.4557	2.3864
KrvskpEW	AVG	13.2667	20.6333	22.8333	11.9667	21.3667	17.6667	19.6333
	STD	3.0667	4.1694	5.1393	2.8335	2.5749	1.7735	1.5873
WaveformEW	AVG	12.6333	24.3667	22.6333	18.3000	29.4333	13.7333	11.3000
	STD	2.0457	3.7064	3.0513	2.8337	3.3963	2.6453	1.7364
WineEW	AVG	2.6333	7.6000	8.4000	7.6333	6.0333	6.2000	7.0667
	STD	1.3056	1.6447	2.3624	2.0154	1.6783	1.9564	1.4358
SonarEW	AVG	18.0000	36.7333	31.9333	32.6667	44.0000	16.2333	29.0000
	STD	2.6247	5.3496	5.0974	5.2167	6.7746	2.4097	3.6379
SpectEW	AVG	6.5333	8.6333	9.6000	7.3000	7.6333	7.7333	8.3667
	STD	1.3678	2.2307	2.8344	2.4837	2.7345	1.9764	2.6843
Tic-tac-toe	AVG	4.3333	8.3667	4.8333	5.6667	9.0000	5.5333	5.2000
	STD	1.5375	2.1453	1.5631	1.8332	2.4637	1.7591	1.7344
M-of-n	AVG	5.6333	7.0000	9.7333	7.2667	7.3333	6.6667	6.6000
	STD	1.0473	1.4752	1.6713	1.4143	1.7726	1.3927	1.5173
PenglungEW	AVG	130.6333	167.4667	178.7333	167.5667	142.0000	116.3000	184.3333
	STD	8.6413	8.8723	9.4157	9.1379	7.5163	7.3827	9.4476

Bold represents the optimal value.

, it can be observed that, compared to the standard HHO, HSHHO obtains the minimum number of selected features on all 18-feature datasets. Compared to GWO, HSHHO obtains the best feature subset on all datasets except the KrvskpEW dataset. SSA obtains the least number of features on the SonarEW and PenglungEW datasets. The GOA finds the optimal number of selected features on the WaveformEW dataset. In total, HSHHO finds the best feature subsets on 14 datasets (Exactly, Exactly2, Breastcancer, Lymphography, Vote, Zoo, BreastEW, CongressEW, HeartEW, IonosphereEW, WineEW, SpectEW, Tic-tac-toe and PenglungEW), and the number of features obtained on the remaining four datasets ranked second.

For average accuracy, the results are reported in

Table 13. Comparison results for average running time.

Datasets	HSHHO	HHO	PSO	GWO	WOA	SSA	GOA
Exactly	34.3726	36.5482	13.3348	13.6872	13.5667	13.4734	14.2581
Exactly2	32.4492	34.1054	11.5381	12.6288	11.6047	12.6749	13.1574
Breastcancer	31.4238	32.6856	13.4655	13.8273	13.1752	13.8574	15.7863
Lymphography	31.5043	28.7248	10.6482	10.7054	10.2477	10.9127	19.5838
Vote	34.8622	29.5712	11.4586	11.8723	11.3375	11.8207	12.6343
Zoo	33.5774	30.1588	10.8792	10.7354	10.3510	10.9367	11.3116
BreastEW	32.7647	33.7821	12.3762	12.6510	12.6344	12.7508	12.8943
CongressEW	29.7340	30.5344	11.0175	11.3738	12.2048	11.4626	11.5746
HeartEW	33.9426	32.5796	12.8216	13.1953	12.4438	13.4693	13.7352
IonosphereEW	32.4257	28.6896	15.1851	10.9683	10.6145	11.2572	11.5113
KrvskpEW	37.2362	35.5392	13.5824	15.7342	13.8733	13.7565	15.3044
WaveformEW	67.4855	56.6837	18.4286	21.4761	19.3480	20.7845	21.3482
WineEW	32.1856	29.8557	10.7308	10.8713	54.6842	10.8761	11.7505
SonarEW	31.7238	30.3580	11.8237	13.9374	12.2836	13.8823	13.7548
SpectEW	32.4804	29.9452	11.6357	11.3648	11.1821	11.6872	11.2387
Tic-tac-toe	35.7163	34.3962	11.4283	12.4835	11.9352	12.4883	13.7543
M-of-n	33.8927	32.2734	12.6704	13.8873	13.7155	13.0489	14.8157
PenglungEW	32.5782	31.2583	12.8733	20.8746	15.1796	15.8438	16.8748

Bold represents the optimal value.

. It is evident that HSHHO demonstrates superior performance. Compared to SSA, HSHHO also achieves the best results on other datasets except the HeartEW, IonosphereEW, and WaveformEW datasets. For PSO, GWO, and GOA, HSHHO achieves the best results on all datasets. Additionally, SSA ranks second for accuracy.

In addition, Table 13 provides detailed information on the average time. The analysis of experimental results shows that, compared with other algorithms, the proposed method requires the longest time. This is due to the introduction of three improvement strategies, which extend the time required to mine the optimal solution. When used for FS, the number of features is a dimension, and the higher the dimension, the longer the required time. Although PSO and WOA require the shortest time in the dataset, their classification accuracy is not the highest and they cannot minimize the number of features. Therefore, considering other metrics, HSHHO still outperforms other algorithms in terms of FS.

3.3.4. Comprehensive Correlation Analysis

To evaluate whether the improvement of fitness is essentially based on classification performance, a three indicators correlation analysis was conducted on Table 8, Table 9 and Table 10. Firstly, the comparison between fitness and accuracy indicates that the optimization of HSHHO is performance-oriented. In 16 out of 18 datasets, a decrease in fitness values was accompanied by a significant increase in accuracy. It is worth noting that on high-dimensional datasets such as KrvskpEW and PenglingEW, HSHHO achieved excellent accuracy while significantly reducing fitness scores. This confirms that the fitness function effectively captures the algorithm’s ability to enhance discriminative power. Secondly, the relationship between accuracy and feature subset size indicates the efficiency of HSHHO in noise filtering. According to Table 10, HSHHO achieved a smaller feature subset than HHO in all 18 datasets. In most cases, a decrease in the feature count leads to a reverse improvement in accuracy, indicating that the eliminated features are mainly redundant or noisy.

In addition to solving quality, the practical usability of the proposed HSHHO framework should also be evaluated based on computational costs. Since HSHHO incorporates three improvement strategies, its processing naturally requires more time than the original HHO. Therefore, the runtime results should be interpreted in conjunction with accuracy results rather than in isolation.

According to Table 7 and Table 13, we can observe that on the feature selection dataset, the average computation time increased from 597.67 s for HHO to 630.35 s for HSHHO; on the benchmark functions, it rose from 16.54 s for HHO to 22.24 s. Although this indicates an increase in computational cost, HSHHO demonstrates superior algorithmic performance.

These results indicate that the additional runtime of HSHO is associated with improved search quality and stronger convergence behavior, rather than an unnecessary computational burden. In other words, HSHO achieves higher optimization accuracy, enhanced robustness, and more reliable final solutions at the cost of a moderate additional runtime. This trade-off is advantageous in applications where solution quality and stability are more critical than minimal execution time. In summary, the improved HSHHO fitness is a collaborative result of achieving higher classification accuracy with a more concise feature set.

4. Conclusions

In this paper, we introduce an enhanced Harris Hawks Optimization algorithm, named HSHHO, which incorporates the SPM with variable logarithmic spiral, nonlinear escape energy mechanism, and Cauchy–Gaussian elite perturbation. By integrating these enhancements, HSHHO improves initial population diversity, global search efficiency, and local refinement accuracy. To evaluate its effectiveness, we first examined the impact of the three strategies on HHO that use 23 classical benchmark functions, revealing that each contributes meaningfully to overall improvement. Next, IEEE CEC 2017 is used to compare HSHHO with eight well-known algorithms. The results indicate that HSHHO dominates a significant competitive advantage at most benchmark functions. Finally, when applied to 18 feature selection tasks, HSHHO delivered outstanding results, notably reducing feature counts by 70.92% while elevating average classification accuracy to 94.47%.

Although HSHHO has demonstrated strong optimization capabilities, the inherent limitations in hybrid strategies need to be addressed. The integration of SPM chaotic mapping and Cauchy–Gaussian mutation significantly improves the search performance of the algorithm, but also enhances its sensitivity by increasing the initial parameter configuration. Meanwhile, experimental results indicate that HSHHO dominates in most benchmark tests, but its performance in specific functions still needs improvement. In addition, the transition between exploration and mining may still experience local stagnation, indicating that a single nonlinear energy model may not be universally applicable to all complex multimodal problems.

From these insights, several avenues for future exploration have emerged. Firstly, the development phase can be improved by integrating search operators from alternative optimizers. Secondly, its application may be extended to real engineering problems, such as railway emergency scenario prediction, medical big data analysis, intelligent risk assessment, etc. Thirdly, future research will compare HSHHO with the latest feature selection algorithms and continuously optimize algorithm performance. In addition, a notable prospect lies in developing a unified framework to interactively orchestrate multiple metaheuristics and machine learning models, leveraging their collective strengths to tackle daunting real-world challenges and generate higher-quality solutions. On the basis of this project, we will improve the HSHHO algorithm and apply it to key point recognition in typical emergency response scenarios of railways. By identifying the core feature subset in the collaborative emergency process of multiple disciplines, a scientific method is provided for railway emergency response operation standards, and an important role is played in risk warning and emergency plan optimization.