Modular Harris Hawks optimization with trend-guided differential evolution and Gaussian exploration for global optimization and engineering design

Abstract

This paper presents DEHHO, a modular and lightweight variant of Harris Hawks Optimization (HHO), engineered to mitigate the stagnation and directional instability often encountered in high-dimensional and structurally complex optimization tasks. The proposed framework integrates two synergistic, phase-specific strategies: a Gaussian-based stochastic perturbation mechanism designed to maintain fine-grained diversity during exploration, and a Trend-Guided Differential Evolution (DE) update incorporating a momentum vector to enforce directional stability for intensified exploitation. A dynamic balancing scheme coordinates these components, ensuring a smooth transition between search phases without incurring excessive computational overhead. Extensive empirical validation on the CEC 2017 and CEC 2020 benchmark suites (up to 100 dimensions) demonstrates that DEHHO statistically outperforms 10 state-of-the-art peer algorithms—comprising 5 advanced HHO variants and 5 mainstream metaheuristics—in terms of convergence accuracy, robustness, and scalability. Furthermore, rigorous ablation studies confirm the individual efficacy and structural complementarity of the proposed mechanisms, establishing DEHHO as a reliable solver for complex numerical and engineering design problems.

Introduction

Solving complex black-box optimization problems—characterized by non-convexity, high dimensionality, and multimodal landscapes—remains a fundamental challenge in computational intelligence. Among the myriad of metaheuristic algorithms proposed to tackle these challenges, Harris Hawks Optimization (HHO), introduced by Heidari et al. in 2019^1,2, has garnered significant attention. Inspired by the cooperative hunting behavior and surprise pounce strategies of Harris hawks, HHO employs a dual-phase structure comprising exploration via random perching and exploitation via adaptive besieging. Due to its gradient-free nature and implementation simplicity, HHO has been successfully applied to diverse domains³, including medical diagnosis, public health modeling, bioinformatics, and industrial engineering design^4,5.

Despite its merits, the canonical HHO framework exhibits inherent structural limitations, particularly when scaling to high-dimensional or structurally complex optimization tasks. A critical deficiency lies in the algorithm’s limited capacity to maintain directional stability during the transition from exploration to exploitation. Specifically, in rugged landscapes, the population tends to suffer from a rapid loss of diversity, leading to premature convergence (stagnation) at local optima. Furthermore, the original besiege strategies rely heavily on the position of the prey (current global best) without sufficient utilization of the population’s historical evolutionary trends, resulting in oscillatory behavior and slow convergence precision in the later stages of the search.

To mitigate these issues, recent research has explored various enhancement strategies, including chaotic initialization, opposition-based learning (OBL), and hybridization with other heuristics. While these methods improve specific performance metrics, many suffer from increased computational complexity or a lack of holistic synergy between the introduced operators and the native HHO structure. Crucially, few studies have addressed the complementarity between HHO’s besieging pressure and the directional guidance mechanisms found in evolutionary algorithms.

To bridge this gap, this paper proposes DEHHO, a lightweight and modular variant of HHO that integrates Differential Evolution (DE) mechanisms to reinforce the search process. The motivation for this hybridization stems from the theoretical complementarity of the two algorithms: while HHO excels at rapid convergence via distinctive besieging phases, DE offers intrinsic directional guidance through difference vectors, which helps navigate flat or rugged valleys where gradient information is unavailable.

DEHHO embeds two coordinated strategies into the standard framework: (a) Gaussian-Stochastic Perturbation: A fine-grained diversity preservation mechanism designed to prevent population stagnation during the early exploration phase by introducing controllable stochasticity. (b) Trend-Guided DE Exploitation: A modified differential evolution operator incorporating a momentum term. This mechanism leverages historical population trends to enhance directional stability, thereby accelerating convergence and reducing oscillation in high-precision exploitation tasks.

By seamlessly integrating these modules, DEHHO aims to achieve a robust balance between global search capability and local refinement accuracy without incurring significant structural overhead. The proposed algorithm is rigorously validated against state-of-the-art competitors on the CEC2017 and CEC2020 benchmark suites (up to 100 dimensions) and applied to constrained engineering design problems.

The remainder of this paper is structured as follows. “Literature review” reviews the related work on HHO variants and differential evolution. “The proposed DEHHO algorithm” details the mathematical formulation and implementation of the proposed DEHHO. “Experimental validation and cross-benchmarkanalysis” presents the experimental results and ablation studies. “Engineering applications and analysis” analyzes the algorithm’s performance on real-world engineering applications. Finally, “Discussion and future work” concludes the paper and outlines directions for future research.

Literature review

This section critically reviews the theoretical foundations and recent advancements of Harris Hawks Optimization (HHO) and Differential Evolution (DE). To provide a comprehensive context for the proposed hybrid framework, we analyze the complementary nature of these two algorithms. Furthermore, we categorize existing HHO variants to systematically identify the research gaps—specifically, the lack of directional stability and synergistic diversity maintenance—that DEHHO aims to address.

HHO fundamentals

Harris Hawks Optimization (HHO) is a population-based metaheuristic introduced by Heidari et al.¹, simulating the cooperative chasing and pouncing behavior of Harris hawks. The core control mechanism is the time-dependent escaping energy parameter, , which governs the transition between global exploration and local exploitation.

The optimization process is distinctly divided into two phases:

Exploration phase (): Hawks observe the search space by perching randomly based on the positions of peers or the prey, ensuring global search coverage.

Exploitation phase (): As the prey’s energy depletes, hawks execute one of four besiege strategies: Soft Besiege, Hard Besiege, Soft Besiege with Progressive Rapid Dives, and Hard Besiege with Progressive Rapid Dives. The selection depends on the magnitude of and a random probability factor.

While the canonical HHO exhibits simplicity, it inherently suffers from late-stage stagnation. The scalar energy parameter , while effective for phase transition, lacks vector-based directional information. This often causes the population to oscillate around the optimum in high-dimensional landscapes without converging efficiently.

Differential evolution: fundamentals and state-of-the-art

Differential Evolution (DE), originally proposed by Storn and Price⁶, stands as a cornerstone of evolutionary computation. Unlike the scalar-pressure mechanism in HHO, DE drives evolution through a vector-based differential mutation operator. By calculating the weighted difference between population vectors, DE naturally encodes gradient-like landscape information into the search step. This capability to estimate descent directions without explicit gradient calculations makes DE theoretically complementary to swarm-based attractors like HHO.

While robust, the canonical DE struggles with parameter sensitivity and stagnation in high-dimensional, multimodal landscapes. Consequently, recent research (2024–2025) has shifted towards adaptive mechanisms, structural enhancements, and knowledge-driven hybridization to unlock its full potential in black-box optimization^7–9.

These studies underscore DE’s robust local refinement capabilities. However, standalone DE variants often require complex adaptive mechanisms or multiple subpopulations, which can increase computational overhead.

State-of-the-art HHO variants

Since its inception, numerous HHO variants have been proposed. We classify these improvements into three categories: initialization/parameter dynamics, hybridization, and diversity maintenance. A comparative summary of representative variants is presented in Table 1.

Table 1.

Summary of representative HHO variants and their limitations.

Category	Algorithm	Core mechanism	Main limitation/key feature
Param. control	EAHHO10	Lyapunov Markov chain	Scalar adjustment; lacks directional guidance
Hybridization	hHHO-SCA11	Sine-cosine operators	High structural complexity; loose coupling
Diversity	HHSC12	Brownian motion	Blind stochasticity; inefficient in high dims
Proposed	DEHHO	Trend-guided DE + Gaussian	Vector-based guidance; Lightweight

Initialization and parameter dynamics

Enhancing the initial population quality and the dynamic control of is a foundational strategy. To address random initialization, Zhu and Fu¹³, Xu et al.¹⁴ and Wei et al.¹⁵ employed chaotic maps and Good Point Set (GPS) strategies, respectively, ensuring a uniform initial distribution. Regarding parameter control, Zhao and Liu¹⁶ combined nonlinear energy control with opposition-based learning, while Uddin et al.¹⁷ proposed an exponential decay function. More sophisticated schemes include the Lyapunov-stable Markov chain adaptation by Mao and Gui¹⁰ and the generation-dependent Lévy scaling by Zhu et al.¹⁸.

While these methods improve the schedule of the search, they rely on scalar adjustments. They extend exploration duration but do not fundamentally alter the directionality of the search trajectory, leaving the algorithm susceptible to oscillation in high-dimensional valleys.

Hybridization with evolutionary operators

Fusing HHO with other metaheuristics aims to leverage complementary behaviors. Zhang et al.¹⁹ integrated mutation-based exploitation in WHHO, while Ghafari and Mansouri²⁰ fused DE with chaos-OBL mechanisms. Habibzadeh-khameneh et al.²¹ integrated ensemble learning for software defect prediction, while CXSHHO²² incorporated cooperative crossover strategies. Other notable designs include the intermittent hunting–rest cycles simulated by Ouyang et al.^23,24, the embedding of sine-cosine operators (SCA) by Kamboj et al.¹¹, and the Brownian motion mutation strategy proposed by Kang et al.²⁵.

Although effective, these hybrids often incur increased structural complexity (e.g., sequential execution of two full algorithms), leading to higher computational costs (often doubling function evaluations per iteration). The integration is frequently “loose,” lacking a unified mechanism to facilitate information flow between the swarm pressure and the operator guidance.

Diversity maintenance and local refinement

Explicit perturbation mechanisms have been widely adopted to resolve population clustering and stagnation. To prevent local optima entrapment, Elgamal et al.²⁶ incorporated simulated annealing (SA) for controlled stochastic perturbations. Similarly, Jiao et al.¹² utilized Brownian motion in HHSC for shallow escapes, while Song et al.²⁷ applied persistent trigonometric differences to maintain variability. Beyond stochastic perturbations, learning-based strategies have been introduced to refine solutions. Mahapatra et al.²⁸ and Li et al.²⁹ utilized dynamic parameter mutation and opposition-based mutation, respectively, while Sihwail et al.³⁰ and Liu et al.³¹ introduced elite opposition learning and rollback recovery to maintain elite diversity. Furthermore, specific real-world applications have driven robust variants; Jia et al.³² and Tang et al.³³ applied adaptive perturbation mechanisms for image segmentation and energy scheduling, while Liu et al.³⁴ and Hussien et al.³⁵ emphasized multi-leader cooperation to reduce the risk of misleading guidance by a single global best.

Many perturbation strategies (e.g., random walks) are “blind”—introducing randomness without trend awareness. In high dimensions (), random perturbations are statistically less efficient than guided evolutionary jumps in locating promising basins of attraction.

Research gaps and motivation

Despite the extensive improvements cited above, critical gaps remain that limit HHO’s efficacy in high-dimensional and structurally complex optimization:

Lack of Directional Stability: Existing variants focus on scalar parameter tuning. They lack a mechanism to exploit the historical momentum of the population, leading to inefficiency when gradient information is implicit.

Inefficient Hybridization: Previous hybrids often suffer from structural rigidity and high computational overhead. There is a need for a framework that seamlessly embeds Trend-Guided DE (for directional acceleration) without significantly increasing the complexity order.

Blind Diversity Mechanisms: Existing diversity strategies often rely on undirected stochasticity. Combining Gaussian diversity (for local cloud exploration) with guided mutation is largely unexplored.

To address these gaps, this paper proposes DEHHO, a modular framework integrating Gaussian-Stochastic exploration and Trend-Guided DE exploitation to ensure robustness and precision across all search phases.

The proposed DEHHO algorithm

Algorithm design and framework

To address the directional instability and stagnation issues inherent in the original HHO, we propose DEHHO, a modular variant that synergizes Gaussian exploration with trend-guided evolutionary exploitation. The framework maintains the canonical dual-phase structure of HHO but embeds two lightweight mechanisms governed by dynamic scheduling.

The core philosophy of DEHHO is “Structured Diversity and Guided Convergence,” realized through the following coupling mechanisms:

Exploration phase (): A conditional Gaussian perturbation is introduced to replace the standard random perching. This mechanism prevents the population from collapsing into local sub-regions too early by maintaining a non-vanishing variance in the search space.

Exploitation phase (): A dual-branch strategy is employed. The primary branch utilizes a Trend-Guided Differential Evolution (TG-DE) operator, incorporating a momentum vector to stabilize the convergence trajectory in high-dimensional valleys. The secondary branch retains HHO’s besiege tactics but is augmented with a Lévy-Annealed mechanism to execute long-range jumps when local stagnation is detected.

The modular structure and interaction flow of DEHHO are illustrated in Fig. 1, and the key control parameters are detailed in Table 2.

Fig. 1.

DEHHO framework flowchart.

Table 2.

The key parameters in DEHHO.

Description	Symbol	Definition/function	Default value
DE scaling factor		Controls the magnitude of the difference vector in DE	0.5～0.75 75 (linearly decayed)
Trend guidance factor		Regulation coefficient for the momentum vector	0.5～1.0 (linearly decayed)
Gaussian probability		Threshold for triggering Gaussian exploration mechanism	0.1 (Fixed)
DE branch probability		Probability of selecting DE strategy over HHO besiege	0.1～0.5 (Linear Increase)
Gaussian Std. Dev.		Standard deviation of the stochastic perturbation noise	0.2 (Fixed)
Lévy index		Power-law exponent for the Lévy flight distribution	1.5 (Fixed)

Exploration phase: stochastic perturbation

In the standard HHO, the exploration phase () relies on random perching strategies that may fail to maintain diversity in complex multimodal landscapes. To mitigate this, DEHHO introduces a Gaussian-based Stochastic Perturbation mechanism.

For a selected individual , the update rule is reformulated to explicitly include a noise term. Instead of a purely random placement, the new position is generated as:

1

Here, represents a randomly selected peer from the population, and is a vector of Gaussian noise with mean 0 and variance . The parameter acts as a diversity control knob: a fixed ensures that the perturbation is localized enough to preserve the structural information of the swarm while providing sufficient stochasticity to expand the search radius. This conditional update is triggered with a fixed probability , ensuring that the algorithm balances between HHO’s global perching and this fine-grained neighbor exploration.

Exploitation phase: trend-guided DE and Lévy annealing

The exploitation phase () is critical for convergence precision. DEHHO enhances this phase through a dual-branch mechanism designed to enforce directional stability.

Trend-guided differential evolution (TG-DE)

Standard HHO exploitation often oscillates around the global best due to the lack of historical information. To smooth this trajectory, we introduce a Momentum Vector, , defined as the displacement of the individual’s historical trajectory.

For a selected individual , the momentum is calculated as . At , is initialized to . The update equation combines the classical “DE/current-to-best/1” strategy with a trend guidance term:

2

where is the current global best position; , are mutually exclusive random individuals; is the scaling factor; and is the trend guidance factor.

Definition 1

(Directional Stability): We define directional stability as the consistency of the update vector’s alignment with the descent direction of the fitness landscape. By incorporating the difference vector , which statistically approximates the gradient in population-based methods, combined with the inertial momentum , DEHHO maximizes this alignment, reducing erratic high-frequency oscillations observed in standard random walks.

To prioritize exploration in the early exploitation phase and convergence in the later phase, the probability of selecting this DE branch, , increases linearly from 0.1 to 0.5.

Lévy-annealed perturbation

For individuals not selected for DE updates (probability ), the algorithm retains the original HHO besiege strategies but augments the outcome with a Lévy-annealed Jump to prevent stagnation in local basins:

3

where denotes element-wise multiplication. The random step is calculated using Mantegna’s algorithm to ensure reproducibility:

4

where .

Crucially, the step size scaling factor follows a Simulated Annealing-like decay:

5

This decay ensures that the algorithm performs long-range jumps (heavy-tailed flights) in early exploitation to escape local optima, while shifting to fine-grained local refinements as .

Computational complexity and limitations

The theoretical time complexity of DEHHO is determined by the population size (), dimension (), and maximum iterations ().

Initialization: .

Fitness evaluation: , where is the cost of the objective function.

Update mechanism: The HHO, Gaussian, and DE updates all involve vector operations with complexity .

Thus, the total complexity is , which is asymptotically equivalent to the original HHO (). Although the generation of Gaussian and Lévy random numbers introduces a marginal constant overhead, the vectorized implementation ensures that DEHHO maintains computational efficiency suitable for resource-constrained engineering tasks.

It is acknowledged that the effectiveness of DEHHO relies on the empirical tuning of parameters (). Additionally, the momentum term is initialized as zero, which may result in a “cold start” delay in directional guidance during the very first iterations.

Algorithm 1. DEHHO optimization framework.

Experimental validation and cross-benchmark analysis

Experimental protocol

To rigorously evaluate the performance and scalability of the proposed DEHHO algorithm, we conducted comprehensive experiments on two widely recognized benchmark suites: CEC 2017 and CEC 2020. These suites were selected to present complementary evaluation challenges: CEC 2017 emphasizes functional diversity across unimodal, multimodal, hybrid, and composition landscapes, while CEC 2020 focuses on structural complexity, including variable dependencies, deceptive optima, and hierarchical interactions. This dual-benchmark setting enables a joint assessment of generalization ability and robustness.

For both benchmarks, experiments were performed under 50-dimensional (50D) and 100-dimensional (100D) configurations to test scalability. The full set of functions was considered: 29 problems from CEC 2017 and 10 hybrid/composition functions from CEC 2020. To benchmark DEHHO, we selected ten state-of-the-art peer algorithms, comprising five representative HHO variants (BGHHO, SHHO, HHSC, CLHHEO, IHAOHHO) and five mainstream metaheuristics (SCSO, JA, SCA, HBA, SSA). The specific parameter settings for these algorithms are detailed in Table 2. All algorithms were implemented in Python 3.9 and executed on a workstation equipped with an Intel Core i7-14700KF processor and 32 GB of RAM.

To ensure a fair and statistically robust comparison, the population size was fixed at for all algorithms. Each algorithm-function combination underwent 30 independent runs to mitigate the stochastic nature of metaheuristics. For statistical validation, the non-parametric Wilcoxon signed-rank test at a significance level of was employed to determine pairwise significance, while the Friedman test was utilized to calculate the overall mean rank across all functions.

It is worth noting that to impartially evaluate the generalization capability of DEHHO, a uniform set of parameter settings (as detailed in Table 3) was maintained across all 38 benchmark functions and engineering problems. We deliberately avoided problem-specific parameter tuning. The consistent superiority of DEHHO across diverse landscapes—despite using fixed default settings—provides intrinsic evidence of the algorithm’s robustness and insensitivity to minor parameter fluctuations³⁶. Furthermore, the dynamic time-varying design of parameters , and enables the algorithm to auto-adapt its search behavior, reducing the reliance on precise initial static value selection.

Table 3.

Key parameters of HHO variants and related algorithms.

Algorithm	Description	Default
BGHHO14	Bernoulli map parameter for chaotic initialization	0.7
SHHO15	Maximum radius of unit hypersphere	1.0
	Minimum radius threshold	1e−3
	Shrinkage rate of perturbation radius	1.5
	Cooperation ratio among leaders	0.25
	Scale factor for Cauchy mutation	1.0
HHSC37	Maximum amplitude	2.0
	Minimum amplitude	0.0
CLHHEO18	Coefficient for equilibrium-based	1.5
IHAOHHO38	Number of elite individuals (used in RH)	5
	Initial σ for RH perturbation	1.0
	Nonlinear exponent for σ	2
SCSO39	Maximum sensitivity range	2.0
JA40	–	–
SCA41	–	–
HBA42	Control the degree of food	6
	Density control constant	6
SSA43	Safety threshold value for danger	0.8
	Proportion of producers	0.2
	Proportion of sparrows perceiving	0.1

In-depth validation on CEC2017: generalization across function types

To comprehensively assess the generalization capability of DEHHO across diverse problem landscapes, this section conducts an extensive evaluation using the CEC 2017 benchmark suite. This suite includes 29 functions (F1–F29, excluding the unstable F2) covering unimodal, multimodal, hybrid, and composition types, representing a wide range of optimization challenges. Experiments were conducted in both 50-dimensional (50D) and 100-dimensional (100D) environments to explicitly test the algorithm’s scalability against the ten state-of-the-art peer algorithms defined in “Experimental protocol”.

Rank stability and dimensional generalization analysis

The comparative performance, summarized through the Friedman Mean Rank in Fig. 2 (50D) and Fig. 3 (100D), reveals a distinct advantage for the proposed method. In the 50-dimensional setting, DEHHO achieves a superior average rank of 2.14, consistently outperforming both advanced HHO variants (e.g., IHAOHHO, CLHHEO) and classic metaheuristics (e.g., SCA, SSA). Remarkably, as the problem complexity increases to 100 dimensions, DEHHO’s average rank further improves to 2.00. This counter-intuitive stability under dimensional expansion suggests that the proposed framework effectively mitigates the “curse of dimensionality”. While comparative algorithms often struggle to navigate the exponentially expanding search space, the Trend-Guided DE mechanism leverages the historical momentum vector to maintain a consistent descent direction, preventing the population from becoming disoriented in high-dimensional voids.

Fig. 2.

Heatmap of algorithm rankings on the CEC2017 benchmark (50D).

Fig. 3.

Heatmap of algorithm rankings on the CEC2017 benchmark (100D).

A closer inspection of performance across specific function landscapes substantiates the efficacy of the proposed coupling mechanisms. On unimodal functions (F1, F3), DEHHO demonstrates sharp convergence precision (as detailed in Tables 4 and 5). This validates that the Momentum-Guided exploitation accelerates the search towards the global optimum once the basin of attraction is identified, thereby avoiding the oscillatory behavior typical of standard HHO. Transitioning to multimodal and hybrid functions (F4–F19), which are characterized by rugged landscapes laden with local optima, DEHHO maintains statistical dominance. This success is attributed to the Gaussian Perturbation during the exploration phase, which ensures sufficient population diversity to cover multiple peaks, working in tandem with Lévy-Annealed jumps to facilitate escape from local traps. Furthermore, on composition functions (F20–F29), which represent the highest level of structural complexity with hierarchical interactions, DEHHO ranks first on the majority of cases (e.g., F22, F23, F28). This confirms the structural synergy of the dual-branch strategy, which dynamically balances exploration pressure and exploitation precision based on the evolutionary state.

Table 4.

Performance comparison of DEHHO and peer algorithms on the CEC2017 benchmark (D50).

		DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	Mean	1.02E + 09	2.19E + 10	9.71E + 09	4.90E + 09	1.61E + 09	3.18E + 10	8.05E + 10	2.62E + 10	1.57E + 11	1.46E + 10	3.78E + 10	1.47E + 10
	Std	2.51E + 08	3.67E + 09	2.85E + 09	1.64E + 09	5.50E + 08	1.01E + 10	8.90E + 09	7.17E + 09	1.57E + 10	2.17E + 09	1.10E + 10	1.43E + 10
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F3	Mean	7.75E + 02	5.27E + 03	2.29E + 03	1.42E + 03	1.08E + 03	6.72E + 03	2.02E + 04	4.00E + 03	3.29E + 04	2.86E + 03	5.71E + 03	8.89E + 03
	Std	1.25E + 02	1.48E + 03	6.21E + 02	3.42E + 02	3.27E + 02	3.48E + 03	4.57E + 03	1.80E + 03	6.61E + 03	4.70E + 02	2.29E + 03	7.56E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F4	Mean	2.90E + 03	3.05E + 04	1.39E + 04	6.79E + 03	3.19E + 03	4.25E + 04	9.83E + 04	3.09E + 04	1.71E + 05	1.70E + 04	4.78E + 04	6.31E + 04
	Std	4.87E + 02	3.60E + 03	2.60E + 03	2.20E + 03	7.77E + 02	1.15E + 04	8.03E + 03	9.93E + 03	1.66E + 04	4.29E + 03	1.05E + 04	1.49E + 04
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F5	Mean	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02
	Std	2.36E−03	2.56E−03	1.60E−03	2.49E−03	1.22E−03	2.23E−03	8.26E−03	1.76E−03	1.50E−02	2.56E−03	2.22E−03	2.10E−03
	=/≈/−		+	−	+	−	+	+	+	+	+	+	+
F6	Mean	6.75E + 04	1.42E + 06	6.57E + 05	2.53E + 05	1.81E + 05	1.73E + 06	3.81E + 06	1.14E + 06	5.32E + 06	7.67E + 05	1.89E + 06	1.54E + 06
	Std	1.72E + 04	2.88E + 05	1.96E + 05	8.11E + 04	6.28E + 04	4.45E + 05	4.99E + 05	3.04E + 05	5.84E + 05	1.23E + 05	4.31E + 05	1.76E + 06
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F7	Mean	1.74E + 03	1.88E + 03	1.80E + 03	1.80E + 03	1.68E + 03	1.79E + 03	1.88E + 03	1.61E + 03	2.50E + 03	1.56E + 03	1.62E + 03	2.27E + 03
	Std	1.26E + 02	1.26E + 02	1.20E + 02	1.73E + 02	1.03E + 02	8.08E + 01	8.60E + 01	1.18E + 02	9.79E + 01	6.45E + 01	1.13E + 02	1.31E + 02
	=/≈/−		+	≈	≈	−	≈	+	−	+	−	−	+
F8	Mean	8.26E + 02	8.35E + 02	8.29E + 02	8.35E + 02	8.23E + 02	8.32E + 02	8.66E + 02	8.33E + 02	9.56E + 02	8.22E + 02	8.35E + 02	8.38E + 02
	Std	5.44E + 00	4.79E + 00	4.73E + 00	7.26E + 00	4.56E + 00	8.59E + 00	1.58E + 01	7.21E + 00	1.67E + 01	5.30E + 00	8.71E + 00	9.47E + 00
	=/≈/−		+	+	+	≈	+	+	+	+	−	+	+
F9	Mean	1.20E + 04	1.20E + 04	1.16E + 04	1.08E + 04	1.05E + 04	1.05E + 04	1.45E + 04	1.09E + 04	1.48E + 04	1.37E + 04	1.17E + 04	1.33E + 04
	Std	1.14E + 03	1.11E + 03	1.09E + 03	1.13E + 03	1.36E + 03	1.14E + 03	8.04E + 02	8.34E + 02	5.04E + 02	5.67E + 02	1.46E + 03	1.99E + 03
	=/≈/−		≈	≈	−	−	−	+	−	+	+	≈	+
F10	Mean	2.16E + 05	7.27E + 07	3.00E + 06	4.22E + 05	2.37E + 05	4.20E + 07	1.46E + 09	1.69E + 08	8.49E + 09	8.50E + 06	3.76E + 08	1.28E + 09
	Std	6.04E + 04	8.30E + 07	5.75E + 06	2.49E + 05	9.91E + 04	8.10E + 07	9.07E + 08	3.75E + 08	2.91E + 09	5.52E + 06	8.03E + 08	5.50E + 09
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F11	Mean	1.58E + 08	9.14E + 08	2.83E + 08	3.29E + 08	9.34E + 07	1.88E + 09	9.00E + 09	1.59E + 09	2.19E + 10	6.27E + 08	2.87E + 09	6.10E + 09
	Std	8.37E + 07	4.40E + 08	1.38E + 08	1.66E + 08	5.79E + 07	2.03E + 09	3.01E + 09	1.21E + 09	3.37E + 09	1.50E + 08	2.34E + 09	6.75E + 09
	=/≈/−		+	+	+	−	+	+	+	+	+	+	+
F12	Mean	6.85E + 07	4.13E + 09	4.35E + 08	1.52E + 08	7.32E + 07	7.16E + 09	2.73E + 10	3.21E + 09	4.87E + 10	1.12E + 09	9.11E + 09	2.60E + 09
	Std	3.09E + 07	2.35E + 09	2.38E + 08	9.36E + 07	3.56E + 07	5.36E + 09	9.50E + 09	2.81E + 09	1.45E + 10	4.80E + 08	6.22E + 09	1.08E + 10
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F13	Mean	8.08E + 05	1.16E + 07	6.99E + 06	8.12E + 06	2.55E + 07	2.95E + 06	1.44E + 07	3.17E + 06	2.30E + 07	2.37E + 06	3.46E + 06	2.63E + 08
	Std	4.06E + 05	7.70E + 06	5.24E + 06	6.18E + 06	3.98E + 07	2.42E + 06	1.26E + 07	2.18E + 06	1.18E + 07	7.90E + 05	1.93E + 06	4.92E + 08
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F14	Mean	1.33E + 07	6.41E + 08	1.60E + 08	5.42E + 07	6.34E + 08	1.51E + 09	6.20E + 09	7.10E + 08	9.83E + 09	3.16E + 08	1.40E + 09	8.95E + 09
	Std	6.13E + 06	5.81E + 08	1.31E + 08	2.79E + 07	1.67E + 09	1.88E + 09	2.68E + 09	7.18E + 08	2.95E + 09	1.93E + 08	1.47E + 09	7.11E + 09
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F15	Mean	2.49E + 05	5.14E + 07	1.31E + 06	2.41E + 05	3.53E + 07	2.42E + 08	3.31E + 08	1.30E + 08	4.28E + 09	1.76E + 06	8.12E + 07	5.38E + 09
	Std	2.01E + 05	1.10E + 08	2.80E + 06	2.11E + 05	1.79E + 08	5.71E + 08	3.13E + 08	2.13E + 08	1.23E + 09	1.38E + 06	1.60E + 08	1.26E + 10
	=/≈/−		+	+	≈	≈	+	+	+	+	+	+	+
F16	Mean	5.67E + 04	2.24E + 08	1.08E + 05	7.28E + 04	5.30E + 04	3.12E + 07	1.36E + 10	5.01E + 04	7.63E + 11	1.69E + 05	4.25E + 04	1.26E + 12
	Std	1.81E + 04	1.10E + 09	1.18E + 05	2.19E + 04	2.56E + 04	1.66E + 08	3.27E + 10	1.30E + 04	1.67E + 12	2.87E + 05	1.24E + 04	3.24E + 12
	=/≈/−		+	+	+	≈	+	+	≈	+	+	−	+
F17	Mean	1.83E + 06	2.11E + 07	1.16E + 07	1.71E + 07	4.60E + 07	6.19E + 06	1.87E + 07	1.04E + 07	1.17E + 08	4.69E + 06	7.45E + 06	1.78E + 08
	Std	1.43E + 06	1.27E + 07	8.36E + 06	1.14E + 07	1.09E + 08	5.36E + 06	1.40E + 07	8.45E + 06	9.89E + 07	2.69E + 06	8.10E + 06	4.05E + 08
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F18	Mean	6.25E + 07	9.16E + 10	2.33E + 10	2.17E + 10	4.46E + 09	1.08E + 12	1.53E + 12	5.94E + 12	1.16E + 15	1.77E + 10	3.77E + 10	1.12E + 10
	Std	1.02E + 08	6.22E + 10	2.14E + 10	1.86E + 10	7.53E + 09	4.44E + 12	3.49E + 12	3.10E + 13	1.03E + 15	6.48E + 09	1.35E + 11	2.02E + 10
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F19	Mean	6.59E + 03	1.20E + 04	9.52E + 03	7.88E + 03	7.72E + 03	8.50E + 03	7.78E + 03	6.51E + 03	1.61E + 04	3.88E + 03	7.57E + 03	1.44E + 04
	Std	1.39E + 03	1.43E + 03	1.59E + 03	2.22E + 03	1.48E + 03	1.62E + 03	9.88E + 02	1.74E + 03	1.91E + 03	3.54E + 02	1.95E + 03	2.87E + 03
	=/≈/−		+	+	+	+	+	+	≈	+		+	+
F20	Mean	4.51E + 03	2.85E + 04	1.28E + 04	6.82E + 03	5.06E + 03	3.59E + 04	5.02E + 04	2.12E + 04	1.36E + 05	1.26E + 04	3.75E + 04	6.51E + 04
	Std	9.32E + 02	1.37E + 04	4.74E + 03	1.61E + 03	1.26E + 03	1.77E + 04	3.17E + 04	1.35E + 04	3.26E + 04	5.21E + 03	1.54E + 04	2.23E + 04
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F21	Mean	7.49E + 03	9.28E + 03	6.91E + 03	6.94E + 03	6.33E + 03	8.36E + 03	5.06E + 03	3.40E + 03	3.75E + 03	2.86E + 03	5.06E + 03	1.07E + 04
	Std	3.23E + 03	1.38E + 03	1.90E + 03	3.06E + 03	1.56E + 03	1.48E + 03	9.47E + 02	1.34E + 03	2.76E + 02	8.41E + 01	1.99E + 03	3.19E + 03
	=/≈/−		≈	≈	≈	≈	≈	−	−	−	−	−	+
F22	Mean	6.53E + 03	4.54E + 04	2.70E + 04	1.35E + 04	1.13E + 04	5.61E + 04	8.37E + 04	4.76E + 04	6.80E + 04	3.00E + 04	5.53E + 04	7.50E + 04
	Std	1.47E + 03	1.05E + 04	7.47E + 03	4.16E + 03	8.25E + 03	1.12E + 04	1.37E + 04	1.25E + 04	1.07E + 04	3.41E + 03	1.05E + 04	9.29E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F23	Mean	5.08E + 03	2.80E + 04	2.06E + 04	7.98E + 03	8.04E + 03	3.82E + 04	5.09E + 04	3.27E + 04	4.19E + 04	1.96E + 04	3.51E + 04	5.17E + 04
	Std	5.14E + 02	6.24E + 03	7.01E + 03	2.23E + 03	2.66E + 03	9.87E + 03	9.58E + 03	7.17E + 03	1.41E + 04	4.09E + 03	7.99E + 03	8.64E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F24	Mean	4.17E + 03	7.89E + 03	5.99E + 03	5.09E + 03	5.14E + 03	7.11E + 03	1.53E + 04	5.94E + 03	1.98E + 04	6.02E + 03	7.30E + 03	1.21E + 04
	Std	2.97E + 02	1.11E + 03	8.44E + 02	6.18E + 02	1.10E + 03	1.39E + 03	3.27E + 03	8.68E + 02	3.87E + 03	3.71E + 02	1.79E + 03	5.36E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F25	Mean	4.94E + 03	1.69E + 04	6.92E + 03	1.05E + 04	5.60E + 03	5.51E + 03	1.06E + 04	5.45E + 03	7.13E + 03	7.37E + 03	7.98E + 03	5.60E + 04
	Std	5.48E + 02	7.50E + 03	1.62E + 03	5.44E + 03	6.05E + 03	8.10E + 02	6.94E + 03	7.41E + 02	6.55E + 02	4.94E + 02	2.80E + 03	2.54E + 04
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F26	Mean	3.94E + 03	5.84E + 03	4.71E + 03	4.18E + 03	4.69E + 03	5.03E + 03	4.09E + 03	3.85E + 03	3.70E + 03	4.14E + 03	4.01E + 03	6.33E + 03
	Std	3.04E + 02	8.24E + 02	4.83E + 02	4.22E + 02	1.35E + 03	8.95E + 02	2.67E + 02	2.18E + 02	9.59E + 01	1.58E + 02	3.61E + 02	8.46E + 02
	=/≈/−		+	+	≈	≈	+	+	≈	−	+	≈	+
F27	Mean	3.20E + 03	3.80E + 03	3.45E + 03	3.33E + 03	3.26E + 03	3.80E + 03	5.75E + 03	3.53E + 03	5.07E + 03	3.46E + 03	3.76E + 03	5.29E + 03
	Std	4.90E + 01	1.90E + 02	6.78E + 01	1.53E + 02	7.71E + 01	2.89E + 02	8.65E + 02	2.02E + 02	2.53E + 02	4.21E + 01	3.36E + 02	2.10E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F28	Mean	9.79E + 07	1.81E + 10	2.66E + 09	2.56E + 09	3.26E + 09	1.04E + 10	2.31E + 11	1.43E + 10	1.07E + 13	1.31E + 09	7.24E + 08	3.03E + 12
	Std	1.20E + 08	3.89E + 10	2.84E + 09	3.05E + 09	1.15E + 10	5.11E + 10	6.01E + 11	6.62E + 10	1.08E + 13	6.92E + 08	8.41E + 08	8.58E + 12
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F29	Mean	8.54E + 08	3.65E + 10	1.41E + 10	8.55E + 09	2.17E + 09	4.74E + 10	4.48E + 11	3.62E + 11	3.69E + 14	6.92E + 09	2.73E + 10	5.68E + 13
	Std	5.83E + 08	2.27E + 10	1.22E + 10	5.30E + 09	2.56E + 09	1.64E + 11	7.67E + 11	1.90E + 12	4.81E + 14	2.54E + 09	1.30E + 11	1.94E + 14
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
	=/≈/−	/	26/2/0	24/3/1	23/4/1	16/8/4	25/2/1	27/0/1	22/3/3	26/0/2	24/0/4	23/3/2	28/0/0

Table 5.

Performance comparison of DEHHO and peer algorithms on the CEC2017 benchmark (D100).

		DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	Mean	2.00E + 10	1.07E + 11	7.39E + 10	5.47E + 10	3.36E + 10	1.26E + 11	2.35E + 11	1.05E + 11	4.75E + 11	1.04E + 11	1.28E + 11	9.79E + 10
	Std	3.55E + 09	8.04E + 09	9.97E + 09	9.46E + 09	6.50E + 09	1.81E + 10	1.99E + 10	1.90E + 10	2.32E + 10	6.95E + 09	1.63E + 10	1.20E + 10
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F3	Mean	3.40E + 03	2.67E + 04	1.27E + 04	8.36E + 03	5.96E + 03	2.04E + 04	8.02E + 04	1.34E + 04	1.74E + 05	1.80E + 04	1.93E + 04	3.82E + 04
	Std	6.37E + 02	4.04E + 03	2.38E + 03	1.82E + 03	4.04E + 03	5.24E + 03	1.54E + 04	2.96E + 03	3.15E + 04	2.51E + 03	5.61E + 03	2.72E + 04
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F4	Mean	2.56E + 04	1.36E + 05	8.78E + 04	5.55E + 04	3.62E + 04	1.40E + 05	2.93E + 05	1.10E + 05	4.84E + 05	1.06E + 05	1.37E + 05	2.30E + 05
	Std	3.13E + 03	1.20E + 04	9.37E + 03	9.47E + 03	6.27E + 03	2.60E + 04	2.00E + 04	1.45E + 04	2.37E + 04	7.55E + 03	2.15E + 04	4.47E + 04
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F5	Mean	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02
	Std	2.27E−03	4.73E−03	1.72E−03	3.07E−03	2.17E−03	3.02E−03	7.71E−03	3.41E−03	9.65E−03	4.81E−03	2.81E−03	9.46E−03
	=/≈/−		+	≈	≈	≈	≈	+	≈	+	+	+	+
F6	Mean	7.97E + 05	4.42E + 06	3.10E + 06	1.94E + 06	1.30E + 06	4.61E + 06	9.17E + 06	3.58E + 06	1.47E + 07	3.89E + 06	4.71E + 06	4.06E + 06
	Std	1.41E + 05	4.77E + 05	4.14E + 05	3.73E + 05	2.06E + 05	5.86E + 05	6.83E + 05	4.59E + 05	9.29E + 05	3.30E + 05	6.44E + 05	1.02E + 06
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F7	Mean	3.34E + 03	3.44E + 03	3.36E + 03	3.59E + 03	3.18E + 03	3.32E + 03	3.55E + 03	3.14E + 03	5.00E + 03	3.07E + 03	3.04E + 03	4.05E + 03
	Std	1.85E + 02	1.28E + 02	2.24E + 02	1.85E + 02	1.40E + 02	2.06E + 02	1.73E + 02	1.45E + 02	1.56E + 02	1.22E + 02	1.96E + 02	1.03E + 02
	=/≈/−		+	+	≈	≈	−	+	−	+	−	−	+
F8	Mean	8.26E + 02	8.35E + 02	8.29E + 02	8.35E + 02	8.23E + 02	8.32E + 02	8.66E + 02	8.33E + 02	9.56E + 02	8.22E + 02	8.35E + 02	8.38E + 02
	Std	5.44E + 00	4.79E + 00	4.73E + 00	7.26E + 00	4.56E + 00	8.59E + 00	1.58E + 01	7.21E + 00	1.67E + 01	5.30E + 00	8.71E + 00	9.47E + 00
	=/≈/−		+	+	+	−	+	+	+	+	−	+	+
F9	Mean	1.20E + 04	1.20E + 04	1.16E + 04	1.08E + 04	1.05E + 04	1.05E + 04	1.45E + 04	1.09E + 04	1.48E + 04	1.37E + 04	1.17E + 04	1.33E + 04
	Std	1.14E + 03	1.11E + 03	1.09E + 03	1.13E + 03	1.36E + 03	1.14E + 03	8.04E + 02	8.34E + 02	5.04E + 02	5.67E + 02	1.46E + 03	1.99E + 03
	=/≈/−		≈	≈	−	−	−	+	+	+	+	≈	+
F10	Mean	2.16E + 05	7.27E + 07	3.00E + 06	4.22E + 05	2.37E + 05	4.20E + 07	1.46E + 09	1.69E + 08	8.49E + 09	8.50E + 06	3.76E + 08	1.28E + 09
	Std	6.04E + 04	8.30E + 07	5.75E + 06	2.49E + 05	9.91E + 04	8.10E + 07	9.07E + 08	3.75E + 08	2.91E + 09	5.52E + 06	8.03E + 08	5.50E + 09
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F11	Mean	1.58E + 08	9.14E + 08	2.83E + 08	3.29E + 08	9.34E + 07	1.88E + 09	9.00E + 09	1.59E + 09	2.19E + 10	6.27E + 08	2.87E + 09	6.10E + 09
	Std	8.37E + 07	4.40E + 08	1.38E + 08	1.66E + 08	5.79E + 07	2.03E + 09	3.01E + 09	1.21E + 09	3.37E + 09	1.50E + 08	2.34E + 09	6.75E + 09
	=/≈/−		+	+	+	−	+	+	+	+	+	+	+
F12	Mean	6.85E + 07	4.13E + 09	4.35E + 08	1.52E + 08	7.32E + 07	7.16E + 09	2.73E + 10	3.21E + 09	4.87E + 10	1.12E + 09	9.11E + 09	2.60E + 09
	Std	3.09E + 07	2.35E + 09	2.38E + 08	9.36E + 07	3.56E + 07	5.36E + 09	9.50E + 09	2.81E + 09	1.45E + 10	4.80E + 08	6.22E + 09	1.08E + 10
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F13	Mean	8.08E + 05	1.16E + 07	6.99E + 06	8.12E + 06	2.55E + 07	2.95E + 06	1.44E + 07	3.17E + 06	2.30E + 07	2.37E + 06	3.46E + 06	2.63E + 08
	Std	4.06E + 05	7.70E + 06	5.24E + 06	6.18E + 06	3.98E + 07	2.42E + 06	1.26E + 07	2.18E + 06	1.18E + 07	7.90E + 05	1.93E + 06	4.92E + 08
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F14	Mean	1.33E + 07	6.41E + 08	1.60E + 08	5.42E + 07	6.34E + 08	1.51E + 09	6.20E + 09	7.10E + 08	9.83E + 09	3.16E + 08	1.40E + 09	8.95E + 09
	Std	6.13E + 06	5.81E + 08	1.31E + 08	2.79E + 07	1.67E + 09	1.88E + 09	2.68E + 09	7.18E + 08	2.95E + 09	1.93E + 08	1.47E + 09	7.11E + 09
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F15	Mean	2.49E + 05	5.14E + 07	1.31E + 06	2.41E + 05	3.53E + 07	2.42E + 08	3.31E + 08	1.30E + 08	4.28E + 09	1.76E + 06	8.12E + 07	5.38E + 09
	Std	2.01E + 05	1.10E + 08	2.80E + 06	2.11E + 05	1.79E + 08	5.71E + 08	3.13E + 08	2.13E + 08	1.23E + 09	1.38E + 06	1.60E + 08	1.26E + 10
	=/≈/−		+	+	≈	≈	+	+	+	+	+	+	≈
F16	Mean	5.67E + 04	2.24E + 08	1.08E + 05	7.28E + 04	5.30E + 04	3.12E + 07	1.36E + 10	5.01E + 04	7.63E + 11	1.69E + 05	4.25E + 04	1.26E + 12
	Std	1.81E + 04	1.10E + 09	1.18E + 05	2.19E + 04	2.56E + 04	1.66E + 08	3.27E + 10	1.30E + 04	1.67E + 12	2.87E + 05	1.24E + 04	3.24E + 12
	=/≈/−		+	+	+	≈	+	+	≈	+	+	−	+
F17	Mean	1.83E + 06	2.11E + 07	1.16E + 07	1.71E + 07	4.60E + 07	6.19E + 06	1.87E + 07	1.04E + 07	1.17E + 08	4.69E + 06	7.45E + 06	1.78E + 08
	Std	1.43E + 06	1.27E + 07	8.36E + 06	1.14E + 07	1.09E + 08	5.36E + 06	1.40E + 07	8.45E + 06	9.89E + 07	2.69E + 06	8.10E + 06	4.05E + 08
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F18	Mean	6.25E + 07	9.16E + 10	2.33E + 10	2.17E + 10	4.46E + 09	1.08E + 12	1.53E + 12	5.94E + 12	1.16E + 15	1.77E + 10	3.77E + 10	1.12E + 10
	Std	1.02E + 08	6.22E + 10	2.14E + 10	1.86E + 10	7.53E + 09	4.44E + 12	3.49E + 12	3.10E + 13	1.03E + 15	6.48E + 09	1.35E + 11	2.02E + 10
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F19	Mean	6.59E + 03	1.20E + 04	9.52E + 03	7.88E + 03	7.72E + 03	8.50E + 03	7.78E + 03	6.51E + 03	1.61E + 04	3.88E + 03	7.57E + 03	1.44E + 04
	Std	1.39E + 03	1.43E + 03	1.59E + 03	2.22E + 03	1.48E + 03	1.62E + 03	9.88E + 02	1.74E + 03	1.91E + 03	3.54E + 02	1.95E + 03	2.87E + 03
	=/≈/−		+	+	+	+	+	+	≈	+	−	+	+
F20	Mean	4.51E + 03	2.85E + 04	1.28E + 04	6.82E + 03	5.06E + 03	3.59E + 04	5.02E + 04	2.12E + 04	1.36E + 05	1.26E + 04	3.75E + 04	6.51E + 04
	Std	9.32E + 02	1.37E + 04	4.74E + 03	1.61E + 03	1.26E + 03	1.77E + 04	3.17E + 04	1.35E + 04	3.26E + 04	5.21E + 03	1.54E + 04	2.23E + 04
	=/≈/−		+	+	+	≈	+	+	+	+	+	+	+
F21	Mean	7.49E + 03	9.28E + 03	6.91E + 03	6.94E + 03	6.33E + 03	8.36E + 03	5.06E + 03	3.40E + 03	3.75E + 03	2.86E + 03	5.06E + 03	1.07E + 04
	Std	3.23E + 03	1.38E + 03	1.90E + 03	3.06E + 03	1.56E + 03	1.48E + 03	9.47E + 02	1.34E + 03	2.76E + 02	8.41E + 01	1.99E + 03	3.19E + 03
	=/≈/−		≈	≈	≈	≈	≈	−	−	−	−	−	+
F22	Mean	6.53E + 03	4.54E + 04	2.70E + 04	1.35E + 04	1.13E + 04	5.61E + 04	8.37E + 04	4.76E + 04	6.80E + 04	3.00E + 04	5.53E + 04	7.50E + 04
	Std	1.47E + 03	1.05E + 04	7.47E + 03	4.16E + 03	8.25E + 03	1.12E + 04	1.37E + 04	1.25E + 04	1.07E + 04	3.41E + 03	1.05E + 04	9.29E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F23	Mean	5.08E + 03	2.80E + 04	2.06E + 04	7.98E + 03	8.04E + 03	3.82E + 04	5.09E + 04	3.27E + 04	4.19E + 04	1.96E + 04	3.51E + 04	5.17E + 04
	Std	5.14E + 02	6.24E + 03	7.01E + 03	2.23E + 03	2.66E + 03	9.87E + 03	9.58E + 03	7.17E + 03	1.41E + 04	4.09E + 03	7.99E + 03	8.64E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F24	Mean	4.17E + 03	7.89E + 03	5.99E + 03	5.09E + 03	5.14E + 03	7.11E + 03	1.53E + 04	5.94E + 03	1.98E + 04	6.02E + 03	7.30E + 03	1.21E + 04
	Std	2.97E + 02	1.11E + 03	8.44E + 02	6.18E + 02	1.10E + 03	1.39E + 03	3.27E + 03	8.68E + 02	3.87E + 03	3.71E + 02	1.79E + 03	5.36E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F25	Mean	4.94E + 03	1.69E + 04	6.92E + 03	1.05E + 04	5.60E + 03	5.51E + 03	1.06E + 04	5.45E + 03	7.13E + 03	7.37E + 03	7.98E + 03	5.60E + 04
	Std	5.48E + 02	7.50E + 03	1.62E + 03	5.44E + 03	6.05E + 03	8.10E + 02	6.94E + 03	7.41E + 02	6.55E + 02	4.94E + 02	2.80E + 03	2.54E + 04
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F26	Mean	3.94E + 03	5.84E + 03	4.71E + 03	4.18E + 03	4.69E + 03	5.03E + 03	4.09E + 03	3.85E + 03	3.70E + 03	4.14E + 03	4.01E + 03	6.33E + 03
	Std	3.04E + 02	8.24E + 02	4.83E + 02	4.22E + 02	1.35E + 03	8.95E + 02	2.67E + 02	2.18E + 02	9.59E + 01	1.58E + 02	3.61E + 02	8.46E + 02
	=/≈/−		+	+	+	+	+	+	≈	−	+	≈	+
F27	Mean	3.20E + 03	3.80E + 03	3.45E + 03	3.33E + 03	3.26E + 03	3.80E + 03	5.75E + 03	3.53E + 03	5.07E + 03	3.46E + 03	3.76E + 03	5.29E + 03
	std	4.90E + 01	1.90E + 02	6.78E + 01	1.53E + 02	7.71E + 01	2.89E + 02	8.65E + 02	2.02E + 02	2.53E + 02	4.21E + 01	3.36E + 02	2.10E + 03
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F28	Mean	9.79E + 07	1.81E + 10	2.66E + 09	2.56E + 09	3.26E + 09	1.04E + 10	2.31E + 11	1.43E + 10	1.07E + 13	1.31E + 09	7.24E + 08	3.03E + 12
	Std	1.20E + 08	3.89E + 10	2.84E + 09	3.05E + 09	1.15E + 10	5.11E + 10	6.01E + 11	6.62E + 10	1.08E + 13	6.92E + 08	8.41E + 08	8.58E + 12
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
F29	Mean	8.54E + 08	3.65E + 10	1.41E + 10	8.55E + 09	2.17E + 09	4.74E + 10	4.48E + 11	3.62E + 11	3.69E + 14	6.92E + 09	2.73E + 10	5.68E + 13
	Std	5.83E + 08	2.27E + 10	1.22E + 10	5.30E + 09	2.56E + 09	1.64E + 11	7.67E + 11	1.90E + 12	4.81E + 14	2.54E + 09	1.30E + 11	1.94E + 14
	=/≈/−		+	+	+	+	+	+	+	+	+	+	+
	=/≈/−	/	26/2/0	25/3/0	23/4/1	17/8/3	24/2/2	27/0/1	22/4/2	26/0/2	24/0/4	23/2/3	27/1/0

Statistical significance is further corroborated by the pairwise Wilcoxon signed-rank test (), as detailed in the bottom rows of Tables 6 and 7. In the 50D case, DEHHO achieves significant wins against the vast majority of competitors (e.g., 28/0/0 against SSA). Crucially, this dominance is sustained in the 100D case without significant performance degradation. Collectively, the consistent low rankings, high convergence accuracy, and robust statistical evidence across both dimensions validate DEHHO as a reliable solver for high-dimensional, structurally complex optimization tasks.

Table 6.

Function-wise Wilcoxon test p-values between DEHHO and peer algorithms on the CEC2017 benchmarks(D50).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1		3.02E−11	3.02E−11	3.02E−11	6.74E−06	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F3		3.02E−11	3.02E−11	2.15E−10	1.16E−07	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F4		3.02E−11	3.02E−11	1.96E−10	1.91E−01	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F5		5.57E−03	5.49E−01	1.70E−02	9.23E−01	5.30E−01	3.34E−11	1.70E−02	3.02E−11	5.00E−09	1.17E−03	5.94E−02
F6		3.02E−11	3.02E−11	3.02E−11	1.96E−10	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F7		3.37E−04	9.05E−02	1.26E−01	4.36E−02	1.91E−01	2.77E−05	3.77E−04	3.02E−11	3.52E−07	3.77E−04	3.02E−11
F8		3.81E−07	1.50E−02	7.22E−06	5.94E−02	3.99E−04	4.98E−11	7.20E−05	3.02E−11	4.03E−03	2.60E−05	7.60E−07
F9		8.42E−01	1.96E−01	3.77E−04	9.51E−06	9.51E−06	5.57E−10	9.52E−04	7.39E−11	3.65E−08	2.84E−01	9.52E−04
F10		3.02E−11	6.70E−11	3.01E−07	3.33E−01	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	4.08E−11
F11		7.39E−11	3.37E−04	8.29E−06	1.52E−03	4.11E−07	3.02E−11	4.62E−10	3.02E−11	3.34E−11	4.98E−11	1.47E−07
F12		3.02E−11	4.62E−10	4.74E−06	8.42E−01	3.02E−11	3.02E−11	3.69E−11	3.02E−11	3.02E−11	3.02E−11	3.47E−10
F13		3.02E−11	1.09E−10	3.69E−11	8.48E−09	2.20E−07	1.21E−10	4.44E−07	3.02E−11	3.82E−10	2.44E−09	5.49E−11
F14		3.02E−11	6.07E−11	1.46E−10	3.03E−02	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	7.39E−11
F15		1.78E−10	4.23E−03	9.47E−01	2.01E−01	1.19E−06	3.02E−11	2.77E−05	3.02E−11	1.33E−10	6.53E−07	7.24E−02
F16		2.39E−08	2.49E−06	3.50E−03	2.23E−01	9.88E−03	3.02E−11	1.19E−01	3.02E−11	6.36E−05	8.12E−04	1.47E−07
F17		2.37E−10	4.57E−09	9.92E−11	5.97E−09	4.94E−05	1.55E−09	1.56E−08	3.02E−11	1.61E−06	1.09E−05	1.10E−08
F18		3.02E−11	7.39E−11	3.02E−11	4.12E−06	6.07E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	4.08E−11	4.62E−10
F19		5.49E−11	4.69E−08	1.91E−02	4.23E−03	5.27E−05	6.20E−04	4.73E−01	3.02E−11	4.98E−11	4.68E−02	4.98E−11
F20		3.02E−11	1.78E−10	1.11E−06	1.76E−01	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F21		8.24E−02	2.12E−01	4.12E−01	7.48E−02	7.96E−01	2.92E−02	3.83E−06	1.77E−03	5.00E−09	2.24E−02	1.60E−03
F22		3.02E−11	3.02E−11	1.61E−10	1.60E−03	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F23		3.02E−11	3.02E−11	1.31E−08	1.25E−07	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F24		3.02E−11	5.49E−11	3.35E−08	2.00E−06	4.98E−11	3.02E−11	2.37E−10	3.02E−11	3.02E−11	3.69E−11	3.02E−11
F25		3.02E−11	1.41E−09	3.16E−10	4.71E−04	7.30E−04	5.08E−03	1.95E−03	8.15E−11	3.69E−11	2.03E−09	3.02E−11
F26		4.50E−11	3.08E−08	1.76E−02	5.75E−02	2.39E−08	4.21E−02	2.77E−01	1.41E−04	9.03E−04	5.49E−01	3.02E−11
F27		3.02E−11	3.02E−11	8.10E−10	8.12E−04	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11	3.02E−11
F28		3.02E−11	6.07E−11	6.07E−11	1.52E−03	2.68E−06	3.02E−11	1.96E−10	3.02E−11	4.50E−11	1.31E−08	3.02E−11
F29		3.02E−11	1.21E−10	1.61E−10	5.57E−03	1.87E−07	3.02E−11	4.20E−10	3.02E−11	3.69E−11	3.83E−06	3.69E−11

Table 7.

Function-wise Wilcoxon test p-values between DEHHO and peer algorithms on the CEC2017 benchmarks (D100).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1		3.02E-11	3.02E-11	3.02E-11	2.15E-10	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F3		3.02E-11	3.02E-11	3.02E-11	1.17E-09	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F4		3.02E-11	3.02E-11	3.02E-11	2.60E-08	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F5		4.86E-03	9.71E-01	1.58E-01	7.73E-01	7.24E-02	3.02E-11	4.83E-01	3.02E-11	3.02E-11	7.66E-05	3.67E-03
F6		3.02E-11	3.02E-11	3.69E-11	1.21E-10	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F7		1.50E-02	8.29E-06	6.84E-01	6.00E-01	3.02E-11	3.77E-04	3.02E-11	4.80E-07	2.00E-06	6.12E-10	3.34E-11
F8		3.81E-07	1.50E-02	7.22E-06	5.94E-02	3.99E-04	4.98E-11	7.20E-05	3.02E-11	4.03E-03	2.60E-05	7.60E-07
F9		8.42E-01	1.96E-01	3.77E-04	9.51E-06	9.51E-06	5.57E-10	9.52E-04	7.39E-11	3.65E-08	2.84E-01	9.52E-04
F10		3.02E-11	6.70E-11	3.01E-07	3.33E-01	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	4.08E-11
F11		7.39E-11	3.37E-04	8.29E-06	1.52E-03	4.11E-07	3.02E-11	4.62E-10	3.02E-11	3.34E-11	4.98E-11	1.47E-07
F12		3.02E-11	4.62E-10	4.74E-06	8.42E-01	3.02E-11	3.02E-11	3.69E-11	3.02E-11	3.02E-11	3.02E-11	3.47E-10
F13		3.02E-11	1.09E-10	3.69E-11	8.48E-09	2.20E-07	1.21E-10	4.44E-07	3.02E-11	3.82E-10	2.44E-09	5.49E-11
F14		3.02E-11	6.07E-11	1.46E-10	3.03E-02	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	7.39E-11
F15		1.78E-10	4.23E-03	9.47E-01	2.01E-01	1.19E-06	3.02E-11	2.77E-05	3.02E-11	1.33E-10	6.53E-07	7.24E-02
F16		2.39E-08	2.49E-06	3.50E-03	2.23E-01	9.88E-03	3.02E-11	1.19E-01	3.02E-11	6.36E-05	8.12E-04	1.47E-07
F17		2.37E-10	4.57E-09	9.92E-11	5.97E-09	4.94E-05	1.55E-09	1.56E-08	3.02E-11	1.61E-06	1.09E-05	1.10E-08
F18		3.02E-11	7.39E-11	3.02E-11	4.12E-06	6.07E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	4.08E-11	4.62E-10
F19		5.49E-11	4.69E-08	1.91E-02	4.23E-03	5.27E-05	6.20E-04	4.73E-01	3.02E-11	4.98E-11	4.68E-02	4.98E-11
F20		3.02E-11	1.78E-10	1.11E-06	1.76E-01	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F21		8.24E-02	2.12E-01	4.12E-01	7.48E-02	7.96E-01	2.92E-02	3.83E-06	1.77E-03	5.00E-09	2.24E-02	1.60E-03
F22		3.02E-11	3.02E-11	1.61E-10	1.60E-03	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F23		3.02E-11	3.02E-11	1.31E-08	1.25E-07	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F24		3.02E-11	5.49E-11	3.35E-08	2.00E-06	4.98E-11	3.02E-11	2.37E-10	3.02E-11	3.02E-11	3.69E-11	3.02E-11
F25		3.02E-11	1.41E-09	3.16E-10	4.71E-04	7.30E-04	5.08E-03	1.95E-03	8.15E-11	3.69E-11	2.03E-09	3.02E-11
F26		4.50E-11	3.08E-08	1.76E-02	5.75E-02	2.39E-08	4.21E-02	2.77E-01	1.41E-04	9.03E-04	5.49E-01	3.02E-11
F27		3.02E-11	3.02E-11	8.10E-10	8.12E-04	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F28		3.02E-11	6.07E-11	6.07E-11	1.52E-03	2.68E-06	3.02E-11	1.96E-10	3.02E-11	4.50E-11	1.31E-08	3.02E-11
F29		3.02E-11	1.21E-10	1.61E-10	5.57E-03	1.87E-07	3.02E-11	4.20E-10	3.02E-11	3.69E-11	3.83E-06	3.69E-11

Convergence trend analysis on CEC2017

To investigate the evolutionary dynamics and temporal behavior of DEHHO, this subsection analyzes the convergence trajectories on six representative functions selected from the CEC2017 suite: F1 (unimodal), F6 (multimodal), F12 and F14 (hybrid), and F22 and F27 (composition). These functions were chosen to span the full spectrum of landscape complexities. The comparative convergence curves for DEHHO and the ten peer algorithms (5 HHO variants and 5 metaheuristics) are illustrated in Fig. 4 (50D) and Fig. 5 (100D).

Fig. 4.

Convergence curves of DEHHO and peer algorithms on CEC2017 functions (D = 50). (a) CEC2017F1D50, (b) CEC2017F6D50, (c) CEC2017F12D50, (d) CEC2017F14D50, (e) CEC2017F22D50, (f) CEC2017F27D50.

Fig. 5.

Convergence curves of DEHHO and peer algorithms on CEC2017 functions (D = 100). (a) CEC2017F1D100, (b) CEC2017F6D100, (c) CEC2017F12D100, (d) CEC2017F14D100, (e) CEC2017F22D100, (f) CEC2017F27D100.

The convergence profile on unimodal functions (e.g., F1) reveals a distinct “L-shaped” trajectory for DEHHO, characterized by a rapid initial descent. This behavior empirically validates the contribution of the Momentum-Guided Exploitation mechanism (“Trend-guided differential evolution (TG-DE)”). Unlike standard HHO, which may oscillate around the optimum, the momentum vector accumulates directional information, accelerating the population towards the global minimum once the basin of attraction is identified.

In contrast, on multimodal functions (e.g., F6), which are riddled with widespread local optima, DEHHO exhibits a smooth and continuous descent curve. While comparative algorithms (e.g., SCA, SSA) often display “staircase-like” stagnation—indicating entrapment in local peaks followed by rare escapes—DEHHO maintains a steady improvement rate. This robustness is attributed to the Gaussian Perturbation in the exploration phase, which sustains population diversity, working in tandem with the Lévy-Annealed jumps to trigger non-local escapes when the search slows down.

Scalability on Complex Landscapes: The advantages of the proposed framework become most pronounced on structurally complex Hybrid (F12, F14) and Composition (F22, F27) functions, particularly under the 100-dimensional setting (Fig. 5). On these non-separable landscapes, DEHHO consistently achieves lower objective values by orders of magnitude compared to competitors. The curves demonstrate that while other algorithms suffer from “late-stage lethargy” (convergence cessation), DEHHO sustains active search pressure throughout the optimization process. This confirms that the Dual-Branch Strategy effectively decouples exploration and exploitation, allowing the Trend-Guided DE operator to navigate twisted valleys where the original HHO’s besiege logic typically fails.

In summary, the convergence analysis provides visual and empirical evidence that DEHHO not only locates superior solutions but does so through a more stable and efficient evolutionary trajectory, effectively mitigating the stagnation risks associated with high-dimensional optimization.

Computational cost analysis

Beyond solution quality, computational efficiency remains a critical metric for evaluating the practical viability of metaheuristic algorithms. To assess this, the average runtime (in seconds) for all algorithms across the 28 CEC 2017 functions was recorded under both 50-dimensional and 100-dimensional settings, as detailed in Tables 8 and 9.

Table 8.

Average runtime comparison of DEHHO and peer algorithms on the CEC2017benchmarks (D50).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	7.59E-01	5.83E-01	7.37E-01	3.40E-01	5.61E-01	7.68E-01	7.77E-01	7.55E + 00	2.98E-01	3.06E + 00	6.18E-01	1.13E + 00
F3	2.29E + 00	1.77E + 00	1.84E + 00	1.00E + 00	1.73E + 00	2.24E + 00	2.43E + 00	2.09E + 01	8.71E-01	8.10E + 00	1.73E + 00	2.97E + 00
F4	2.10E + 00	1.65E + 00	1.69E + 00	9.33E-01	1.66E + 00	2.12E + 00	2.26E + 00	2.07E + 01	8.12E-01	8.09E + 00	1.68E + 00	2.82E + 00
F5	4.76E + 00	3.89E + 00	1.08E + 01	5.12E + 00	1.01E + 01	1.19E + 01	1.39E + 01	4.73E + 01	6.25E + 00	2.05E + 01	6.67E + 00	1.31E + 01
F6	1.10E + 00	8.74E-01	8.95E-01	4.92E-01	8.55E-01	1.08E + 00	1.18E + 00	7.65E + 00	4.31E-01	3.37E + 00	7.70E-01	1.37E + 00
F7	1.69E + 00	1.34E + 00	1.30E + 00	6.90E-01	1.25E + 00	1.49E + 00	1.74E + 00	7.64E + 00	6.49E-01	3.53E + 00	1.01E + 00	1.85E + 00
F8	1.06E + 00	8.43E-01	8.68E-01	5.08E-01	8.39E-01	1.04E + 00	1.15E + 00	7.42E + 00	4.23E-01	3.25E + 00	7.78E-01	1.38E + 00
F9	1.59E + 00	1.26E + 00	1.26E + 00	6.92E-01	1.21E + 00	1.46E + 00	1.84E + 00	7.64E + 00	6.36E-01	3.47E + 00	9.79E-01	1.74E + 00
F10	1.28E + 00	1.05E + 00	1.06E + 00	5.69E-01	1.00E + 00	1.22E + 00	1.35E + 00	7.40E + 00	5.07E-01	3.30E + 00	8.19E-01	1.49E + 00
F11	1.70E + 00	1.47E + 00	1.48E + 00	8.11E-01	1.43E + 00	1.71E + 00	1.99E + 00	8.03E + 00	7.59E-01	3.71E + 00	1.11E + 00	1.95E + 00
F12	1.36E + 00	1.15E + 00	1.13E + 00	6.08E-01	1.09E + 00	1.31E + 00	1.48E + 00	7.47E + 00	5.53E-01	3.33E + 00	8.68E-01	1.58E + 00
F13	1.70E + 00	1.46E + 00	1.38E + 00	7.43E-01	1.34E + 00	1.56E + 00	1.84E + 00	7.60E + 00	6.92E-01	3.47E + 00	1.02E + 00	1.88E + 00
F14	1.37E + 00	1.12E + 00	1.12E + 00	6.10E-01	1.07E + 00	1.30E + 00	1.48E + 00	7.40E + 00	5.98E-01	3.33E + 00	8.74E-01	1.62E + 00
F15	3.49E + 00	2.90E + 00	3.00E + 00	1.60E + 00	2.92E + 00	3.22E + 00	3.97E + 00	8.42E + 00	1.56E + 00	4.37E + 00	1.90E + 00	3.58E + 00
F16	1.39E + 01	1.19E + 01	1.21E + 01	6.47E + 00	1.20E + 01	1.25E + 01	1.63E + 01	1.39E + 01	6.47E + 00	9.32E + 00	6.80E + 00	1.30E + 01
F17	1.57E + 00	1.31E + 00	1.32E + 00	7.10E-01	1.28E + 00	1.50E + 00	1.74E + 00	7.50E + 00	6.79E-01	3.43E + 00	9.81E-01	1.82E + 00
F18	6.19E + 00	5.48E + 00	5.71E + 00	3.07E + 00	5.49E + 00	5.91E + 00	7.49E + 00	9.83E + 00	2.93E + 00	5.79E + 00	3.28E + 00	6.24E + 00
F19	1.88E + 01	1.21E + 01	1.25E + 01	6.79E + 00	1.23E + 01	1.29E + 01	1.67E + 01	1.41E + 01	6.67E + 00	9.46E + 00	7.27E + 00	1.34E + 01
F20	1.88E + 00	1.55E + 00	1.58E + 00	8.45E-01	1.54E + 00	1.78E + 00	2.10E + 00	7.75E + 00	7.87E-01	3.62E + 00	1.16E + 00	2.10E + 00
F21	2.50E + 00	2.01E + 00	2.09E + 00	1.11E + 00	2.03E + 00	2.31E + 00	2.80E + 00	8.16E + 00	1.10E + 00	3.94E + 00	1.41E + 00	2.58E + 00
F22	2.73E + 00	2.36E + 00	2.38E + 00	1.27E + 00	2.32E + 00	2.62E + 00	3.21E + 00	8.25E + 00	1.26E + 00	4.09E + 00	1.57E + 00	2.90E + 00
F23	2.09E + 00	1.80E + 00	1.84E + 00	9.87E-01	1.80E + 00	2.08E + 00	2.46E + 00	7.82E + 00	9.35E-01	3.75E + 00	1.28E + 00	2.35E + 00
F24	2.35E + 00	2.01E + 00	2.04E + 00	1.09E + 00	2.01E + 00	2.26E + 00	2.73E + 00	7.98E + 00	1.04E + 00	3.86E + 00	1.41E + 00	2.58E + 00
F25	1.06E + 01	9.46E + 00	9.43E + 00	5.05E + 00	9.47E + 00	9.88E + 00	1.28E + 01	1.20E + 01	5.07E + 00	7.91E + 00	5.41E + 00	1.03E + 01
F26	1.10E + 01	9.65E + 00	9.66E + 00	5.19E + 00	9.62E + 00	1.00E + 01	1.31E + 01	1.21E + 01	5.19E + 00	8.03E + 00	5.52E + 00	1.05E + 01
F27	1.00E + 01	9.11E + 00	9.16E + 00	4.95E + 00	9.11E + 00	9.54E + 00	1.24E + 01	1.23E + 01	1.30E + 01	2.02E + 01	1.38E + 01	9.89E + 00
F28	1.74E + 01	1.57E + 01	1.56E + 01	8.44E + 00	1.66E + 01	1.62E + 01	2.13E + 01	1.53E + 01	8.49E + 00	2.82E + 01	9.32E + 00	1.68E + 01
F29	2.21E + 01	1.98E + 01	1.97E + 01	3.98E + 00	7.58E + 00	8.01E + 00	1.04E + 01	1.08E + 01	3.95E + 00	6.80E + 00	4.33E + 00	1.36E + 01

Table 9.

Average runtime comparison of DEHHO and peer algorithms on the CEC2017benchmarks (D100).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	3.17E + 00	2.55E + 00	2.71E + 00	1.57E + 00	2.56E + 00	3.15E + 00	7.27E + 00	4.36E + 01	2.71E + 00	1.72E + 01	3.67E + 00	6.99E + 00
F3	3.27E + 00	2.75E + 00	2.84E + 00	1.58E + 00	2.78E + 00	3.35E + 00	4.62E + 00	2.22E + 01	1.60E + 00	1.26E + 01	2.08E + 00	4.22E + 00
F4	3.59E + 00	2.67E + 00	2.70E + 00	1.62E + 00	2.93E + 00	3.27E + 00	3.80E + 00	2.14E + 01	1.49E + 00	1.18E + 01	1.93E + 00	4.13E + 00
F5	1.44E + 01	1.19E + 01	1.29E + 01	6.93E + 00	1.28E + 01	1.33E + 01	1.69E + 01	3.93E + 01	6.19E + 00	2.09E + 01	8.38E + 00	1.42E + 01
F6	3.92E + 00	3.15E + 00	3.28E + 00	1.80E + 00	3.26E + 00	3.65E + 00	4.48E + 00	2.12E + 01	1.73E + 00	1.19E + 01	2.13E + 00	4.52E + 00
F7	4.93E + 00	3.98E + 00	4.04E + 00	2.12E + 00	3.98E + 00	4.56E + 00	5.40E + 00	2.18E + 01	2.10E + 00	1.23E + 01	2.54E + 00	5.30E + 00
F8	1.06E + 00	8.43E-01	8.68E-01	5.08E-01	8.39E-01	1.04E + 00	1.15E + 00	7.42E + 00	4.23E-01	3.25E + 00	7.78E-01	1.38E + 00
F9	1.59E + 00	1.26E + 00	1.26E + 00	6.92E-01	1.21E + 00	1.46E + 00	1.84E + 00	7.64E + 00	6.36E-01	3.47E + 00	9.79E-01	1.74E + 00
F10	1.28E + 00	1.05E + 00	1.06E + 00	5.69E-01	1.00E + 00	1.22E + 00	1.35E + 00	7.40E + 00	5.07E-01	3.30E + 00	8.19E-01	1.49E + 00
F11	1.70E + 00	1.47E + 00	1.48E + 00	8.11E-01	1.43E + 00	1.71E + 00	1.99E + 00	8.03E + 00	7.59E-01	3.71E + 00	1.11E + 00	1.95E + 00
F12	1.36E + 00	1.15E + 00	1.13E + 00	6.08E-01	1.09E + 00	1.31E + 00	1.48E + 00	7.47E + 00	5.53E-01	3.33E + 00	8.68E-01	1.58E + 00
F13	1.70E + 00	1.46E + 00	1.38E + 00	7.43E-01	1.34E + 00	1.56E + 00	1.84E + 00	7.60E + 00	6.92E-01	3.47E + 00	1.02E + 00	1.88E + 00
F14	1.37E + 00	1.12E + 00	1.12E + 00	6.10E-01	1.07E + 00	1.30E + 00	1.48E + 00	7.40E + 00	5.98E-01	3.33E + 00	8.74E-01	1.62E + 00
F15	3.49E + 00	2.90E + 00	3.00E + 00	1.60E + 00	2.92E + 00	3.22E + 00	3.97E + 00	8.42E + 00	1.56E + 00	4.37E + 00	1.90E + 00	3.58E + 00
F16	1.39E + 01	1.19E + 01	1.21E + 01	6.47E + 00	1.20E + 01	1.25E + 01	1.63E + 01	1.39E + 01	6.47E + 00	9.32E + 00	6.80E + 00	1.30E + 01
F17	1.57E + 00	1.31E + 00	1.32E + 00	7.10E-01	1.28E + 00	1.50E + 00	1.74E + 00	7.50E + 00	6.79E-01	3.43E + 00	9.81E-01	1.82E + 00
F18	6.19E + 00	5.48E + 00	5.71E + 00	3.07E + 00	5.49E + 00	5.91E + 00	7.49E + 00	9.83E + 00	2.93E + 00	5.79E + 00	3.28E + 00	6.24E + 00
F19	1.88E + 01	1.21E + 01	1.25E + 01	6.79E + 00	1.23E + 01	1.29E + 01	1.67E + 01	1.41E + 01	6.67E + 00	9.46E + 00	7.27E + 00	1.34E + 01
F20	1.88E + 00	1.55E + 00	1.58E + 00	8.45E-01	1.54E + 00	1.78E + 00	2.10E + 00	7.75E + 00	7.87E-01	3.62E + 00	1.16E + 00	2.10E + 00
F21	2.50E + 00	2.01E + 00	2.09E + 00	1.11E + 00	2.03E + 00	2.31E + 00	2.80E + 00	8.16E + 00	1.10E + 00	3.94E + 00	1.41E + 00	2.58E + 00
F22	2.73E + 00	2.36E + 00	2.38E + 00	1.27E + 00	2.32E + 00	2.62E + 00	3.21E + 00	8.25E + 00	1.26E + 00	4.09E + 00	1.57E + 00	2.90E + 00
F23	2.09E + 00	1.80E + 00	1.84E + 00	9.87E-01	1.80E + 00	2.08E + 00	2.46E + 00	7.82E + 00	9.35E-01	3.75E + 00	1.28E + 00	2.35E + 00
F24	2.35E + 00	2.01E + 00	2.04E + 00	1.09E + 00	2.01E + 00	2.26E + 00	2.73E + 00	7.98E + 00	1.04E + 00	3.86E + 00	1.41E + 00	2.58E + 00
F25	1.06E + 01	9.46E + 00	9.43E + 00	5.05E + 00	9.47E + 00	9.88E + 00	1.28E + 01	1.20E + 01	5.07E + 00	7.91E + 00	5.41E + 00	1.03E + 01
F26	1.10E + 01	9.65E + 00	9.66E + 00	5.19E + 00	9.62E + 00	1.00E + 01	1.31E + 01	1.21E + 01	5.19E + 00	8.03E + 00	5.52E + 00	1.05E + 01
F27	1.00E + 01	9.11E + 00	9.16E + 00	4.95E + 00	9.11E + 00	9.54E + 00	1.24E + 01	1.23E + 01	1.30E + 01	2.02E + 01	1.38E + 01	9.89E + 00
F28	1.74E + 01	1.57E + 01	1.56E + 01	8.44E + 00	1.66E + 01	1.62E + 01	2.13E + 01	1.53E + 01	8.49E + 00	2.82E + 01	9.32E + 00	1.68E + 01
F29	2.21E + 01	1.98E + 01	1.97E + 01	3.98E + 00	7.58E + 00	8.01E + 00	1.04E + 01	1.08E + 01	3.95E + 00	6.80E + 00	4.33E + 00	1.36E + 01

It is observed that the integration of Gaussian perturbations and Trend-Guided DE operators inevitably introduces a marginal computational overhead compared to the canonical HHO. For instance, on simpler unimodal functions, DEHHO exhibits a slight increase in runtime due to the generation of random number vectors and additional fitness comparisons in the greedy selection step. However, when juxtaposed with other advanced hybrid variants such as IHAOHHO, CLHHEO, and BGHHO, DEHHO demonstrates a significantly more favorable runtime profile. This efficiency is attributed to the lightweight, vector-based design of the proposed modules, which avoids the heavy computational burden associated with multi-population frameworks, chaotic map iterations, or complex surrogate models often employed in peer algorithms.

Crucially, the runtime analysis reveals a stable linear growth pattern as the problem dimensionality doubles from 50D to 100D. DEHHO consistently ranks among the top tier in terms of speed on the majority of functions, confirming that the internal mechanisms—specifically the Momentum Calculation and Lévy-Annealed Jumps—do not incur exponential time complexity. From a “performance-to-cost” perspective, the slight additional runtime (typically sub-second differences) is arguably negligible compared to the substantial improvements in convergence accuracy (often orders of magnitude). Therefore, DEHHO offers a superior engineering trade-off, exchanging a minimal fraction of computational resources for a maximal gain in solution reliability and robustness in high-dimensional tasks.

In-width validation: performance under structural complexity on CEC2020

While the CEC2017 suite validates functional diversity, the CEC2020 benchmark suite introduces a higher level of difficulty through increased structural complexity, characterized by strong variable dependencies, deceptive bias, and hierarchical sub-components. To assess the robustness of DEHHO in these challenging environments, experiments were conducted on the 10 hybrid and composition functions of CEC 2020 under both 50D and 100D configurations.

Performance under Structural Complexity:

The comparative results, detailed in Tables 10 and 11, reveal that DEHHO maintains superior performance even as the landscape complexity intensifies. In the 50D setting, DEHHO ranks first (or ties for first) on 9 out of 10 functions. This dominance is particularly evident on hybrid functions (e.g., F5, F6), which simulate real-world problems with non-separable variables. The algorithm’s ability to navigate these coupled variables is attributed to the Trend-Guided DE mechanism, where the difference vector implicitly captures the correlation between dimensions, a capability lacking in the component-wise updates of standard HHO. As dimensionality scales to 100D, DEHHO sustains its lead, securing top ranks on 7 functions. The overall superiority is visually summarized by the radar charts in Fig. 6, where DEHHO consistently encompasses the smallest area (indicating the lowest rank) across both dimensions compared to the ten peer algorithms.

Table 10.

Performance comparison of DEHHO and peer algorithms on the CEC2020 benchmark (50D).

		DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	Mean	1.01E + 09	2.22E + 10	1.04E + 10	4.07E + 09	2.04E + 09	3.57E + 10	8.94E + 10	2.65E + 10	1.61E + 11	1.49E + 10	3.82E + 10	1.31E + 10
	Std	2.67E + 08	3.36E + 09	3.04E + 09	1.63E + 09	7.49E + 08	1.10E + 10	1.14E + 10	6.33E + 09	1.56E + 10	2.00E + 09	7.07E + 09	1.06E + 10
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F2	Mean	1.15E + 04	1.19E + 04	1.10E + 04	1.10E + 04	1.12E + 04	1.09E + 04	1.43E + 04	1.08E + 04	1.53E + 04	1.39E + 04	1.20E + 04	1.35E + 04
	Std	1.18E + 03	1.37E + 03	1.28E + 03	9.14E + 02	1.47E + 03	1.15E + 03	7.86E + 02	7.61E + 02	4.97E + 02	6.00E + 02	1.28E + 03	2.32E + 03
	=/≈/-		≈	≈	≈	≈	≈		-
F3	Mean	4.01E + 04	8.62E + 05	3.36E + 05	1.53E + 05	5.65E + 04	9.96E + 05	3.01E + 06	8.10E + 05	5.69E + 06	4.91E + 05	1.06E + 06	6.92E + 05
	Std	9.55E + 03	1.76E + 05	7.68E + 04	3.99E + 04	1.63E + 04	2.91E + 05	3.99E + 05	2.60E + 05	4.71E + 05	1.09E + 05	3.31E + 05	6.98E + 05
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F4	Mean	2.09E + 03	5.99E + 04	1.29E + 04	6.22E + 03	2.77E + 03	5.39E + 04	1.19E + 06	3.21E + 04	1.06E + 07	1.16E + 04	4.20E + 04	2.66E + 05
	Std	1.09E + 02	3.58E + 04	1.26E + 04	4.19E + 03	7.61E + 02	5.52E + 04	7.91E + 05	2.63E + 04	6.43E + 06	4.99E + 03	3.09E + 04	5.85E + 05
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F5	Mean	7.24E + 06	7.24E + 07	2.20E + 07	2.19E + 07	8.93E + 07	1.75E + 07	6.03E + 07	1.56E + 07	1.58E + 08	1.11E + 07	1.29E + 07	3.06E + 08
	Std	5.23E + 06	4.16E + 07	9.88E + 06	1.20E + 07	1.22E + 08	1.33E + 07	3.01E + 07	1.47E + 07	4.87E + 07	4.94E + 06	9.98E + 06	2.10E + 08
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F6	Mean	7.26E + 05	7.08E + 06	1.57E + 06	9.72E + 05	8.62E + 05	2.69E + 06	3.02E + 08	4.66E + 06	3.55E + 09	1.53E + 06	4.55E + 07	7.49E + 08
	Std	5.39E + 05	1.65E + 07	1.04E + 06	7.40E + 05	5.99E + 05	4.09E + 06	3.76E + 08	7.43E + 06	1.27E + 09	7.31E + 05	1.46E + 08	1.26E + 09
	=/≈/-		+	+	≈	≈	+	+	+	+	+	+	+
F7	Mean	1.41E + 07	2.90E + 08	5.88E + 07	6.44E + 07	1.07E + 09	3.69E + 07	1.04E + 09	5.78E + 07	3.77E + 09	6.30E + 07	7.56E + 07	1.25E + 10
	Std	8.12E + 06	2.79E + 08	4.28E + 07	4.07E + 07	3.25E + 09	4.98E + 07	5.37E + 08	1.03E + 08	1.95E + 09	2.24E + 07	1.33E + 08	8.80E + 09
	=/≈/-		+	+	+	+	≈	+	+	+	+	+	+
F8	Mean	7.16E + 03	9.92E + 03	7.22E + 03	7.25E + 03	6.40E + 03	8.63E + 03	5.03E + 03	3.93E + 03	4.65E + 03	2.96E + 03	5.03E + 03	1.11E + 04
	Std	3.22E + 03	1.54E + 03	1.98E + 03	3.03E + 03	1.35E + 03	1.29E + 03	8.51E + 02	1.69E + 03	2.97E + 03	1.00E + 02	1.69E + 03	2.62E + 03
	=/≈/-			≈	≈	≈	≈	≈	-	≈	-	≈	+
F9	Mean	5.10E + 03	2.98E + 04	2.07E + 04	7.67E + 03	8.24E + 03	3.74E + 04	5.40E + 04	3.18E + 04	4.55E + 04	2.12E + 04	3.38E + 04	4.89E + 04
	Std	5.31E + 02	6.86E + 03	7.10E + 03	1.28E + 03	4.97E + 03	9.69E + 03	7.69E + 03	8.91E + 03	1.86E + 04	5.11E + 03	9.49E + 03	1.19E + 04
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F10	Mean	4.32E + 03	7.90E + 03	5.95E + 03	5.17E + 03	5.29E + 03	7.29E + 03	1.58E + 04	6.34E + 03	1.94E + 04	6.33E + 03	7.05E + 03	9.48E + 03
	Std	2.84E + 02	9.55E + 02	7.07E + 02	5.56E + 02	1.42E + 03	1.47E + 03	2.63E + 03	1.22E + 03	2.84E + 03	3.06E + 02	1.37E + 03	3.86E + 03
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
	=/≈/-		9/1/0	8/2/0	7/3/0	7/3/0	7/3/0	9/1/0	8/0/2	9/1/0	9/0/1	9/1/0	10/0/0

Table 11.

Performance comparison of DEHHO and peer algorithms on the CEC2020 benchmark (100D).

		DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1	Mean	1.92E + 10	1.09E + 11	7.81E + 10	5.32E + 10	2.95E + 10	1.26E + 11	2.40E + 11	1.02E + 11	4.70E + 11	1.00E + 11	1.24E + 11	1.01E + 11
	Std	3.46E + 09	1.02E + 10	9.06E + 09	1.20E + 10	4.95E + 09	1.99E + 10	1.78E + 10	1.98E + 10	3.28E + 10	8.75E + 09	1.64E + 10	2.74E + 10
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F2	Mean	2.64E + 04	2.69E + 04	2.55E + 04	2.34E + 04	2.46E + 04	2.37E + 04	3.20E + 04	2.26E + 04	3.28E + 04	3.03E + 04	2.57E + 04	2.98E + 04
	Std	2.27E + 03	2.38E + 03	2.34E + 03	1.39E + 03	1.78E + 03	1.94E + 03	9.62E + 02	1.76E + 03	7.22E + 02	8.71E + 02	2.34E + 03	2.74E + 03
	=/≈/-		≈	≈	-	-	-	+	-	+	+	-	+
F3	Mean	6.98E + 05	3.97E + 06	2.56E + 06	1.68E + 06	1.10E + 06	4.46E + 06	8.85E + 06	3.28E + 06	1.60E + 07	3.38E + 06	4.12E + 06	5.14E + 06
	Std	1.28E + 05	3.81E + 05	3.52E + 05	3.28E + 05	1.49E + 05	5.66E + 05	6.39E + 05	5.09E + 05	1.08E + 06	3.08E + 05	6.57E + 05	2.10E + 06
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F4	Mean	1.10E + 04	3.74E + 05	1.27E + 05	1.49E + 05	2.14E + 04	3.57E + 05	4.45E + 06	1.52E + 05	7.06E + 07	2.74E + 05	2.43E + 05	7.63E + 05
	Std	5.43E + 03	1.40E + 05	5.76E + 04	8.35E + 04	7.05E + 03	2.00E + 05	1.45E + 06	7.66E + 04	1.93E + 07	8.70E + 04	9.41E + 04	1.06E + 06
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F5	Mean	7.03E + 07	3.35E + 08	2.02E + 08	1.93E + 08	1.35E + 08	1.19E + 08	5.84E + 08	1.10E + 08	1.17E + 09	1.83E + 08	1.21E + 08	1.01E + 09
	Std	2.60E + 07	1.31E + 08	5.59E + 07	5.80E + 07	1.07E + 08	3.30E + 07	1.92E + 08	3.76E + 07	2.02E + 08	3.98E + 07	3.89E + 07	6.96E + 08
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F6	Mean	1.21E + 06	1.22E + 09	1.32E + 08	1.21E + 07	4.62E + 06	1.48E + 09	1.21E + 10	1.05E + 09	3.42E + 10	1.91E + 08	1.70E + 09	5.20E + 09
	Std	1.86E + 06	7.92E + 08	1.57E + 08	1.42E + 07	8.40E + 06	1.53E + 09	7.43E + 09	8.96E + 08	1.28E + 10	1.13E + 08	1.19E + 09	7.93E + 09
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F7	Mean	1.96E + 08	2.84E + 09	8.49E + 08	5.40E + 08	2.93E + 09	8.16E + 08	1.12E + 10	6.09E + 08	1.55E + 10	1.28E + 09	7.06E + 08	2.21E + 10
	Std	6.56E + 07	1.35E + 09	3.85E + 08	2.31E + 08	5.31E + 09	5.38E + 08	4.64E + 09	4.90E + 08	3.30E + 09	3.02E + 08	3.99E + 08	1.07E + 10
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F8	Mean	1.80E + 04	2.08E + 04	1.84E + 04	1.65E + 04	1.57E + 04	2.04E + 04	1.84E + 04	9.91E + 03	1.02E + 04	5.61E + 03	1.29E + 04	2.62E + 04
	Std	4.90E + 03	1.40E + 03	2.38E + 03	5.21E + 03	3.08E + 03	1.77E + 03	3.64E + 03	5.32E + 03	7.50E + 03	4.00E + 02	5.22E + 03	3.83E + 03
	=/≈/-		+	≈	≈	-	+	≈	-	-	-	-	+
F9	Mean	4.53E + 04	1.16E + 05	9.73E + 04	7.43E + 04	6.12E + 04	1.27E + 05	1.76E + 05	1.20E + 05	1.85E + 05	1.17E + 05	1.26E + 05	1.53E + 05
	Std	1.72E + 04	6.35E + 03	9.68E + 03	2.06E + 04	1.50E + 04	9.47E + 03	6.29E + 03	7.18E + 03	2.33E + 04	5.35E + 03	1.18E + 04	8.15E + 03
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+
F10	Mean	5.74E + 03	1.24E + 04	9.15E + 03	7.64E + 03	6.21E + 03	1.12E + 04	3.19E + 04	9.75E + 03	9.81E + 04	1.12E + 04	1.09E + 04	1.37E + 04
	Std	5.59E + 02	1.13E + 03	7.54E + 02	7.44E + 02	4.52E + 02	1.88E + 03	5.46E + 03	1.42E + 03	1.22E + 04	7.99E + 02	1.66E + 03	4.33E + 03
	=/≈/-		+	+	+	+	+	+	+	+	+	+	+

Fig. 6.

Friedman ranking of EDEHHO and peer algorithms on the CEC2020. (a) CEC2020D50, (b) CEC2020D100.

To further scrutinize the evolutionary behavior, convergence trajectories for representative functions (F1, F3, F6, F9) are illustrated in Fig. 7 (50D) and Fig. 8 (100D). A notable observation on the composition function F9 is DEHHO’s ability to escape “deep deceptive valleys”—regions where the gradient points away from the global optimum. While comparative algorithms such as IHAOHHO and SCA exhibit early stagnation (flat-lining), DEHHO displays a multi-stage descent pattern. This confirms the efficacy of the dual-branch strategy: the Gaussian perturbation maintains diversity to resist deceptive attractors, while the Lévy-Annealed jumps provide the necessary impulse to traverse barriers between nested sub-functions.

Fig. 7.

Convergence curves of DEHHO and peer algorithms on CEC2020 functions (D = 50). (a) CEC2020F1D50, (b)CEC2020F3D5, (c) CEC2020F6D50, (d) CEC2020F9D50.

Fig. 8.

Convergence curves of DEHHO and peer algorithms on CEC2020 functions (D = 100). (a) CEC2020F1D100, (b) CEC2020F3D100, (c) CEC2020F6D100, (d) CEC2020F9D100.

The robustness of these findings is underpinned by rigorous statistical testing. The pairwise Wilcoxon signed-rank test () results, presented in the heatmaps of Tables 12 and 13, indicate widespread significant improvements. Specifically, DEHHO achieves at least 7 significant wins against all competitors in the 50D case, including robust variants like CLHHEO. This statistical evidence reinforces that the performance gains are not artifacts of stochastic luck but stem from the algorithmic synergy designed to handle structural complexity.

Table 12.

Function-wise Wilcoxon test p-values between DEHHO and peer algorithms on the CEC 2020 benchmarks (D50).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1		3.02E-11	3.02E-11	3.34E-11	1.31E-08	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F2		2.71E-01	1.15E-01	5.37E-02	5.37E-02	7.48E-02	2.15E-10	9.47E-03	3.02E-11	7.38E-10	2.77E-01	4.46E-04
F3		3.02E-11	3.02E-11	3.02E-11	8.88E-06	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F4		3.02E-11	3.02E-11	3.02E-11	3.20E-09	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F5		6.07E-11	5.09E-08	2.78E-07	7.09E-08	1.53E-05	4.50E-11	4.86E-03	3.02E-11	2.75E-03	5.32E-03	5.49E-11
F6		1.87E-07	5.56E-04	2.23E-01	3.40E-01	2.62E-03	3.02E-11	9.03E-04	3.02E-11	1.25E-05	1.68E-04	1.01E-08
F7		3.02E-11	5.53E-08	5.46E-09	4.51E-02	5.19E-02	3.02E-11	3.16E-05	3.02E-11	1.33E-10	9.53E-07	3.34E-11
F8		1.95E-03	9.59E-01	9.00E-01	2.84E-01	2.23E-01	7.48E-02	1.37E-03	1.12E-01	1.01E-08	7.01E-02	4.94E-05
F9		3.02E-11	3.02E-11	5.57E-10	9.51E-06	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F10		3.02E-11	3.34E-11	9.06E-08	1.41E-04	4.50E-11	3.02E-11	3.16E-10	3.02E-11	3.02E-11	3.69E-11	3.69E-11

Table 13.

Function-wise Wilcoxon test p-values between DEHHO and peer algorithms on the CEC 2020 benchmarks (D100).

	DEHHO	HHO	BGHHO	SHHO	HHSC	CLHHEO	IHAOHHO	SCSO	JA	SCA	HBA	SSA
F1		3.02E-11	3.02E-11	3.02E-11	8.89E-10	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F2		3.79E-01	1.91E-01	1.61E-06	1.52E-03	1.25E-05	5.49E-11	4.31E-08	3.02E-11	3.20E-09	1.71E-01	5.09E-06
F3		3.02E-11	3.02E-11	3.02E-11	1.96E-10	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F4		3.02E-11	3.02E-11	3.02E-11	2.03E-07	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F5		3.02E-11	1.21E-10	1.33E-10	4.46E-04	1.39E-06	4.50E-11	6.77E-05	3.02E-11	3.69E-11	8.84E-07	3.02E-11
F6		3.02E-11	3.34E-11	2.44E-09	3.57E-06	3.02E-11	3.02E-11	3.34E-11	3.02E-11	3.02E-11	3.02E-11	3.69E-11
F7		3.02E-11	3.34E-11	8.89E-10	7.66E-05	2.61E-10	3.02E-11	1.36E-07	3.02E-11	3.02E-11	3.82E-10	3.02E-11
F8		4.43E-03	4.20E-01	1.33E-01	1.78E-04	4.68E-02	5.59E-01	3.37E-05	3.83E-05	9.06E-08	5.56E-04	3.20E-09
F9		3.02E-11	9.92E-11	1.19E-06	8.66E-05	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F10		3.02E-11	3.02E-11	8.15E-11	2.62E-03	3.02E-11	3.02E-11	3.69E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11

In conclusion, the results on CEC 2020 demonstrate that DEHHO is not merely an optimizer for standard functions but a robust solver capable of disentangling complex variable interactions and overcoming deceptive landscape features in high-dimensional spaces.

Ablation study: mechanism contribution analysis on CEC2017 50D

To systematically isolate and quantify the contribution of each algorithmic component within the DEHHO framework, a rigorous ablation study was conducted on the CEC 2017 benchmark (50D). Ideally, the performance gain of a hybrid algorithm should stem from the synergy of its parts rather than a single dominant operator. To verify this, four distinct variants were developed by disabling specific core modules while maintaining all other experimental settings strictly ceteris paribus. Specifically, the DEHHO-NoDE variant replaces the Trend-Guided DE operator entirely with the original HHO besiege strategy to isolate the impact of evolutionary hybridization. The DEHHO-NoTrend variant retains the DE operator but removes the momentum term (), assessing the specific value of historical directional guidance. Furthermore, to evaluate the diversity mechanisms, DEHHO-NoLevy replaces the adaptive Lévy-annealed jump with a standard uniform random walk, while DEHHO-NoGauss removes the Gaussian perturbation during the exploration phase.

The comparative results, detailed in Tables 14and summarized by the Friedman ranking in Fig. 9, provide compelling evidence for the synergistic integration of all proposed modules. The complete DEHHO framework achieves the lowest mean rank of 1.48, significantly outperforming all ablated variants. A granular analysis of the performance degradation reveals the hierarchy of mechanism importance. The most severe deterioration is observed in DEHHO-NoDE (Rank 4.12), which confirms that the canonical HHO logic lacks the vector-based directionality required for high-precision convergence. The introduction of the DE difference vector is thus identified as the cornerstone of the proposed improvement. Following this, the inferior performance of DEHHO-NoTrend compared to the full model highlights that DE alone is insufficient without trend guidance; the momentum term is proven effectively suppresses oscillation, particularly on unimodal functions where directional stability is paramount.

Fig. 9.

Ablation ranking of CEC 2017benchmarks (D50).

Regarding the diversity preservation mechanisms, while DEHHO-NoGauss and DEHHO-NoLevy perform comparably to the full model on simple landscapes, their failure rate notably increases on complex multimodal and composition functions (e.g., F20–F30). This divergence underscores that while Gaussian perturbation and Lévy jumps may contribute only marginally to local precision, they are indispensable for escaping basins of attraction and preventing premature stagnation in rugged environments. These observations are statistically corroborated by the pairwise comparison in Table 13, where the full DEHHO achieves 25 significant wins against the NoDE variant and 14 wins even against the closest competitor (NoGauss). In conclusion, the ablation study confirms that DEHHO is not merely a collection of operators but a holistic system where Gaussian noise ensures coverage, Lévy flights enable escape, and Trend-Guided DE ensures precision.

Table 14.

Performance comparison of DEHHO and its ablations on the CEC 2017 benchmark (50D).

		DEHHO	DEHHO-NoDE	EDEHHO-NoTrend	DEHHO-NoLevy	DEHHO-NoGauss
F1	Mean	1.02E + 09	1.32E + 09	9.67E + 08	7.73E + 10	1.71E + 09
	Std	2.51E + 08	4.23E + 08	3.47E + 08	1.16E + 10	3.70E + 08
	=/≈/-		-	≈	+	+
F3	Mean	7.75E + 02	8.95E + 02	7.56E + 02	2.21E + 04	8.91E + 02
	Std	1.25E + 02	1.71E + 02	1.02E + 02	5.26E + 03	1.80E + 02
	=/≈/-		+	-	+	+
F4	Mean	2.90E + 03	2.98E + 03	3.12E + 03	1.05E + 05	3.58E + 03
	Std	4.87E + 02	4.37E + 02	5.38E + 02	1.04E + 04	5.95E + 02
	=/≈/-		≈	≈	+	+
F5	Mean	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02	5.00E + 02
	Std	2.36E-03	1.69E-03	5.24E-03	5.91E-03	2.00E-03
	=/≈/-		≈	+	+	+
F6	Mean	6.75E + 04	1.03E + 05	6.69E + 04	4.41E + 06	9.55E + 04
	Std	1.72E + 04	3.20E + 04	1.38E + 04	7.14E + 05	2.63E + 04
	=/≈/-		+	≈	+	+
F7	Mean	1.74E + 03	1.70E + 03	1.85E + 03	2.05E + 03	1.82E + 03
	Std	1.26E + 02	1.24E + 02	1.79E + 02	9.92E + 01	1.08E + 02
	=/≈/-		≈	+	+	+
F8	Mean	8.26E + 02	8.27E + 02	8.34E + 02	8.58E + 02	8.28E + 02
	Std	5.44E + 00	8.85E + 00	1.30E + 01	1.13E + 01	8.18E + 00
	=/≈/-		≈	+	+	+
F9	Mean	1.20E + 04	1.14E + 04	1.15E + 04	1.43E + 04	1.21E + 04
	Std	1.14E + 03	1.19E + 03	1.56E + 03	1.02E + 03	9.96E + 02
	=/≈/-		≈	≈		≈
F10	Mean	2.16E + 05	3.21E + 05	2.32E + 05	2.41E + 09	3.39E + 05
	Std	6.04E + 04	1.40E + 05	5.78E + 04	2.20E + 09	1.78E + 05
	=/≈/-		+	≈	+	+
F11	Mean	1.58E + 08	2.05E + 08	1.55E + 08	9.07E + 09	1.69E + 08
	Std	8.37E + 07	1.18E + 08	9.14E + 07	2.69E + 09	8.85E + 07
	=/≈/-		≈	≈	+	≈
F12	Mean	6.85E + 07	1.12E + 08	6.74E + 07	3.18E + 10	1.10E + 08
	Std	3.09E + 07	3.61E + 07	3.80E + 07	1.45E + 10	4.20E + 07
	=/≈/-		+	≈	+	+
F13	Mean	8.08E + 05	4.86E + 06	9.70E + 05	3.76E + 06	6.90E + 05
	Std	4.06E + 05	3.39E + 06	6.36E + 05	3.35E + 06	4.52E + 05
	=/≈/-		+	≈	+	≈
F14	Mean	1.33E + 07	3.85E + 07	1.44E + 07	9.19E + 09	2.48E + 07
	Std	6.13E + 06	2.07E + 07	4.46E + 06	4.22E + 09	1.53E + 07
	=/≈/-		+	≈	+	+
F15	Mean	2.49E + 05	3.00E + 05	2.24E + 05	1.79E + 09	2.53E + 05
	Std	2.01E + 05	3.40E + 05	2.09E + 05	1.95E + 09	2.17E + 05
	=/≈/-		≈	≈		≈
F16	Mean	5.67E + 04	7.03E + 04	6.09E + 04	1.18E + 10	6.12E + 04
	Std	1.81E + 04	1.83E + 04	2.13E + 04	3.19E + 10	2.02E + 04
	=/≈/-		+	≈	+	≈
F17	Mean	1.83E + 06	9.96E + 06	2.32E + 06	1.46E + 07	9.30E + 05
	Std	1.43E + 06	5.98E + 06	2.23E + 06	3.24E + 07	6.83E + 05
	=/≈/-		+	≈	+	-
F18	Mean	6.25E + 07	3.57E + 09	3.92E + 07	3.23E + 13	2.64E + 08
	Std	1.02E + 08	5.36E + 09	5.20E + 07	1.24E + 14	3.17E + 08
	=/≈/-		+	≈	+	+
F19	Mean	6.59E + 03	6.68E + 03	6.86E + 03	1.21E + 04	6.67E + 03
	Std	1.39E + 03	1.01E + 03	1.35E + 03	1.20E + 03	1.17E + 03
	=/≈/-		≈	≈	+	≈
F20	Mean	4.51E + 03	4.88E + 03	4.63E + 03	8.37E + 04	5.36E + 03
	Std	9.32E + 02	8.50E + 02	8.32E + 02	1.64E + 04	1.12E + 03
	=/≈/-		≈	≈	+	+
F21	Mean	7.49E + 03	7.45E + 03	6.97E + 03	1.23E + 04	7.05E + 03
	Std	3.23E + 03	3.30E + 03	2.93E + 03	1.71E + 03	3.24E + 03
	=/≈/-		≈	≈	+	≈
F22	Mean	6.53E + 03	7.26E + 03	6.15E + 03	9.52E + 04	6.31E + 03
	Std	1.47E + 03	1.34E + 03	8.32E + 02	8.11E + 03	1.20E + 03
	=/≈/-		+	≈	+	≈
F23	Mean	5.08E + 03	5.43E + 03	5.05E + 03	6.34E + 04	5.59E + 03
	Std	5.14E + 02	7.58E + 02	5.59E + 02	3.86E + 03	9.21E + 02
	=/≈/-		≈	≈	+	+
F24	Mean	4.17E + 03	4.47E + 03	4.04E + 03	1.69E + 04	4.14E + 03
	Std	2.97E + 02	4.85E + 02	3.17E + 02	4.34E + 03	2.52E + 02
	=/≈/-		+	-	+	≈
F25	Mean	4.94E + 03	5.62E + 03	5.37E + 03	1.33E + 04	5.09E + 03
	Std	5.48E + 02	1.02E + 03	1.03E + 03	7.27E + 03	6.10E + 02
	=/≈/-		+	+	+	≈
F26	Mean	3.94E + 03	4.07E + 03	3.98E + 03	6.08E + 03	3.97E + 03
	Std	3.04E + 02	4.78E + 02	2.66E + 02	1.11E + 03	3.64E + 02
	=/≈/-		≈	≈	+	≈
F27	Mean	3.20E + 03	3.23E + 03	3.20E + 03	5.60E + 03	3.22E + 03
	Std	4.90E + 01	3.58E + 01	2.24E + 01	9.25E + 02	1.98E + 01
	=/≈/-		+	≈	+	+
F28	Mean	9.79E + 07	3.59E + 08	7.39E + 07	1.94E + 12	1.01E + 08
	Std	1.20E + 08	4.75E + 08	9.53E + 07	5.15E + 12	1.13E + 08
	=/≈/-		+	≈	+	+
F29	Mean	8.54E + 08	1.67E + 09	7.59E + 08	5.96E + 12	7.82E + 08
	Std	5.83E + 08	1.40E + 09	4.51E + 08	2.61E + 13	6.06E + 08
	=/≈/-		+	≈	+	≈
	+/≈/-	/	16/12/0	4/22/2	28/0/0	2015/12/1

Engineering applications and analysis

To empirically validate the practical applicability and constraint-handling capability of DEHHO, this section investigates its performance on classical engineering design problems. These tasks—specifically the Three-Bar Truss, Welded Beam, and Speed Reducer design problems—pose significant challenges due to their non-convex objective functions, mixed-variable search spaces, and highly constrained feasible regions.

Experimental design

Consistent with the benchmark experiments, DEHHO is compared against the ten state-of-the-art peer algorithms. The population size is set to, and the maximum iteration count is to ensure convergence.

Constraint Handling Mechanism:

Handling complex constraints is pivotal in engineering optimization. Unlike simple static penalties, DEHHO employs a Hybrid Barrier-Penalty Method strictly enforce feasibility.

For a problem minimizing subject to inequality constraints and equality constraints , the transformed objective function is formulated as:

6

where is the barrier parameter that decays over time (allowing the search to approach the boundary from the feasible side), and is the penalty coefficient for equality violations. This specific formulation ensures that DEHHO prioritizes the feasible interior during early exploration while precisely converging to the active constraint boundaries—where optimal solutions often lie—during the exploitation phase.

Three-bar truss design problem⁴⁴

The Three-Bar Truss Design problem aims to minimize the total weight of a truss structure subject to stress, deflection, and buckling constraints. The problem involves two decision variables representing the cross-sectional areas. The mathematical formulation is detailed in.

The comparative results, summarized in Table 15, demonstrate the precision of the proposed method. DEHHO identifies the minimum weight of 263.8959, surpassing standard HHO (264.42) and advanced variants like IHAOHHO (264.48). Notably, the variables found by DEHHO () strictly satisfy all non-linear buckling constraints. While algorithms like HHSC and HBA achieve competitive results near the optimal value, DEHHO exhibits higher stability in locating the global optimum without violating the constraints boundaries, validating the efficacy of the Momentum-Guided search in constrained continuous spaces.

Table 15.

Comparison of results for the Three-Bar truss design problem.

	Best
DEHHO	263.8959	0.7887	0.4083
HHO	264.4239	0.7852	0.4234
BGHHO	264.5982	0.7893	0.4136
SHHO	263.9117	0.7892	0.4070
HHSC	263.8972	0.7900	0.4044
CLHHEO	263.8967	0.7888	0.4079
IHAOHHO	264.4892	0.8186	0.3298
SCSO	264.5285	0.7959	0.3941
JA	268.3068	0.8074	0.3994
SCA	263.9084	0.7883	0.4095
HBA	263.8962	0.7887	0.4083
SSA	263.9826	0.7895	0.4069

The mathematical formulation of the problem is as follows:

Function:

7

Subject to:

8

Parameters:

Weld beam design problem⁴⁵

This problem presents a rigorous test of an algorithm’s ability to navigate a highly coupled constraint space. The objective is to minimize the fabrication cost of a welded beam, governed by four design variables (weld thickness , bar length , height , and beam thickness ) and seven constraints involving shear stress, bending stress, and buckling load .

As presented in Table 16, DEHHO achieves the lowest cost of 1.67022, significantly outperforming the original HHO (2.7589) and other variants like BGHHO (2.8222). The substantial performance gap suggests that standard swarm-based methods struggle to maintain feasibility along the active constraint boundaries of this problem. In contrast, DEHHO leverages its Trend-Guided DE operator to perform fine-grained steps along the constraint edge. The optimal design variables (, , , ) indicate that DEHHO successfully balances material reduction with structural integrity, avoiding the premature convergence into infeasible regions observed in competitors like JA and SHHO.

Table 16.

Comparison of results for the welded beam design problem.

	Best
DEHHO	1.6702	0.1988	3.3374	9.1900	0.19990
HHO	2.7589	0.2328	5.1194	6.5800	0.5030
BGHHO	2.8222	0.3677	3.3974	6.2600	0.5780
SHHO	2.2075	0.1907	5.7104	8.2100	0.2930
HHSC	1.7221	0.1968	3.2464	9.6900	0.1970
CLHHEO	1.7549	0.2204	3.2769	8.9600	0.2230
IHAOHHO	1.7908	0.1414	4.9656	8.6900	0.1870
SCSO	1.6774	0.1929	3.4599	9.1900	0.19990
JA	2.2147	0.1078	7.1341	9.6200	0.2580
SCA	1.6835	0.1941	3.4557	9.1900	0.1990
HBA	1.7860	0.2184	3.2739	8.7000	0.2300
SSA	2.0540	0.1960	4.4991	8.3200	0.2570

The mathematical model is formulated as follows:

Function:

9

Subject to:

10

Parameters:

Speed reducer design problem⁴⁶

The Speed Reducer problem is a high-dimensional mechanical design task involving seven decision variables (face width, teeth module, etc.) and eleven constraints. The goal is to minimize the total weight of the reducer while satisfying geometric and stress requirements.

The results in Table 17 highlight DEHHO’s capability in high-dimensional engineering spaces. DEHHO achieves a minimized weight of 1341.6221, which is marginally superior to CLHHEO (1341.6265) and significantly better than HHO (1526.51). A critical observation is the stability of the integer-like variables (e.g., number of teeth ). While algorithms like BGHHO () fail to converge to the optimal discrete integer vicinity, DEHHO consistently locks onto the optimal subspace (). This precision is attributed to the Gaussian perturbation, which allows for localized exploration around discrete steps, coupled with the Lévy-Annealed mechanism that prevents stagnation in suboptimal gear configurations.

Table 17.

Comparison of results for the speed reducer design problem.

	Best
DEHHO	1341.6221	3.5000	0.7000	17.0000	8.1406	8.2433	3.9000	5.4984
HHO	1526.5176	3.5019	0.7000	17.8723	7.8231	7.8621	3.6466	5.3789
BGHHO	1770.9319	3.5116	0.7012	18.9005	7.8126	7.9495	3.5950	5.3411
SHHO	1342.0326	3.5010	0.7000	17.0000	8.1760	8.2349	3.8981	5.4991
HHSC	1350.2554	0.3400	0.6613	18.4828	7.9492	7.8403	3.8986	5.4002
CLHHEO	1341.5265	3.5000	0.7000	17.0000	8.0554	8.1562	3.9000	5.5000
IHAOHHO	1353.9743	3.5000	0.7000	17.0000	7.8552	7.8181	3.8177	5.3410
SCSO	1341.5393	3.5000	0.7000	17.0000	8.2153	8.2196	3.9000	5.5000
JA	1344.0843	3.5067	0.7000	17.0000	8.2716	8.2939	3.9000	5.5000
SCA	1404.3101	3.5445	0.7021	17.0959	7.8618	8.0354	3.5945	5.3876
HBA	1341.5265	3.5000	0.7000	17.0000	8.3000	8.3000	3.9000	5.5000
SSA	1403.3005	3.5286	0.7019	17.0950	7.7688	7.9617	3.5159	5.3395

The mathematical model is detailed as below.

Function:

11

Subject to:

12

Parameters:

Discussion and future work

Mechanism synthesis and critical analysis

The consistent superiority of DEHHO across high-dimensional and structurally complex landscapes is attributed to the structural complementarity of its dual components. While canonical HHO relies on scalar energy parameters for contraction pressure, it fundamentally lacks the directional cues required to navigate high-dimensional () ridges. DEHHO remedies this via the Trend-Guided DE operator, which utilizes difference vectors to approximate local gradients, thereby providing the directional stability needed for precise convergence. Furthermore, the Gaussian-Stochastic Perturbation overcomes the “blindness” of standard random walks. By offering tunable, fine-grained diversity maintenance, it works in tandem with the evolutionary momentum to break the curse of dimensionality, a critical capability where single-mechanism algorithms often falter.

Positioning against bayesian optimization

Positioning DEHHO within the broader optimization spectrum is essential. While surrogate-assisted methods like Bayesian Optimization (BO) excel in low-dimensional () problems with expensive evaluation costs, DEHHO targets a distinct problem class: high-dimensional landscapes where function evaluations are computationally cheap but the search space is exponentially vast. Unlike BO, which suffers from cubic complexity scaling, DEHHO maintains linear complexity . This scalability makes DEHHO a more viable candidate for large-scale engineering design, whereas BO remains superior for data-scarce scenarios.

Limitations and future directions

Despite its robustness, DEHHO is not without limitations. First, it relies on empirically tuned hyperparameters (e.g., ), and the zero-initialized Momentum Vector may induce a “cold start” delay in early iterations. Second, as a continuous optimizer, discretization errors may arise when applied to combinatorial tasks.

Future research will address these gaps through three strategic avenues: (1) integrating Reinforcement Learning (RL) for adaptive parameter control based on real-time landscape feedback; (2) extending the framework to Multi-Objective Optimization (MOO) to handle conflicting engineering criteria; and (3) exploring Lightweight Surrogate-Assisted variants to bridge the gap between evolutionary computation and data-efficient optimization in expensive scenarios.

Author contributions

F K: Software, visualization, writing—original draft, X S: Conceptualization, methodology, formal analysis, investigation, writing—original draft, writing—review and editing. All authors read and approved the final manuscript.

Funding

The authors declare that no grants, funding, or financial support were received for this work. The study was entirely self-funded by the authors.

Data availability

The datasets generated during and analysed during the current study are not publicly available due to privacy restrictions but are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.