Auto-Tuning Memory-Based Adaptive Local Search Gaining–Sharing Knowledge-Based Algorithm for Solving Optimization Problems

Abstract

The Gaining–Sharing Knowledge-based (GSK) algorithm is a human-inspired metaheuristic that models how people learn and disseminate knowledge across their lifetime. It has shown promising results across a range of engineering optimization problems. However, one of its major limitations lies in the use of fixed parameters to guide the search process, which often causes the algorithm to get stuck in local optima. To address this challenge, we propose an Auto-Tuning Memory-based Adaptive Local Search (ATMALS) empowered GSK, that is, ATMALS-GSK. This enhanced version of GSK introduces two key improvements: adaptive local search and memory-driven automatic tuning of parameters. Rather than relying on fixed values, ATMALS-GSK continuously adjusts its parameters during the optimization process. This is achieved through a Gaussian distribution mechanism that iteratively updates the likelihood of selecting different parameter values based on their historical impact on the fitness function. This selection process is guided by a weighted moving average that tracks each parameter’s contribution to fitness improvement over time. To further reduce the risk of premature convergence, an adaptive local search strategy is embedded, facilitating the algorithm’s escape from local traps and guiding it toward more optimal regions within the search domain. To validate the effectiveness of the ATMALS-GSK algorithm, it is evaluated on the CEC 2011 and CEC 2017 benchmarks. The results indicate that the ATMALS-GSK algorithm outperforms the original GSK, its variants, and other metaheuristics by delivering greater robustness, quicker convergence, and superior solution quality.

1. Introduction

Optimization seeks to identify the most effective solutions while adhering to specific constraints and conditions. It plays a vital role in both theoretical research and real-world engineering applications, where achieving peak performance, efficiency, or minimal cost is often the goal. As technology advances and we take on more complex challenges, the nature of optimization problems has evolved significantly. These problems have grown not only in scale but also in complexity, making the quest for optimal solutions increasingly difficult. While many optimization algorithms exist, reaching the global optimum, especially in large-scale or highly dynamic scenarios, remains a daunting task [1].

In general, numerical optimization algorithms are typically classified into two main groups: deterministic algorithms (also known as exact methods) and stochastic algorithms (also referred to as approximate methods) [2]. Traditional optimization methods are often grounded in exact mathematical formulations and rely on gradient-based information to navigate the solution space. These deterministic methods are well-suited to problems where all parameters are known, the objective function is smooth, and the search space is manageable. However, when faced with non-deterministic polynomial-time hardness (i.e., NP-hard problems), which are common in engineering and logistics, these traditional approaches struggle. High computational costs, long execution times, and a tendency to get trapped in local optima make them impractical for many real-world applications. This has opened the door for approximate optimization algorithms that favor speed, adaptability, and robustness over theoretical precision [3].

Metaheuristic algorithms have gained prominence as promising alternatives to conventional exact methods, especially for solving NP-hard problems. These algorithms are inherently flexible, employing probabilistic search mechanisms that do not require derivative information [4]. They can be applied across a wide array of applications, including those with noisy, nonlinear, or multi-modal objective functions. Their effectiveness mainly comes from their ability to strike a balance between exploring the search space widely (exploration) and focusing on the most promising areas (exploitation), which helps them avoid getting stuck too soon and discover high-quality solutions more quickly. This balance is key to the effectiveness of any metaheuristic approach [5].

Generally, metaheuristics can be grouped into four categories considering their underlying inspiration: evolutionary-based, swarm intelligence-based, physics-inspired, and human behavior-based techniques [6]. These encompass evolutionary approaches inspired by natural selection [7], swarm intelligence based on the collective behavior of animals [8], and physics-inspired models that simulate natural processes like gravity and thermodynamics [9]. A newer branch includes human-inspired algorithms, which model decision-making, negotiation, learning, or social behavior [10]. These human-based algorithms have shown considerable promise in solving complex optimization problems due to their intuitive mechanisms and adaptability, qualities that are especially relevant in today’s dynamic problem spaces [11,12,13,14].

Despite their versatility, existing metaheuristics face several limitations that hinder their widespread adoption and performance in highly demanding environments. One major challenge lies in their reliance on static control parameters, which often fail to adapt as the search progresses. This can result in premature convergence or inefficient exploration, especially in dynamic or high-dimensional problems [15]. Additionally, most current algorithms are not designed to fully utilize memory-based learning or historical performance, which limits their capacity to make intelligent, context-aware decisions [6]. Another critical gap is the lack of universal frameworks that can systematically tune the trade-off between exploitation and exploration based on problem complexity or runtime feedback. As real-world problems become more interconnected and volatile, there is a growing need for metaheuristics that can self-adapt, learn over time, and operate efficiently in uncertain, dynamic, and large-scale environments. Addressing these gaps is key to pushing the boundaries of what modern optimization algorithms can achieve.

The Gaining–Sharing Knowledge-based (GSK) algorithm is a relatively new member of the human-inspired metaheuristic family, originally proposed by Ali Wagdy et al. in 2020 [16]. It models the natural process of human learning and knowledge transfer, structured into two main stages: the junior phase, where knowledge is accumulated, and the senior phase, where it is shared. This intuitive and behaviorally grounded design has allowed GSK to tackle various real-world problems with promising results. However, like many algorithms, it has its shortcomings. GSK tends to suffer from premature convergence, often getting stuck in local optima, which limits its exploration capabilities. Moreover, its performance is highly sensitive to a set of manually tuned parameters. These parameters are usually determined through trial and error, a process that can be time-consuming, inconsistent, and difficult to generalize across different types of optimization challenges.

To overcome the parameter sensitivity issue in the original GSK algorithm, this paper introduces an Auto-Tuning Memory-based Adaptive Local Search (ATMALS) empowered GSK, that is, ATMALS-GSK. This improved version enhances GSK through two main innovations: memory-based automatic parameter tuning and an adaptive local search strategy. Unlike the traditional GSK, ATMALS-GSK dynamically adjusts its key parameters during the optimization process using a Gaussian-based mechanism. This mechanism updates the probability of selecting different parameter values based on their historical contribution to improving fitness. A memory-driven roulette wheel selection (RWS) method, powered by a weighted moving average, guides this process, ensuring the algorithm intelligently favors more effective parameter settings over time. In addition, an adaptive local search strategy is applied to the senior solutions, helping the algorithm better balance exploration and exploitation, escape local optima, and improve overall solution quality. These enhancements work to improve robustness and reduce premature convergence. The performance of ATMALS-GSK is evaluated through experiments on the CEC 2017 benchmark functions. Specifically, the key contributions of this paper can be mentioned as follows:

• Proposing an adaptive Auto-Tuning Memory-based GSK (ATM-GSK) algorithm, which enhances the original GSK algorithm with dynamic parameter adaptation.
• Integration of historical memory-based adaptiveness using a Gaussian-based weighted moving average mechanism that evaluates and adjusts parameter values based on their impact on the fitness function over time.
• Development of a novel selection mechanism using a memory-driven RWS strategy that leverages exponential memory-based knowledge and weighted moving averages to prioritize more effective parameter values. This adaptive tuning strategy improves the ability of the algorithm in escaping local optima and enhancing the exploration–exploitation balance.
• Incorporating an adaptive local search strategy to improve the exploitation capabilities of the algorithm. This approach enhances solution quality, preserves population diversity, and increases the ability of the algorithm to avoid trapping in local optima.
• Validation of the ATM-GSK algorithm through experiments on CEC 2011 and CEC 2017 benchmarks. Results show that ATM-GSK outperforms the original GSK and other metaheuristic algorithms in terms of convergence accuracy and computational robustness.

The remainder of this study is structured as follows. Section 2 presents a review of the related literature and highlights the existing approaches in the field. Section 3 introduces the ATMALS-GSK algorithm, outlining its key mechanisms and innovations. Section 4 provides a comprehensive performance evaluation through testing on CEC 2011 and CEC 2017 benchmark functions. Finally, Section 5 concludes the study and suggests potential directions for future research.

2. Literature Review

2.1. Metaheuristic Optimization Algorithms

Numerical optimization algorithms are generally classified into two broad categories: deterministic and stochastic approaches [2], as illustrated in Figure 1. Deterministic algorithms operate under the assumption that the problem model is completely defined, with a clear and explicit link between potential solutions and their associated outcomes. Prominent examples of such algorithms include gradient descent, conjugate gradient methods, mixed-integer nonlinear programming, and the branch and bound technique. These methods typically leverage gradient information to navigate the solution space and are known for their robustness. Exact algorithms can guarantee optimal solutions, provided the problem is not NP-hard, in which case they may become computationally impractical. For complex, NP-hard problems commonly encountered in the real world, approximate methods (especially metaheuristic algorithms) have become increasingly popular. These approaches introduce randomness and probabilistic reasoning to efficiently explore the solution space. Rather than seeking the perfect solution, their goal is to find a sufficiently good one within a reasonable timeframe.

As depicted in Figure 1, metaheuristic algorithms are commonly grouped into four categories: evolutionary, swarm intelligence, physics-based, and human-based methods. Among evolutionary algorithms, traditional techniques such as Genetic Algorithms (GA) and Differential Evolution (DE) remain widely used. More recent innovations include the Covariance Matrix Adaptation Evolution Strategy with Adaptive Population (CMAES-APOP) [17], the Love Evolution Algorithm (LEA) [18], and the Multi-Adaptation DE (MadDE) [19].

Swarm intelligence-based algorithms have also found widespread use in engineering and optimization domains. Notable methods in this group include the Particle Swarm Optimization (PSO) [20], Whale Optimization Algorithm (WOA) [21], Quantum-based Avian Navigation Optimizer (QANA) [22], Starling Murmuration Optimizer (SMO) [23], and a modified version of Artificial Rabbits Optimization called MNEARO [24].

Physics-inspired algorithms draw their strategies from physical phenomena. These include Nuclear Reaction Optimization (NRO), modeled after nuclear interactions [25]; the Flow Regime Algorithm (FRA), inspired by principles of fluid dynamics [26]; Plasma Generation Optimization (PGO), based on plasma generation processes [27]; and Colliding Bodies Optimization (CBO), which mimics the physics of object collisions [28]. Another example is the Doppler Effect–Mean Euclidean Distance Threshold (DE-MEDT), which draws from the Doppler shift phenomenon [29].

Human-based metaheuristics are another fascinating class that has shown promise across various disciplines, including engineering, healthcare, and data analysis. These methods are inspired by social behaviors and cognitive processes. Some widely recognized human-based algorithms are Brain Storm Optimization (BSO) [30], Poor and Rich Optimization (PRO) [31], Teamwork Optimization Algorithm (TOA) [32], Skill Optimization Algorithm (SOA) [33], and the Gaining–Sharing Knowledge-Based Algorithm (GSK) [16].

2.2. GSK and Its Variants

GSK stands as a notable human-inspired metaheuristic algorithm that has successfully tackled numerous complex real-world optimization problems. However, it does have limitations. It often struggles to escape local optimal points and shows high sensitivity to parameter settings. In response, researchers have proposed various enhancements, which can be grouped into four categories: population improvement, hybridization techniques, strategy enhancement, and composite improvements. Each of these directions has brought meaningful contributions, as detailed below.

2.2.1. Population Enhancement Techniques

To boost GSK’s convergence and diversity, Jalali et al. [34] integrated opposition-based learning and introduced Cauchy mutation to diversify the search. Ma et al. [35] developed an enhanced version of the GSK algorithm, named mGSK-DPMO, which incorporates a dual-population framework and multiple operators to adaptively improve the algorithm’s performance. Bakir et al. [36] introduced FDBAGSK—an adaptive variant that balances fitness and distance from the best solution to enhance search capabilities. In another development, Pan et al. [37] proposed the Parallel Opposition-based GSK (POGSK), segmenting the population into multiple groups and employing the Taguchi method for cross-group knowledge sharing. Similarly, Kamarposhti et al. [38] proposed the Adaptive Population GSK-Improved Multi-Operator DE (APGSK-IMODE) algorithm to identify the optimal locations and operational states of switches for these enhancements. Furthermore, Liang and Wang introduced the EPD-GSK framework [39], an enhanced version of the GSK algorithm designed to prevent premature convergence due to loss of population diversity. EPD-GSK improves diversity through three key strategies: using Sobol sequences for diverse initialization, applying Cauchy mutation to the junior stage to broaden the search space, and employing reverse learning in the senior phase to help escape local optima.

2.2.2. Hybrid Algorithms

A growing trend in GSK improvement involves hybridization with other algorithms. Nabahat et al. [40] proposed integrating the GSK and WOA algorithms for noise reduction filter design. Nunes et al. [41] introduced a fusion of GSK and PSO, named GSKPSO, for the parameter estimation of lithium-ion batteries based on impedance data. Zhong et al. [42] hybridized GSK with the Harris Hawks Optimization (HHO), incorporating mutation operators to balance exploration and exploitation. In a similar vein, Krishnan and Thiyagarajan [43] combined GSK with DE, using DE to escape local optima after the senior sharing phase. Liang et al. [44] blended the WOA with GSK to enhance exploration and minimize early convergence.

2.2.3. Strategy Enhancement Techniques

To make GSK more effective in binary search spaces, several strategy-based improvements have been introduced. For example, Wagdy Mohamed et al. [45] proposed an enhanced GSK with adaptive parameters (APGSK) by introducing an adaptive parameter control mechanism. This modified version dynamically adjusts the knowledge ratio and knowledge factor based on the population’s diversity and progression, thereby simulating knowledge acquisition over time. Additionally, the algorithm employs a nonlinear population size reduction and starts with a larger initial population to strengthen exploration.

2.2.4. Composite Enhancement Techniques

The Adaptive GSK (AGSK) algorithm [46] utilizes adaptive controls for knowledge factor (k_f) and knowledge ratio (k_r), and adopted a population reduction mechanism for performance tuning. This variant enhances GSK performance in many ways. However, the strategy improvement variants are designed to solve the binary space problems and do not significantly improve the single-objective optimization performance of GSK. Numerous studies have focused on improving search strategies and achieved excellent performance [47]. Furthermore, Xie et al. 2025 [6] have introduced an adaptive variant named HPE-AGSK, which integrates historical probability expansion to enhance both local and global search capabilities. The key points include the implementation of an expansion sharing strategy in the junior phase, a historical probability expansion strategy in the senior phase, and a reverse gaining strategy in the early iterations to promote population diversity and prevent premature convergence.

2.3. Our Contributions Against the Existing Literature

While the existing literature presents numerous advances in metaheuristic algorithms, most continue to suffer from three critical limitations: the static nature of their control parameters, the lack of historical memory-based adaptability, and the lack of local search mechanisms in continuous-based search-space metaheuristics. In particular, the original GSK algorithm, although promising in concept and application, relies on fixed parameter settings that limit its performance to balance exploration and exploitation in a dynamic manner. This static design of the controllable parameters in GSK typically leads to premature convergence and suboptimal performance in complex or high-dimensional optimization problems. Furthermore, the absence of mechanisms to utilize historical information prevents GSK from automatic tuning during the search process.

To address these gaps, the proposed ATMALS-GSK algorithm introduces a novel framework that integrates auto-tuning, memory-based learning, and adaptive local search within the GSK paradigm. Unlike prior efforts that focused on trial-and-error tuning or simplistic parameter adaptation, ATMALS-GSK employs a Gaussian-driven, memory-aware tuning mechanism that dynamically updates parameter probabilities based on their historical impact on the objective function. Additionally, a RWS strategy, powered by exponential memory and a weighted moving average, intelligently prioritizes parameter configurations that have proven effective over time. Complementing these enhancements is an adaptive local search mechanism applied during the senior knowledge-sharing phase, enabling deeper exploration of promising regions and escape from local optima. Together, these contributions not only overcome the limitations of traditional GSK but also advance the field of human-inspired metaheuristics by introducing a robust, self-adaptive model that significantly outperforms existing methods on both benchmark functions and real-world engineering problems.

3. Proposed ATMALS-GSK Algorithm

In this section, at first, the original GSK algorithm is described (Section 3.1), and then, the proposed ATMALS-GSK algorithm is presented in detail (Section 3.2). The proposed ATMALS-GSK algorithm has three innovations compared to the GSK for enabling it with auto-tuning of the controllable parameters, memory-based knowledge enhancement, and adaptive local search-empowered search strategies. These enhancements are proposed in Section 3.2.1, Section 3.2.2, Section 3.2.3, Section 3.2.4 and Section 3.2.5.

3.1. GSK

The GSK algorithm begins with a randomly generated population where individuals, seen as “juniors,” lack prior experience. As the optimization evolves, these individuals interact with their environment. Through these interactions, they gain knowledge and eventually reach a “senior” stage, where they can share what they have learned with others, enhancing the overall intelligence of the population. This learning process is divided into the junior phase (beginner-intermediate) and the senior phase (intermediate-expert).

During the search process, individuals (i.e., solutions) continuously interact and influence one another through cooperation and competition. In the early stage of life, knowledge is mostly acquired from close, familiar networks, while the ability to judge others is still underdeveloped. As individuals engage with broader social and professional networks, they gain diverse experiences and develop the capacity to assess, categorize, and share knowledge with others. This transition enhances both personal learning and the collective performance of the population. The mathematical formulation of this process is detailed below. The overall pseudocode of GSK can be seen in Algorithm 1.

Let $x_j,\forall j=\mathrm{1,2},3,\dots .N$ represent feasible solutions (i.e., individuals), where N is the population size of the algorithm. Each solution $x_j$ is characterized using a vector $x_{ij}=(x_{j1,} x_{j2,}\dots ,x_{jD}$), where D denotes the number of knowledge domains or dimensions assigned to that person, i.e., decision variables. The corresponding fitness value for each individual j is given by $f_j$ $\forall j=\mathrm{1,2},3,\dots .N$. The vector representation of an individual j during the junior phase (early-middle stage) is illustrated in Figure 2a, while its representation during the senior phase (middle-later stage) is depicted in Figure 2b.

Figure 2 illustrates that in the junior gaining and sharing phase (early-middle stage), more dimensions of each vector are updated compared to the senior phase. In contrast, during the senior phase (middle-later stage), the senior scheme updates more dimensions than the junior scheme. The number of dimensions updated in both phases depends on the knowledge rate, which controls how much knowledge is transferred between generations. The specific number of updated dimensions in each phase is determined using a nonlinear formula that adjusts over time, which can be formulated as follows:

$$D(Junior)=ProbSize\times {\left(1-{\displaystyle \frac{G}{MaxGen}}\right)}^K$$

(1)

$$D(Senior)=ProbSize-D(Junior)=ProbSize\left(1-{\left(1-{\displaystyle \frac{G}{MaxGen}}\right)}^K\right)$$

(2)

where K (K > 0) is a real number denoting the knowledge rate, ProbSize is the problem size (dimensions), G is the iteration number, and MaxGen is the maximum number of iterations.

Algorithm 1. Overall operation of original GSK Algorithm

Inputs: - N: Population size - D: Number of dimensions (decision variables) - K: Knowledge rate (controls the volume of knowledge exchange) - Kf: Knowledge factor (controls intensity of learning) - Kr: Knowledge ratio (probability of applying knowledge update) - MaxGen: Maximum number of generations - P: Proportion for grouping in senior phase Output: Global best solution (Xbest) 1   Initialize generation counter G ← 0 2   Generate an initial population of N individuals: 3      for 1 each i = 1:N 4         Initialize individual xi randomly in D-dimensional space 5         Evaluate fitness f (xi) for each individual i 6         for 2 each generation G = 1 to MaxGen: 7            Compute knowledge-sharing dimensions for both phases 8            Determine D(Junior) and D(Senior) using Equations (1) and (2). 9               Apply Junior Gaining–Sharing Knowledge Phase: 10              - For each individual i: 11              - Select nearest better (X_better) and worse (X_worse) individuals 12              - Choose a random peer (xr) for exploration 13              - Update D(Junior) dimensions based on the rules in Algorithm 2 14             Apply Senior Gaining–Sharing Knowledge Phase: 15              - Sort the population by fitness into Top P%, Middle N − 2P%, and Bottom P% 16              - For each individual i: 17             Select one peer from each group: X_best, X_mid, and X_worst 18             Update D(Senior) dimensions based on the rules in Algorithm 3 19             Evaluate new fitness values for all individuals 20             Update individuals based on new fitness: 21             Update the global best solution if a better one is found: 22       end for 2 23    end for 1 24  Return the global best solution found over all generations (X_best)

3.1.1. Junior Phase

In the junior phase, each solution primarily acquires knowledge from nearby individuals within small, familiar groups while also sharing with an outsider—someone not part of any group—driven by curiosity and a desire to explore. To update an individual’s position, the entire population is first sorted considering their objective function values in ascending order: $x_{best}$, …, $x_{worst}$. For a given individual x_i, the nearest better person ($x_{i-1}$) and the nearest worse person ($x_{i+1}$) are selected as knowledge sources, and a random individual x_r is chosen for sharing. Special handling is applied for the best and worst solutions: the best individual uses the next two best as references, while the worst uses the two just above it. This process is formalized in the pseudo-code of Algorithm 2, which outlines the junior knowledge update mechanism.

Algorithm 2. Junior phase of GSK
1	for 1 each solution i = 1:PopSize
2	for 2 each dimension j = 1:D
3	Generate a random number RND ∈ [0, 1]
4	if 1 RND ≤ Kr (Knowledge Ratio)
5	if 2 ${f(x}_i)>{f(x}_r)$
6	$\mathrm{Update} x_{ij}^{new}$$= x_i$$+ K_f$$\times \left[\right(x_{r }$$- x_{i }$$) + (x_{i-1}$$- x_{i+1})$];
7	else if 2
8	$\mathrm{Update} x_{ij}^{new}$$= x_i$$+ K_f$$\times [{(x}_i$$- x_r$$) + (x_{i-1}$$- x_{i+1})$];
9	end if 2
10	else if 1
11	$\mathrm{Keep} \mathrm{current} \mathrm{value}: x_{ij}^{new}=x_{ij}^{old}$;
12	end if 1
13	end for 2
14	end for 1

3.1.2. Senior Phase

In the senior phase, individuals utilize knowledge from various groups within the population, specifically, the best, middle, and worst individuals. This phase emphasizes how individuals are influenced by both high- and low-performing peers, simulating real-world judgment and selective knowledge exchange. First, the solutions are sorted based on their objective function values. Then, they are divided into three categories: the top P% as the best, the bottom P% as the worst, and the remaining N − 2 × P% as middle-performing individuals. The senior phase is illustrated in the pseudo-code of Algorithm 3.

For each individual x_i, the knowledge update involves three sources: two vectors randomly selected from the best and worst groups (to simulate learning from extremes) and one from the middle group (to share or refine knowledge). If a random value RND is less than or equal to the knowledge ratio K_r, and the individual’s fitness is worse than that of the selected middle peer, its position is updated based on both extremes and the difference with the middle peer. Otherwise, the sharing component is reversed. If the random value RND exceeds K_r, no update occurs, and the individual’s value remains unchanged. The knowledge factor K_f regulates the amount of knowledge transferred, while the knowledge ratio K_r determines the likelihood of an update at each generation. Furthermore, P $\in$ [0, 1] represents the percentage of the individuals.

Algorithm 3. Senior phase of GSK
1	for 1 each individual i = 1:PopSize
2	for 2 each dimension j = 1:D
3	Generate a random number RND ∈ [0, 1]
4	if 1  $\mathit{RND} \le K_r$ (Knowledge Ratio)
5	if 2 ${f(x}_i)>{f(x}_r)$
6	$\mathrm{Update} x_{ij}^{new}$$= x_i$$+ K_f$$\times [{(x}_m - x_i$$) + (x_{p-best} - x_{p-worst})$];
7	else if 2
8	$\mathrm{Update} x_{ij}^{new}$$= x_i$$+ K_f$$\times \left[\right(x_i - x_m$$) + (x_{p-best} - x_{p-worst})$];
9	end if 2
10	else if 1
11	$\mathrm{Keep} \mathrm{current} \mathrm{value}: x_{ij}^{new}=x_{ij}^{old}$;
12	end if 1
13	end for 2
14	end for 1

3.2. ATMALS-GSK

As previously discussed, the original GSK algorithm is highly sensitive to a set of manually defined controllable parameters: K_f, K_r, K, and P. These parameters were originally set to fixed values based on empirical tuning as K = 10, K_f = 0.5, K_r = 0.9, and P = 0.1. However, this trial-and-error approach is not only time-consuming and tedious but also inconsistent and difficult to generalize across different problems. One of the key limitations of the original GSK is its reliance on these static parameter settings, which inherently restrict the ability of the algorithm to adaptively balance exploitation and exploration throughout the search process. This rigid configuration can result in suboptimal performance, particularly when tackling complex problems. Several underlying issues contribute to this limitation:

• Lack of generalizability across problem domains: Fixed parameter values might yield strong results for a particular problem but often fail to perform well across different problem settings or objective landscapes. What works for a scheduling problem, for instance, may underperform in a continuous optimization task.
• Inflexibility across search stages: The same parameter values are used throughout the entire execution of the algorithm, from the initial exploration phase to the final convergence phase. However, different phases of the search process typically benefit from different behaviors—strong exploration in early generations and refined exploitation in later ones. Static parameters fail to accommodate this shift, potentially leading to premature convergence or excessive wandering.
• No feedback from historical performance: The original GSK lacks a learning mechanism to leverage past search experience. Without any form of memory or adaptive control, the algorithm is unable to assess the effectiveness of different parameter configurations over time. This absence of self-tuning prevents the algorithm from evolving and improving its own strategy dynamically.

To overcome the mentioned challenges, there is a need to design an adaptive historical memory-based version of GSK that can iteratively adjust and update its parameters in response to the current state of the search process, based on the problem specifications. This kind of smart tuning would make the algorithm more flexible, robust, and capable of handling a wider range of problems more effectively. To fill this gap, the proposed ATMALS-GSK algorithm introduces a new framework that combines automatic parameter tuning, memory-based learning, and adaptive local search into the GSK structure. The key idea is to boost GSK’s performance by learning from past experiences and adjusting its behavior accordingly.

Unlike the original GSK algorithm, ATMALS-GSK does not rely on fixed parameter settings. Instead, it uses a Gaussian-based mechanism to dynamically update the selection probabilities of different parameter values, based on how well they have performed in the past. A memory-guided RWS method, using a weighted moving average, helps the algorithm gradually favor more effective parameter settings over time. It is important to note that the ATMALS-GSK algorithm uses a memory-guided RWS strategy to adapt parameter values—not to select individuals, as is typically performed in traditional algorithms like GA. Specifically, this RWS utilizes a Gaussian-weighted moving average to intelligently guide the selection of optimal values for five key parameters: K_f, K_r, K, P, and P_LS. By integrating memory-driven adaptation into the parameter control process, the algorithm enhances its ability to fine-tune performance in a data-driven and responsive manner.

In addition, ATMALS-GSK applies an adaptive local search to the top-performing (senior) solutions. This helps avoid getting stuck in local optima and ultimately leads to better results. The overall operation of the proposed ATMALS-GSK algorithm can be seen in Figure 3.

The initial value, range of variation, and discretized level for each parameter in the ATMALS-GSK algorithm are provided in

Table 1. Comparison of fixed and adaptive parameters in GSK and ATMALS-GSK algorithms.

Parameter	GSK	ATMALS-GSK	Range of Variation	Discretized Levels
	Fixed Value	Initial Value
K	10	10	[5, 30]	1
Kf	0.5	0.5	[0.2, 0.8]	0.1
Kr	0.9	0.9	[0.85, 1]	0.01
P	0.1	0.1	[0.05, 0.15]	0.01
PLS	N/A	1	[1, 2]/ProbSize	0.1
NLS	N/A	0.05 × PopSize	N/A	N/A
RPG	N/A	5	[3, 5]	N/A

. In this table, the parameter P_LS represents the probability of applying a local search change for each decision variable of the N_LS randomly selected solutions at every iteration of the ATMALS-GSK algorithm.

3.2.1. Adaptive K

The knowledge rate K serves as a key parameter in the GSK algorithm, controlling how much experience an individual accumulates throughout generations in the junior and senior phases. It directly affects the percentage of dimensions updated by each phase over time. As shown in Figure 4 and detailed in

Table 2. Comparison of the number of junior and senior dimensions for different values of K (problem size = 100).

K	G	D (Junior)	D (Senior)
0.5	0	100	0
	0.25 MaxGen	87	13
	0.5 MaxGen	71	29
	0.75 MaxGen	50	50
	MaxGen	0	100
1	0	100	0
	0.25 MaxGen	75	25
	0.5 MaxGen	50	50
	0.75 MaxGen	25	75
	MaxGen	0	100
2	0	100	0
	0.25 MaxGen	56	44
	0.5 MaxGen	25	75
	0.75 MaxGen	6	94
	MaxGen	0	100
5	0	100	0
	0.25 MaxGen	24	76
	0.5 MaxGen	3	97
	0.75 MaxGen	1	99
	MaxGen	0	100

, different values of K result in distinct learning dynamics. When K = 1, the transition between junior and senior phases is smooth and linear. The number of junior dimensions decreases steadily, while the number of senior dimensions increases at the same rate, perfectly balancing exploration early on and gradually shifting to exploitation.

However, this linear behavior changes when K ≠ 1: For K < 1 (e.g., K = 0.5), the decline in junior updates is slow and non-linear, meaning the algorithm relies more heavily on exploration for a longer time. For instance, at 50% of the total generations, 71% of the dimensions are still updated by the junior scheme, with only 29% by the senior scheme. This results in gradual knowledge acquisition, allowing broader search of the solution space. Furthermore, for K > 1 (e.g., K = 2 or 5), the shift from junior to senior updates is faster and more aggressive. At K = 2, by halfway through the run, only 25% of dimensions are updated by the junior phase, and 75% by the senior. With K = 5, this shift is even more dramatic—only 3% of dimensions are updated via the junior scheme at the halfway point. This leads to faster convergence but may risk premature exploitation.

These patterns reveal that K functions as a learning rate controller within the GSK algorithm. Lower values of K promote slower, more cautious learning by extending the influence of the junior phase and encouraging prolonged exploration of the solution space. In contrast, higher K values accelerate the learning process by shifting focus to the senior phase earlier, thus enabling faster and more targeted exploitation. The selected K value directly influences how updated dimensions are allocated between the junior and senior schemes in each generation. These distributions are initialized at the start and adapt automatically as the algorithm evolves, underscoring the critical role of K in striking the right balance between exploration and exploitation based on the specific characteristics of the problem at hand.

In the proposed ATMALS-GSK algorithm, the range of K is considered within [5, 30] with a discretized level of 1. A historical weighted moving average function is used to adaptively tune the value of K based on an exponential function of the improvement rates of the different values of K in previous iterations. Weighted moving average [48] is a trend-following index used for smoothing the short-term variation of the parameter, which can be obtained on the basis of the past effectiveness of the different values of the parameter. The exponential weighted moving average function considers the weights that are assigned exponentially reducing for previous values, in which the closest values are of more importance.

After fitness evaluation for all individuals at every generation G, a Gaussian enhancement rate is applied to the different values of K as follows:

$$ER(K_{VALUES},G)=\left\{\begin{array}{l}GF(K_{sel})\times \exp \left(-{\displaystyle \frac{F_{new}-F_{old}}{F_{old}}}\right) if F_{new}<F_{old}\\ 0 if F_{new}=F_{old}\\ \left(1-GF(K_{sel})\right)\times \exp \left(-{\displaystyle \frac{F_{new}-F_{old}}{F_{old}}}\right) if F_{new}>F_{old}\end{array}\right.$$

(3)

where F_new and F_old are the average fitness (cost) values of the top 50% of the population in the current generation G and previous generation G − 1, respectively. Furthermore, G is the current generation (iteration), and GF(K_sel) is a Gaussian function with a central point at the current selected value of K (i.e., K_sel) and standard deviation (STD) of δ, which can be calculated as follows:

$$GF(K_{sel})=\exp \left(-{\displaystyle \frac{{\left(K_{VALUES}-K_{sel}\right)}^2}{2{\left(\delta \times \left(max(K_{VALUES})-min(K_{VALUES})\right)\right)}^2}}\right)$$

(4)

where K_sel is the current selected value of K (peak of the Gaussian function), and K_VALUES are the discretized values of K within [5, 30], i.e., K_VALUES = {5, 6, 7, …, 30}. In our experiments, we specified the STD of the Gaussian distribution as δ = 0.05 × (max(K_VALUES) − min(K_VALUES)). As an example for K_sel = 5, GF(K_sel) with different STDs can be illustrated as Figure 5.

Accordingly, the selection probability for each value v for K at generation G is calculated according to the weighted moving average of the Gaussian enhancement rates at the previous generations, which can be formulated as follows:

$$\begin{array}{l}Prob(K_{VALUES},G)={\displaystyle \frac{{\displaystyle \sum _{t=1}^G\alpha tER(K_{VALUES},G-t)}}{{\displaystyle \sum _{t=1}^G{\alpha }^t}}}\\ ={\displaystyle \frac{\alpha ER(K_{VALUES},G-1)+{\alpha }^{2}ER(K_{VALUES},G-2)+\dots +{\alpha }^{G}ER(K_{VALUES},0)}{\alpha +{\alpha }^2+\dots +{\alpha }^G}}\end{array}$$

(5)

In Equation (5), α is a constant (0 < α ≤ 1), which tunes the exponential type in the moving average method. The maximum value of α = 1 means simple averaging is applied to the previous enhancement rates. The higher the value of α, the more previous generations will be accounted for in the calculations. Typically, this parameter is set close to 1 [48]. An example considering α = 0.97 is illustrated in Figure 6.

At every generation during the optimization process, once the probability of the different values of K is computed according to their historical contributions to fitness improvement, a powered RWS mechanism is applied to guide the choice of K for the next update. According to this selection strategy, the probability of selecting the value of v for K (${Val}_v^K$) is calculated as follows:

$$ProbRP({Val}_v^K,G)={\left({\displaystyle \frac{Prob(Va{l}_v^K,G)}{{\displaystyle \sum _{j\in K_{VALUES}}Prob(Va{l}_j^K,G)}}}\right)}^{RP(G)}$$

(6)

where RP_G (RP_G > 1) is the RWS power at generation G. If RP_G = 1, the selection strategy is a simple RWS. The higher the RP, the more chance of elitism selection. To refine this mechanism over time, a gradually increasing RP is introduced according to Equation (7). In the early stages, RP_G starts low, resulting in a flatter probability distribution that allows for broader exploration across a wider range of K values, even those with lower past performance. As the algorithm progresses and gains more insight into what works best for the problem at hand, RP_G is gradually increased, sharpening the selection focus toward more historically successful values. This dynamic adjustment shifts the balance toward exploitation in later stages, helping the algorithm converge more effectively while still avoiding premature stagnation. It makes the proposed ATMALS-GSK algorithm more intelligent throughout the search process.

$$R{P}_G=R{P}_{\min }+\left(R{P}_{\max }-R{P}_{\min }\right)\times \left(1-{\displaystyle \frac{G}{MaxGen}}\right)$$

(7)

3.2.2. Adaptive K_r

The knowledge ratio (K_r) is a key parameter in the GSK algorithm that determines the proportion of individuals participating in the senior (exploitative) versus junior (explorative) phases. Its value typically lies within the range (0, 1], and it’s often set around 0.9 in the original GSK. When K_r is low (e.g., 0.3–0.6), more individuals are assigned to the junior phase, leading to broader exploration. This supports preserving variety within the population and is particularly useful for complex or multimodal problems. However, it may also slow down convergence. As K_r increases (e.g., 0.7–0.9), the algorithm shifts focus toward the senior phase, improving exploitation and accelerating convergence. This is effective for problems where refining known good solutions is more valuable than broad search. With very high values of K_r (close to 1), nearly the entire population engages in the senior scheme. Although this enhances exploitation and can lead to quicker convergence in simpler or well-known problems, it significantly limits diversity and raises the chances of converging too early to suboptimal solutions.

Finding the right balance for the knowledge ratio K_r is essential, as it directly influences how much of an individual’s knowledge is retained versus how much is gained from others during the learning process. To address the limitations of setting a fixed value for the knowledge ratio, the ATMALS-GSK algorithm adaptively adjusts K_r during the optimization run, allowing it to dynamically respond to the algorithm’s progress and the problem landscape. This adaptive adjustment is driven by historical improvement trends mirroring the mechanisms used for tuning K in Equations (3)–(7), which ensures that K_r evolves based on the success rate of previous generations. By allowing K_r to adaptively vary within [0.85, 1] with a discretized level of 0.01, the ATMALS-GSK algorithm intelligently balances knowledge retention and acquisition, improving both convergence speed and solution quality across diverse optimization problems.

3.2.3. Adaptive K_f

The knowledge factor (K_f) in the GSK algorithm governs the intensity with which individuals move toward their knowledge sources during the learning process. Serving as a scaling factor, K_f directly influences the magnitude of updates in both the junior and senior phases. With its range defined over (0, ∞), different values of K_f induce diverse search behaviors. Smaller values (e.g., 0.1–0.4) encourage fine-grained, cautious movements, favoring local refinement and stability—particularly beneficial in complex or noisy problem spaces. Moderate settings (around 0.5–1) offer a balanced trade-off, promoting steady convergence without excessive risk, which is often ideal for general-purpose optimization tasks. On the other hand, higher values of K_f (e.g., 2, 5, or even 10+) induce more aggressive movements toward knowledge sources, enabling faster exploration and improving the algorithm’s chances of escaping local optima early in the search. However, such large steps can lead to instability or overshooting if not regulated.

In the proposed ATMALS-GSK framework, K_f is adaptively tuned within a practical range of [0.2, 0.8] with a discretized level of 0.1, allowing the algorithm to adjust its aggressiveness over time. To support this adaptivity, the same Gaussian-based probability function used in Equations (3)–(7) is applied to K_f as well, enabling intelligent learning from historical performance and dynamically steering the search behavior. This adaptive mechanism ensures that K_f evolves with the problem landscape, contributing to a robust balance between effective exploration and reliable convergence.

3.2.4. Adaptive P

The parameter P, ranging within [0, 1], determines the probability of selecting individuals to participate in knowledge sharing during both the junior and senior learning phases. In practice, P is typically set to values less than 0.5, as lower values promote more selective and targeted interactions. This cautious approach helps the algorithm avoid excessive imitation, maintain diversity, and support broader exploration of the search space. In the proposed ATMALS-GSK algorithm, P is adaptively adjusted during the execution of the algorithm, typically within a refined range of 0.01 to 0.5, depending on the evolving needs of the search. Keeping P relatively small ensures that only a selective subset of individuals engages in knowledge exchange at each generation, preventing premature convergence and preserving healthy exploratory behavior, especially in early iterations.

To adaptively control P, the same Gaussian-based probability function used for tuning K in Equations (3)–(7) is applied, enabling the algorithm to learn from historical success and gradually fine-tune P as optimization proceeds, considering a range of variation for P as [0.05, 0.15] with a discretized level of 0.01. By restricting knowledge diffusion to high-quality individuals early on and allowing more sharing in later stages, the algorithm achieves a smooth transition from diverse exploration to focused exploitation, resulting in improved convergence reliability and solution quality across diverse problem domains.

3.2.5. Adaptive P_LS

In the proposed ATMALS-GSK algorithm, an adaptive local search mechanism is incorporated to improve solution quality through controlled mutation. This local refinement is designed to balance exploitation and exploration dynamically as the algorithm progresses. At each iteration G, N_LS solutions are randomly chosen from the entire population, and their variables are considered for local changing. The probability of mutating each decision variable in a selected individual (i.e., P_LS/ProbSize) is governed by the same Gaussian-based probability function used in Equations (3)–(7), considering a variation range of [1, 2] with a discretized value of 0.1.

Simultaneously, the range of local search, denoted Range_LS, shrinks linearly over generations, as shown in Equation (8), which means that mutations at the beginning are more exploratory, enabling the algorithm to explore broader areas of the search space. Over time, the range contracts, guiding the search toward local regions around promising solutions, which enhances the exploitation capabilities of the algorithm. This dynamic adjustment promotes intensive search and fine-tuning near convergence.

$$Rang{e}_{LS}(G)=Rang{e}_{LS,\max }-\left(Rang{e}_{LS,\max }-Rang{e}_{LS,\min }\right)\times {\displaystyle \frac{G}{MaxGen}}$$

(8)

Together, these two adaptive mechanisms (i.e., auto-tuning adaptiveness of the local search probability and shrinking the local search range) help the proposed ATMALS-GSK algorithm evolve from automatic adaptive broad global search to focused local refinement, improving its ability to find high-quality, well-converged solutions.

3.2.6. Overall Operation of ATMALS-GSK

The overall pseudo-code of the ATMALS-GSK algorithm is outlined in Algorithm 4. It integrates adaptive parameter tuning, memory-guided learning, and a probabilistic local search mechanism to enhance solution quality over generations. Key parameters (K, K_f, K_r, P, and P_LS) are dynamically selected using a Gaussian-enhanced probability scheme and powered RW S. The algorithm iteratively partitions the population into Junior and Senior groups, applies corresponding update rules, and performs local refinements to exploit promising areas of the solution space.

Algorithm 4. Proposed ATMALS-GSK Algorithm

Inputs:   - MaxGen: Maximum generations   - PopSize: Population size   - ProbSize: Problem dimension   - Fitness(): Objective function to minimize   - KVALUES ← [1, 2, 3, …, 30]   - KfVALUES ← [0.2, 0.3, …, 0.8]   - KrVALUES ← [0.8, 0.81, …, 1.0]   - PVALUES ← [0.05, 0.06, …, 0.2]   - PLSVALUES ← [0.1, 0.11, …, 0.2]   Initialize Probabilities for K, Kf, Kr, P, and PSL, using uniform distribution   Initialize     RPG ← 1,     NLS ← 0.05 × PopSize,     α ← 0.97,     δ ← 0.05 × (max(VALUES) − min(VALUES)) Main ATMALS-GSK Algorithm: 1  Initialize population of PopSize individuals randomly 2  for 1 generation G = 1 to MaxGen do: 3    Evaluate Fitness of all individuals 4    for 2 each parameter in {K, Kf, Kr, P, PLS} do: 5      Compute Gaussian enhancement rate based on Equations (3) and (4) 6      Update selection probabilities using Equation (5) 7      Apply powered RWS using Equations (6) and (7) to choose parameter values 8    end for 2 9    RPG ← Adjust RPG based on generation G (gradual increase) 10     Partition individuals into Junior and Senior based on adaptive Kr value 11     for 3 each individual i in population do: 12        if i is in Junior group then: 13           Update D(junior) using junior learning rule (based on K) 14        else if i is in Senior group then: 15           Update D(senior) using senior learning rule (based on K) 16        end if 17      end for 3 18      Update RangeLS(G) using Equation (8) 19      Select NLS individuals randomly 20      for 4 each selected individual in NLS do: 21        for 5 each dimension d do: 22           Apply adaptive local search with probability of PLS/ProbSize 23        end for 5 24      end for 4 25      Update historical memory of parameters’ fitness improvements 26      Update the global best solution found so far 27    end for 1 Output: Best solution found and its fitness value

4. Performance Evaluation

In this section, the detailed results of the ATMALS-GSK algorithm are provided and compared against the original GSK [16] and advanced GSK versions, including AGSK [46], APGSK [45], FDBAGSK [36], and eGSK [49], as well as recent metaheuristic algorithms, including IADE [50], GJO-JOS [51], QCSCA [52], cSM [53], and FOWFO [54].

4.1. Settings

In the original GSK algorithm, the parameters K, K_f, K_r, and P were tuned empirically using a trial-and-error process. This involved running multiple experiments on benchmark functions and selecting the parameter values that yielded the best average performance. However, this method is problem-specific and lacks generalizability. Therefore, the values tuned for one problem may not transfer well to different optimization problems, especially technical or real-world ones. In contrast, the ATMALS-GSK algorithm adaptively tunes these parameters during runtime based on historical performance, making it more robust and broadly applicable across different problem domains.

The effectiveness of the proposed ATMALS-GSK algorithm is assessed using the 22 functions from the CEC 2011 and 30 functions from the CEC 2017 test suites (including unimodal functions, multimodal functions, hybrid functions, and composition functions). The full details of these benchmark problems can be found in the technical reports of CEC 2011 [55] and CEC 2017 [56].

In all simulations on the CEC 2011 benchmark suite, we follow the official recommendation by setting the maximum number of function evaluations (FEs) to 150,000 [55] to ensure a fair comparison with other methods. With a fixed population size of 100, this results in 1500 iterations for each function (since FEs = population size × iterations). Each algorithm is run independently 25 times on each function to ensure statistical reliability. All settings for simulations on CEC 2011 have been considered based on the technical report of the CEC 2011 benchmark in [55].

To assess the algorithm’s performance on the CEC 2017 benchmarks, solution error is used as the evaluation metric, defined as f(x) − f(x*), where x is the best solution found in a single run and x* is the known global optimum. As recommended in [56], errors smaller than 10⁻⁸ are considered zero. Experiments are conducted across four problem dimensions: D = 10, 30, 50, and 100, with FEs set to 10,000 × D. Each algorithm is run independently 51 times per function for every dimension. These settings are based on the technical report in [56].

In the following, to provide a thorough evaluation, the experimental results are organized into four main subsections. Section 4.2 focuses on analyzing the performance of ATMALS-GSK on the CEC 2011 and CEC 2017 benchmark functions. Section 4.3 presents a comparison between ATMALS-GSK, the original GSK algorithm, and its other modified variants. In Section 4.4, ATMALS-GSK is benchmarked against several recent metaheuristics. Furthermore, Section 4.5 includes statistical tests using the Wilcoxon signed-rank and Friedman methods to verify the significance of the results.

4.2. Results of ATMALS-GSK

4.2.1. CEC 2011

The comparative analysis between the original GSK algorithm [16] and the proposed ATMALS-GSK algorithm on the CEC 2011 benchmark suite reveals several noteworthy improvements brought about by the proposed enhancements.

Table 3. Results of GSK and ATMALS-GSK for CEC 2011 benchmark functions.

Function	GSK [16]					ATMALS-GSK (Proposed)
	Best	Median	Worst	Mean	STD	Best	Median	Worst	Mean	STD
1	0.00	1.85 × 10−21	1.49 × 101	3.28	5.21	0.14 × 10−3	1.01 × 101	1.54 × 101	6.63	6.10
2	−1.35 × 101	−1.14 × 101	−9.25	−1.13 × 101	1.03	−2.40 × 101	−1.69 × 101	−1.19 × 101	−1.76 × 101	3.13
3	1.15 × 10−5	1.15 × 10−5	1.15 × 10−5	1.15 × 10−5	9.12 × 10−13	1.15 × 10−5	1.15 × 10−5	1.15 × 10−5	1.15 × 10−5	1.13× 10−20
4	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
5	−2.39 × 101	−2.08 × 101	−1.87 × 101	−2.06 × 101	1.21	−3.59 × 101	−2.89 × 101	−2.42 × 101	−2.97 × 101	3.31
6	−1.16 × 101	−6.85	0.00	−6.94	2.48	−2.91 × 101	−1.85 × 101	−7.69	−1.79 × 101	4.85
7	1.59	1.78	1.97	1.78	1.08 × 10−1	9.99 × 10−1	1.2	1.49	1.29	1.25 × 10−1
8	2.20 × 102	2.20 × 102	2.20 × 102	2.20 × 102	0.00	2.20 × 102	2.20 × 102	2.20 × 102	2.20 × 102	0.00
9	1.29 × 103	2.07 × 103	3.09 × 103	2.11 × 103	5.02 × 102	1.52 × 103	2.43 × 103	5.31 × 103	2.67 × 103	9.43 × 102
10	−2.18 × 101	−2.16 × 101	−2.14 × 101	−2.16 × 101	1.19 × 10−1	−2.18 × 101	−1.75 × 101	−1.30 × 101	−1.81 × 101	3.07
11	5.09 × 104	5.24 × 104	5.39 × 104	5.24 × 104	6.88 × 102	5.15 × 104	5.23 × 104	5.31 × 104	5.23 × 104	4.29 × 102
12	1.07 × 106	1.07 × 106	1.08 × 106	1.07 × 106	1.73 × 103	1.06 × 106	1.07 × 106	1.07 × 106	1.07 × 106	2.10 × 103
13	1.54 × 104	1.54 × 104	1.55 × 104	1.54 × 104	2.44	1.54 × 104	1.54 × 104	1.54 × 104	1.54 × 104	1.29 × 10−5
14	1.82 × 104	1.84 × 104	1.86 × 104	1.84 × 104	1.22 × 102	1.80 × 104	1.81 × 104	1.84 × 104	1.82 × 104	1.02 × 102
15	3.28 × 104	3.28 × 104	3.28 × 104	3.28 × 104	1.55 × 101	3.27 × 104	3.27 × 104	3.28 × 104	3.27 × 104	3.13 × 101
16	1.32 × 105	1.35 × 105	1.39 × 105	1.35 × 105	2.22 × 103	1.25 × 105	1.27 × 105	1.29 × 105	1.27 × 105	9.35 × 102
17	1.97 × 106	2.03 × 106	2.36 × 106	2.09 × 106	1.20 × 105	1.87 × 106	1.89 × 106	1.93 × 106	1.90 × 106	1.61 × 104
18	1.17 × 106	1.27 × 106	1.57 × 106	1.27 × 106	7.56 × 104	9.36 × 105	9.42 × 105	9.46 × 105	9.42 × 105	2476.651
19	1.75 × 106	2.03 × 106	2.25 × 106	2.00 × 106	1.36 × 105	9.37 × 105	9.47 × 105	9.76 × 105	9.47 × 105	7.47 × 103
20	1.12 × 106	1.29 × 106	1.46 × 106	1.29 × 106	9.20 × 104	9.35 × 105	9.41 × 105	9.47 × 105	9.41 × 105	3.28 × 103
21	1.34 × 101	1.61 × 101	2.49 × 101	1.70 × 101	3.11	1.52 × 101	1.95 × 101	2.40 × 101	1.92 × 101	2.11
22	8.61	1.28 × 101	2.09 × 101	1.29 × 101	2.93	8.61	1.63 × 101	2.39 × 101	1.61 × 101	3.97

reports the performance results in terms of Best, Median, Worst, Mean, and Standard Deviation (STD) across 25 independent runs, while

Table 4. Wilcoxon signed-rank test of ATMALS-GSK along with the original GSK for CEC 2011.

Algorithms	R+	R−	p Value	+	=	−	Dec.
ATMALS−GSK vs. GSK	127	26	0.0168	12	5	5	+

summarizes the statistical significance through the Wilcoxon signed-rank test.

According to the results obtained in Table 3, ATMALS-GSK demonstrates superior robustness and accuracy on a majority of the functions. For instance, in F2, the improvement is particularly noticeable; ATMALS-GSK reached a better minimum of −2.40 × 10¹ versus −1.35 × 10¹ by GSK, with a wider performance range indicating better exploration capacity. For multimodal functions like F6 and F7, which are known for their local optima traps, ATMALS-GSK consistently outperformed GSK in both mean and worst-case scenarios, showcasing its ability to avoid premature convergence. The mean value for F6 improved significantly from −6.94 to −1.79 × 10¹, and for F7, both mean and worst-case values decreased, reflecting more consistent convergence behavior. Notably, in some functions such as F3, F4, and F8, both techniques achieved identical or near-identical performances, indicating that these functions were relatively easy for both methods, leaving little room for improvement. The superiority of ATMALS-GSK becomes even more evident in complex hybrid and composition functions like F17 through F20. These functions typically combine the challenges of high dimensionality, non-separability, and rugged landscapes. On F17, the mean objective value decreased from 2.09 × 10⁶ to 1.90 × 10⁶, and the standard deviation dropped sharply, highlighting more reliable convergence. Similar improvements can be seen across F18 to F20, where ATMALS-GSK consistently provided better mean performance and significantly lower standard deviations.

The Wilcoxon signed-rank test in Table 4 confirms the overall effectiveness of ATMALS-GSK. With a P-value of 0.0168, the improvement is statistically significant at the 5% level. The test outcomes include 12 functions where ATMALS-GSK outperformed GSK, 5 where the two algorithms tied, and only 5 cases where GSK was better. The positive ranking sum (R+ = 127) far exceeds the negative sum (R− = 26), clearly supporting the conclusion that ATMALS-GSK provides a significant and consistent performance gain over the original GSK across diverse benchmark functions. The detailed analysis confirms that ATMALS-GSK is a promising enhancement over the baseline GSK.

As mentioned earlier, the proposed ATMALS-GSK algorithm incorporates adaptive parameters. Figure 7 shows the adaptive value of K across iterations for functions 1–6 as examples, along with the average value over all functions. Figure 8 illustrates the variations of the parameters K, K_F, K_R, P, and PLS during the algorithm’s execution. Additionally, Figure 9 presents the average convergence curves (over 25 runs) for the GSK and ATMALS-GSK algorithms on various functions from the CEC 2011 benchmark.

4.2.2. CEC 2017

The experimental results of the original GSK algorithm [16] and the proposed ATMALS-GSK algorithm over 51 independent runs are provided in

Table 5. Results of GSK and ATMALS-GSK for CEC 2017 benchmark functions in D = 10.

Function	GSK [16]					ATMALS-GSK (Proposed)
	Best	Median	Worst	Mean	STD	Best	Median	Worst	Mean	STD
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
3	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
4	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
5	1.53 × 101	2.01 × 101	2.58 × 101	2.03 × 101	2.79	2.99	8.47	1.00 × 101	7.57	2.71
6	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
7	2.46 × 101	3.10 × 101	3.74 × 101	3.07 × 101	3.08	1.27 × 101	1.69 × 101	1.85 × 101	1.63 × 101	2.21
8	1.45 × 101	1.99 × 101	2.58 × 101	2.02 × 101	2.92	3.98	7.97	1.09 × 101	7.57	2.11
9	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
10	7.54 × 102	1.10 × 103	1.29 × 103	1.06 × 103	1.33 × 102	1.40 × 102	5.07 × 102	6.57 × 102	4.55 × 102	1.93 × 102
11	0.00	0.00	1.02× 10−8	0.00	0.00	6.40 × 10−1	1.77	2.67	1.67	7.10 × 10−1
12	1.42 × 101	7.57 × 101	2.69 × 102	8.93 × 101	7.26 × 101	2.12 × 10−1	1.05 × 101	3.45 × 101	1.12 × 101	9.45
13	9.19 × 10−1	6.56	9.73	6.56	1.41	3.32 × 10−1	1.80	5.70	2.65	2.18
14	5.19 × 10−3	5.42	1.19 × 101	5.88	3.06	3.57 × 10−1	2.94	3.56	2.78	9.25 × 10−1
15	2.96 × 10−3	1.75 × 10−1	5.00 × 10−1	2.22 × 10−1	2.13 × 10−1	1.45 × 10−2	1.30 × 10−1	2.63 × 10−1	1.47 × 10−1	8.60 × 10−2
16	3.29 × 10−1	9.48 × 10−1	1.18 × 101	4.27	4.95	4.89 × 10−1	1.11	1.80	1.12	4.70 × 10−1
17	9.76 × 10−1	9.94	2.52 × 101	1.17 × 101	7.09	2.26	1.26 × 101	2.27 × 101	1.25 × 101	9.02 × 101
18	2.05 × 10−2	4.30 × 10−1	5.01 × 10−1	3.20 × 10−1	1.92 × 10−1	3.94 × 10−2	3.32 × 10−1	4.95 × 10−1	3.04 × 10−1	1.85 × 10−1
19	2.81 × 10−2	6.95 × 10−2	1.80	1.55 × 10−1	3.51 × 10−1	5.36 × 10−2	1.27 × 10−1	2.20 × 10−1	1.39 × 10−1	5.66 × 10−2
20	3.12 × 10−1	3.12 × 10−1	2.03 × 101	1.19	3.99	1.16	1.97	4.55	2.51	1.14
21	1.00 × 102	2.20 × 102	2.31 × 102	1.93 × 102	5.07 × 101	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	0.00
22	1.00 × 102	1.00 × 102	1.02 × 102	1.00 × 102	4.87 × 10−1	0.00	1.00 × 102	1.00 × 102	6.38 × 101	4.73 × 101
23	3.11 × 102	3.18 × 102	3.29 × 102	3.18 × 102	3.99	2.24 × 10−1	3.06 × 102	3.10 × 102	2.76 × 102	9.72 × 101
24	2.78 × 102	3.50 × 102	3.57 × 102	3.44 × 102	1.87 × 101	1.00 × 102	3.35 × 102	3.39 × 102	3.11 × 102	7.44 × 101
25	3.98 × 102	4.34 × 102	4.46 × 102	4.27 × 102	2.05 × 101	3.97 × 102	3.97 × 102	3.98 × 102	3.97 × 102	1.32 × 10−1
26	3.00 × 102	3.00 × 102	3.00 × 102	3.00 × 102	0.00	3.00 × 102	3.00 × 102	3.00 × 102	3.00 × 102	0.00
27	3.89 × 102	3.90 × 102	3.90 × 102	3.89 × 102	2.17 × 10−1	3.89 × 102	3.89 × 102	3.89 × 102	3.89 × 102	1.98 × 10−1
28	3.00 × 102	3.00 × 102	3.97 × 102	3.12 × 102	3.20 × 101	3.00 × 102	3.00 × 102	3.00 × 102	3.00 × 102	0.00
29	2.39 × 102	2.48 × 102	2.57 × 102	2.48 × 102	4.76	2.32 × 102	2.39 × 102	2.43 × 102	2.38 × 102	3.33
30	3.96 × 102	4.65 × 102	5.02 × 102	4.56 × 102	3.44 × 101	3.94 × 102	4.42 × 102	4.42 × 102	4.29 × 102	2.09 × 101

,

Table 6. Results of GSK and ATMALS-GSK for CEC 2017 benchmark functions in D = 30.

Function	GSK [16]					ATMALS-GSK (Proposed)
	Best	Median	Worst	Mean	STD	Best	Median	Worst	Mean	STD
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
3	0.00	5.66 × 10−8	8.54 × 10−6	7.07 × 10−7	1.82 × 10−6	0.00	0.00	0.00	0.00	0.00
4	9.28 × 10−3	4.00	7.25 × 101	1.11 × 101	2.17 × 101	0.00	1.99	3.98	1.99	2.10
5	1.36 × 102	1.62 × 102	1.73 × 102	1.60 × 102	8.87	1.08 × 101	2.67 × 101	9.72 × 101	1.53 × 101	1.90 × 101
6	0.00	5.47 × 10−7	8.42 × 10−6	1.52 × 10−6	2.36 × 10−6	2.26 × 10−5	5.71 × 10−5	1.39 × 10−4	6.82 × 10−5	3.90 × 10−5
7	1.64 × 102	1.87 × 102	1.99 × 102	1.87 × 102	8.40	2.52 × 101	5.60 × 101	9.83 × 101	4.33 × 101	1.15 × 101
8	1.24 × 102	1.56 × 102	1.73 × 102	1.55 × 102	1.11 × 101	1.07 × 101	2.34 × 101	5.06 × 101	1.35 × 101	1.29 × 101
9	0.00	0.00	0.00	0.00	0.00	0.00	0.00	8.95 × 10−2	1.79 × 10−2	3.77 × 10−2
10	5.84 × 103	6.68 × 103	7.16 × 103	6.69 × 103	3.54 × 102	1.63 × 103	2.96 × 103	3.49 × 103	2.80 × 103	5.90 × 102
11	6.00 × 10−1	8.57	9.85 × 101	3.30 × 101	3.83 × 101	9.94	2.04 × 101	7.19 × 101	1.46 × 101	1.73 × 101
12	1.03 × 103	5.32 × 103	1.91 × 104	6.62 × 103	4.59 × 103	9.02 × 102	3.12 × 103	8.19 × 103	3.84 × 103	2.62 × 103
13	4.61 × 101	9.11 × 101	1.82 × 102	9.83 × 101	3.42 × 101	2.06 × 101	3.91 × 101	6.34 × 101	3.01 × 101	1.12 × 101
14	4.72 × 101	5.73 × 101	6.65 × 101	5.69 × 101	5.49	1.04 × 101	1.79 × 101	4.57 × 101	1.78 × 101	4.26
15	2.13	9.17	7.48 × 101	1.45 × 101	1.49 × 101	5.88	1.75 × 101	1.92 × 101	1.43 × 101	5.56
16	4.35 × 102	7.76 × 102	1.15 × 103	7.96 × 102	1.94 × 102	1.97 × 101	5.47 × 102	7.50 × 102	2.53 × 102	2.81 × 102
17	6.28 × 101	2.04 × 102	3.77 × 102	1.89 × 102	9.44 × 101	2.92 × 101	6.37 × 101	1.41 × 102	4.71 × 101	2.30 × 101
18	2.25 × 101	3.69 × 101	4.80 × 101	3.68 × 101	5.42	2.25 × 101	2.74 × 101	4.02 × 101	3.02 × 101	6.67
19	3.42	1.13 × 101	2.22 × 101	1.29 × 101	6.02	8.03	1.37 × 101	1.65 × 101	1.30 × 101	3.34
20	1.34	5.51 × 101	4.51 × 102	1.08 × 102	1.14 × 102	3.73 × 101	5.58 × 101	1.62 × 102	4.74 × 101	3.35 × 101
21	3.20 × 102	3.49 × 102	3.55 × 102	3.46 × 102	8.30	2.51 × 102	2.65 × 102	2.79 × 102	2.66 × 102	9.12
22	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	0.00	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	0.00
23	3.50 × 102	4.86 × 102	5.07 × 102	4.70 × 102	4.49 × 101	3.97 × 102	4.28 × 102	4.43 × 102	3.44 × 102	1.70 × 101
24	5.29 × 102	5.70 × 102	5.86 × 102	5.68 × 102	1.45 × 101	4.52 × 102	4.77 × 102	4.91 × 102	4.75 × 102	1.48 × 101
25	3.87 × 102	3.87 × 102	3.87 × 102	3.87 × 102	2.11 × 10−1	3.83 × 102	3.86 × 102	3.87 × 102	3.86 × 102	1.07
26	6.93 × 102	9.56 × 102	2.12 × 103	9.87 × 102	2.49 × 102	1.45 × 103	1.87 × 103	1.97 × 103	1.79 × 103	1.93 × 102
27	4.80 × 102	4.92 × 102	5.14 × 102	4.93 × 102	8.02	4.87 × 102	4.97 × 102	5.06 × 102	4.96 × 102	6.31
28	3.00 × 102	3.00 × 102	4.04 × 102	3.21 × 102	4.22 × 101	3.00 × 102	3.00 × 102	4.03 × 102	3.10 × 102	3.26 × 101
29	4.23 × 102	5.92 × 102	7.69 × 102	5.77 × 102	1.01 × 102	4.09 × 102	5.13 × 102	5.68 × 102	4.32 × 102	5.95 × 101
30	1.94 × 103	2.09 × 103	2.36 × 103	2.08 × 103	9.27 × 101	1.97 × 103	2.06 × 103	2.12 × 103	2.06 × 103	5.55 × 101

,

Table 7. Results of GSK and ATMALS-GSK for CEC 2017 benchmark functions in D = 50.

Function	GSK [16]					ATMALS-GSK (Proposed)
	Best	Median	Worst	Mean	STD	Best	Median	Worst	Mean	STD
1	1.71 × 101	4.73 × 102	4.61 × 103	1.09 × 103	1.24 × 103	9.10 × 10−4	1.25 × 10−1	3.41 × 10−1	1.25 × 10−1	2.15 × 10−1
3	1.61 × 103	3.79 × 103	6.73 × 103	3.85 × 103	1.51 × 103	1.12 × 102	7.85 × 102	1.46 × 103	7.85 × 102	6.73 × 102
4	1.33 × 10−2	7.21 × 101	1.46 × 102	8.33 × 101	5.00 × 101	8.20	9.80	2.70 × 101	9.80	1.82 × 101
5	2.65 × 102	3.25 × 102	3.45 × 102	3.20 × 102	1.79 × 101	2.98 × 101	4.21 × 101	9.48 × 101	3.21 × 101	1.27 × 101
6	3.11 × 10−7	2.80 × 10−6	1.64 × 10−5	3.78 × 10−6	3.52 × 10−6	1.17 × 10−7	7.14 × 10−6	1.55 × 10−5	7.14 × 10−6	8.31 × 10−6
7	3.29 × 102	3.74 × 102	3.86 × 102	3.70 × 102	1.41 × 101	9.68 × 101	1.08 × 102	1.19 × 102	1.08 × 102	1.12 × 101
8	2.90 × 102	3.28 × 102	3.38 × 102	3.24 × 102	1.36 × 101	5.16 × 101	6.15 × 101	7.14 × 101	3.85 × 101	9.92
9	0.00	0.00	8.95 × 10−2	1.07 × 10−2	2.79 × 10−2	1.50 × 10−2	1.91 × 10−1	3.97 × 10−1	1.91 × 10−1	2.06 × 10−1
10	1.21 × 104	1.28 × 104	1.37 × 104	1.30 × 104	4.50 × 102	7.88 × 103	8.80 × 103	9.73 × 103	8.80 × 103	9.25 × 102
11	2.34 × 101	2.96 × 101	1.45 × 102	3.45 × 101	2.32 × 101	2.59 × 101	3.11 × 101	3.63 × 101	2.91 × 101	5.17
12	2.16 × 103	7.71 × 103	3.17 × 104	9.46 × 103	7.01 × 103	1.53 × 104	3.65 × 104	5.77 × 104	3.65 × 104	2.12 × 104
13	7.41 × 101	6.24 × 102	8.98 × 103	1.49 × 103	2.16 × 103	6.72 × 101	1.19 × 102	1.71 × 102	1.19 × 102	5.17 × 101
14	5.76 × 101	1.28 × 102	1.42 × 102	1.24 × 102	1.87 × 101	2.77 × 101	3.69 × 101	4.60 × 101	3.69 × 101	9.02
15	2.52 × 101	3.62 × 101	1.00 × 102	4.20 × 101	1.68 × 101	3.31 × 101	4.45 × 101	5.59 × 101	4.45 × 101	1.14 × 101
16	1.30 × 102	2.01 × 103	2.70 × 103	1.83 × 103	6.59 × 102	4.56 × 102	7.91 × 102	1.13 × 103	4.91 × 102	3.35 × 102
17	7.75 × 102	1.39 × 103	1.63 × 103	1.35 × 103	1.90 × 102	4.77 × 102	7.08 × 102	9.39 × 102	7.08 × 102	2.31 × 102
18	1.78 × 102	5.01 × 102	1.45 × 103	5.98 × 102	3.37 × 102	4.40 × 101	1.17 × 102	1.90 × 102	1.17 × 102	7.30 × 101
19	1.84 × 101	2.92 × 101	5.13 × 101	3.05 × 101	9.59	2.05 × 101	2.83 × 101	3.62 × 101	2.83 × 101	7.93
20	1.17 × 103	1.40 × 103	1.62 × 103	1.37 × 103	1.28 × 102	5.44 × 102	6.91 × 102	8.38 × 102	6.91 × 102	1.48 × 102
21	4.90 × 102	5.25 × 102	5.46 × 102	5.21 × 102	1.31 × 101	1.91 × 102	1.98 × 102	2.05 × 102	1.98 × 102	6.81
22	1.00 × 102	1.31 × 104	1.35 × 104	1.10 × 104	4.85 × 103	6.29 × 103	8.71 × 103	1.11 × 104	8.71 × 103	2.41 × 103
23	4.20 × 102	4.46 × 102	7.49 × 102	5.42 × 102	1.39 × 102	4.66 × 102	4.79 × 102	4.92 × 102	4.39 × 102	1.27 × 101
24	5.01 × 102	5.24 × 102	8.11 × 102	6.34 × 102	1.39 × 102	5.41 × 102	5.54 × 102	5.68 × 102	5.14 × 102	1.35 × 101
25	4.60 × 102	5.64 × 102	6.11 × 102	5.56 × 102	4.62 × 101	5.18 × 102	5.45 × 102	5.72 × 102	5.45 × 102	2.74 × 101
26	1.06 × 103	1.29 × 103	1.42 × 103	1.27 × 103	9.18 × 101	9.95 × 102	1.35 × 103	1.71 × 103	1.35 × 103	3.55 × 102
27	5.18 × 102	5.64 × 102	9.16 × 102	5.92 × 102	8.29 × 101	5.43 × 102	5.71 × 102	5.99 × 102	5.71 × 102	2.78 × 101
28	4.59 × 102	4.97 × 102	5.70 × 102	4.94 × 102	2.24 × 101	4.64 × 102	4.88 × 102	5.12 × 102	4.88 × 102	2.39 × 101
29	3.28 × 102	3.55 × 102	4.09 × 102	3.60 × 102	2.23 × 101	4.59 × 102	5.01 × 102	5.43 × 102	5.01 × 102	4.19 × 101
30	5.79 × 105	5.81 × 105	6.52 × 105	5.96 × 105	2.24 × 104	5.74 × 105	5.92 × 105	6.11 × 105	5.92 × 105	1.85 × 104

and

Table 8. Results of GSK and ATMALS-GSK for CEC 2017 benchmark functions in D = 100.

Function	GSK [16]					ATMALS-GSK (Proposed)
	Best	Median	Worst	Mean	STD	Best	Median	Worst	Mean	STD
1	3.96 × 10−1	5.38 × 103	2.11 × 104	5.80 × 103	4.63 × 103	1.96 × 10−1	1.98	7.98	2.55	4.79
3	6.01 × 104	1.15 × 105	1.48 × 105	1.15 × 105	2.15 × 104	4.87 × 104	4.91 × 104	5.30 × 104	4.91 × 104	1.26 × 104
4	8.46 × 101	2.17 × 102	2.89 × 102	2.05 × 102	4.57 × 101	1.81 × 102	1.97 × 102	2.03 × 102	1.98 × 102	1.45 × 101
5	7.26 × 101	7.70 × 102	8.14 × 102	5.31 × 102	3.39 × 102	3.70 × 102	5.30 × 102	5.90 × 102	5.40 × 102	2.17 × 101
6	1.25 × 10−5	6.83 × 10−5	3.14 × 10−2	1.72 × 10−3	6.46 × 10−3	1.80 × 10−6	6.20 × 10−3	3.90 × 10−2	1.24 × 10−2	2.73 × 10−2
7	8.33 × 102	8.76 × 102	9.16 × 102	8.75 × 102	1.83 × 101	6.99 × 102	8.21 × 102	8.99 × 102	8.20 × 102	1.91 × 101
8	5.90 × 101	7.36 × 102	8.18 × 102	4.92 × 102	3.41 × 102	3.95 × 102	4.91 × 102	5.88 × 102	4.92 × 102	1.12 × 101
9	5.45	7.69	1.72 × 101	8.46	3.41	4.75	8.90	1.60 × 101	8.80	3.14
10	2.87 × 104	2.94 × 104	3.04 × 104	2.95 × 104	4.44 × 102	2.28 × 104	2.45 × 104	2.85 × 104	2.44 × 104	1.01 × 103
11	1.61 × 102	2.65 × 102	4.64 × 102	2.78 × 102	6.85 × 101	1.30 × 102	1.70 × 102	2.80 × 102	1.76 × 102	6.28 × 101
12	2.25 × 104	6.53 × 104	3.23 × 105	8.34 × 104	7.52 × 104	1.20 × 104	2.92 × 105	5.88 × 105	3.78 × 105	1.69 × 105
13	5.17 × 101	2.85 × 103	9.31 × 103	3.20 × 103	2.64 × 103	1.06 × 102	1.30 × 102	2.00 × 102	1.31 × 102	5.43 × 101
14	3.30 × 102	2.53 × 103	1.69 × 104	4.64 × 103	4.47 × 103	6.20 × 102	2.49 × 103	3.50 × 103	2.57 × 103	2.31 × 103
15	3.20 × 101	4.23 × 102	5.20 × 103	7.33 × 102	1.09 × 103	5.90 × 101	9.90 × 101	1.40 × 102	1.04 × 102	4.05 × 101
16	2.87 × 102	8.25 × 102	7.04 × 103	2.27 × 103	2.61 × 103	8.60 × 102	1.80 × 103	2.30 × 103	1.89 × 103	4.24 × 102
17	2.14 × 103	4.12 × 103	4.59 × 103	3.91 × 103	6.68 × 102	8.00 × 102	1.30 × 103	2.80 × 103	1.57 × 103	6.92 × 102
18	1.93 × 104	4.43 × 104	1.87 × 105	5.73 × 104	3.63 × 104	9.50 × 103	1.13 × 104	1.30 × 104	1.14 × 104	4.25 × 103
19	5.04 × 101	8.41 × 102	3.25 × 103	1.00 × 103	8.30 × 102	9.00 × 101	1.55 × 102	2.30 × 102	1.61 × 102	7.63 × 101
20	3.97 × 103	4.52 × 103	4.80 × 103	4.46 × 103	2.22 × 102	3.10 × 103	2.80 × 103	2.90 × 103	2.84 × 103	5.21 × 102
21	2.83 × 102	3.29 × 102	1.02 × 103	6.07 × 102	3.37 × 102	2.10 × 102	2.85 × 102	3.10 × 102	2.89 × 102	2.11 × 101
22	2.86 × 104	3.01 × 104	3.09 × 104	3.00 × 104	4.57 × 102	2.10 × 104	2.35 × 104	3.80 × 104	2.43 × 104	1.92 × 103
23	5.86 × 102	6.11 × 102	6.49 × 102	6.11 × 102	1.58 × 101	5.25 × 102	5.85 × 102	6.45 × 102	5.87 × 102	1.45 × 101
24	8.89 × 102	9.33 × 102	9.69 × 102	9.32 × 102	1.70 × 101	8.40 × 102	9.10 × 102	9.50 × 102	9.11 × 102	1.77 × 101
25	7.61 × 102	8.20 × 102	8.99 × 102	8.21 × 102	4.34 × 101	7.20 × 102	7.50 × 102	7.80 × 102	7.53 × 102	4.18 × 101
26	3.33 × 103	3.63 × 103	3.96 × 103	3.66 × 103	1.78 × 102	3.12 × 103	3.35 × 103	3.65 × 103	3.42 × 103	1.28 × 103
27	6.20 × 102	6.46 × 102	7.30 × 102	6.57 × 102	3.07 × 101	6.10 × 102	6.78 × 102	7.90 × 102	6.87 × 102	3.47 × 101
28	4.99 × 102	5.57 × 102	6.09 × 102	5.53 × 102	3.25 × 101	3.10 × 102	3.30 × 102	3.85 × 102	3.51 × 102	1.10 × 102
29	9.12 × 102	1.21 × 103	1.56 × 103	1.21 × 103	1.74 × 102	1.01 × 103	1.37 × 103	1.78 × 103	1.39 × 103	2.14 × 102
30	2.41 × 103	3.02 × 103	3.72 × 103	2.99 × 103	2.73 × 102	2.40 × 103	3.21 × 103	3.40 × 103	3.23 × 103	8.70 × 102

for D = 10, 30, 50, and 100, respectively. The results obtained highlight the improvements brought by ATMALS-GSK over GSK, particularly in terms of robustness, accuracy, and stability across most benchmark functions in the 10-dimensional test setting. It can be seen in Table 5 that both algorithms perform identically on some simpler or unimodal functions such as F1, F3, F4, F6, and F9. In these cases, all metrics are consistently zero, indicating that both algorithms successfully find the global optimum with perfect reliability. This suggests that the difficulty level of these functions is relatively low for both approaches.

However, when it comes to more complex and multimodal functions, ATMALS-GSK demonstrates a significant edge. For instance, in F5, the mean fitness value achieved by ATMALS-GSK is 7.57, substantially lower than the 20.3 reported for GSK, with a reduced standard deviation. This trend is consistently observed in F7 and F8, where ATMALS-GSK yields much better best and mean results, indicating not only better convergence but also enhanced robustness against local optima. The improvements are even more evident in challenging composite and hybrid functions. Take F10 as an example: while GSK results in a mean error of 1060, ATMALS-GSK drastically reduces this to 455, cutting the error by more than half. A similar pattern is seen in F12, F13, and F14, where ATMALS-GSK not only achieves better average fitness values but also shows much lower variability (i.e., lower standard deviation), signifying higher consistency across multiple runs.

One striking result is observed in F21, where GSK shows a mean of 193 with a high deviation of 50.7, while ATMALS-GSK consistently scores 100 across all runs with zero variance, reflecting a perfectly stable and optimal outcome. Likewise, in Function F25, ATMALS-GSK closely matches the best performance of GSK but with remarkably improved stability (STD = 0.132 compared to GSK’s 20.5), indicating better reliability in difficult scenarios. In some functions such as F17 and F22, while ATMALS-GSK does not always outperform GSK in all statistics, it still offers highly competitive or comparable results with a favorable trade-off between accuracy and stability. This suggests that the adaptive and learning-based mechanisms integrated into ATMALS-GSK enhance its generalization ability across a broad spectrum of optimization landscapes.

Figure 10 shows the adaptive value of K across iterations for functions F1 (as a unimodal function), F4 (as a multimodal function), F11 (as a hybrid function), and F21 (as a composition function), along with the average value over all functions. Figure 11 illustrates the variations of all parameters of the ATMALS-GSK algorithm over iterations during the execution of the algorithm. Additionally, Figure 12 illustrates the average convergence behavior of the original GSK algorithm and the proposed ATMALS-GSK algorithm across various CEC 2017 functions, with a problem dimensionality of D = 10. The results, averaged over 51 independent runs, demonstrate the optimization efficiency of both algorithms in terms of their function error value over iterations. As shown, ATMALS-GSK consistently outperforms the original GSK algorithm by achieving faster convergence rates and lower final error values across most test functions.

4.3. Comparison with GSK and Modified GSKs

The comparisons of the ATMALS-GSK algorithm with the original GSK and its variants (AGSK, APGSK, FDBAGSK, and eGSK) in terms of the average and standard deviation results (Mean ± STD) over 51 runs for different dimensions D = 10, 30, 50, and 100 on CEC 2017 are provided in

Table 9. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with GSK and its modifications over 51 runs for D = 10 on CEC 2017.

Fun.	GSK [16]	AGSK [46]	APGSK [45]	FDBAGSK [36]	eGSK [49]	ATMALS-GSK
1	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00
3	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00
4	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00 × 101	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00
5	2.03 × 101 ± 2.79	6.33 ± 1.89	8.04 ± 2.41	5.83 ± 1.64	4.82 ± 1.90	7.57 ± 2.71
6	0.00 ± 0.00	9.52 × 10−6 ± 2.42 × 10−5	4.45 × 10−5 ± 1.33 × 10−4	5.59 × 10−6 ± 1.35 × 10−5	0.00 ± 0.00	0.00 ± 0.00
7	3.07 × 101 ± 3.08	1.80 × 101 ± 2.39	1.90 × 101 ± 3.50	1.81 × 101 ± 3.57	1.54 × 101 ± 2.43	1.63 × 101 ± 2.21
8	2.02 × 101 ± 2.92	8.12 ± 2.29	9.16 ± 3.07	7.54 ± 2.30	5.29 ± 2.08	7.57 ± 2.11
9	0.00 ± 0.00	1.42 × 10−2 ± 6.64 × 10−2	1.76 × 10−3 ± 1.25 × 10−2	0.00 ± 0.00	0.00 ± 0.00	0.00 ± 0.00
10	1.06 × 103 ± 1.33 × 102	3.20 × 102 ± 8.64 × 101	3.05 × 102 ± 1.05 × 102	3.02 × 102 ± 1.15 × 102	7.59 × 102 ± 2.35 × 102	4.55 × 102 ± 1.93 × 102
11	0.00 ± 0.00	3.99 × 10−1 ± 8.06 × 10−1	1.16 ± 9.76 × 10−1	9.30 × 10−1 ± 9.07 × 10−1	0.00 ± 0.00	1.67 ± 7.10 × 10−1
12	8.93 × 101 ± 7.26 × 101	4.46 × 101 ± 5.34 × 101	9.50 × 101 ± 8.64 × 101	7.06 × 101 ± 5.51 × 101	5.58 × 101 ± 6.17 × 101	1.12 × 101 ± 9.45
13	6.56 ± 1.41	3.35 ± 2.16	4.29 ± 2.70	4.15 ± 1.96	5.50 ± 6.56 × 10−1	2.65 ± 2.18
14	5.88 ± 3.06	3.91 × 10−2 ± 1.95 × 10−1	5.73 × 10−1 ± 6.10 × 10−1	1.41 × 10−1 ± 3.70 × 10−1	2.20 × 10−1 ± 4.61 × 10−1	2.78 ± 9.25 × 10−1
15	2.22 × 10−1 ± 2.13 × 10−1	1.97 × 10−2 ± 2.88 × 10−2	1.48 × 10−1 ± 2.02 × 10−1	3.33 × 10−2 ± 4.30 × 10−2	3.06 × 10−1 ± 2.18 × 10−1	1.47 × 10−1 ± 8.60 × 10−2
16	4.27 ± 4.95	7.01 × 10−1 ± 3.37 × 10−1	7.44 × 10−1 ± 7.35 × 10−1	1.52 ± 1.01	4.37 ± 5.17	1.12 ± 4.70 × 10−1
17	1.17 × 101 ± 7.09	1.48 ± 9.38 × 10−1	1.08 ± 8.74 × 10−1	1.41 ± 1.11	6.01 ± 6.22	1.25 × 101 ± 9.02 × 101
18	3.20 × 10−1 ± 1.92 × 10−1	9.94 × 10−2 ± 1.56 × 10−1	3.97 × 10−1 ± 3.95 × 10−1	1.25 × 10−1 ± 1.09 × 10−1	6.95 × 10−1 ± 2.84	3.04 × 10−1 ± 1.85 × 10−1
19	1.55 × 10−1 ± 3.51 × 10−1	6.96 × 10−2 ± 4.46 × 10−2	8.32 × 10−2 ± 9.37 × 10−2	8.91 × 10−2 ± 4.13 × 10−2	3.75 × 10−2 ± 4.31 × 10−2	1.39 × 10−1 ± 5.66 × 10−2
20	1.19 ± 3.99	4.28 × 10−2 ± 1.08 × 10−1	9.79 × 10−2 ± 2.84 × 10−1	4.28 × 10−2 ± 1.08 × 10−1	8.16 × 10−1 ± 2.79	2.51 ± 1.14
21	1.93 × 102 ± 5.07 × 101	1.02 × 102 ± 1.53 × 101	1.03 × 102 ± 3.34 × 101	9.80 × 101 ± 1.40 × 101	1.77 × 102 ± 4.80 × 101	1.00 × 102 ± 0.00
22	1.00 × 102 ± 4.87 × 10−1	8.00 × 101 ± 3.74 × 101	5.24 × 101 ± 4.54 × 101	5.87 × 101 ± 4.55 × 101	9.82 × 101 ± 1.40 × 101	6.38 × 101 ± 4.73 × 101
23	3.18 × 102 ± 3.99	2.97 × 102 ± 5.75 × 101	3.00 × 102 ± 5.77 × 101	2.98 × 102 ± 4.42 × 101	3.04 × 102 ± 2.17	2.76 × 102 ± 9.72 × 101
24	3.44 × 102 ± 1.87 × 101	1.04 × 102 ± 1.99 × 101	9.33 × 101 ± 2.36 × 101	1.00 × 102 ± 3.45 × 101	3.05 × 102 ± 7.57 × 101	3.11 × 102 ± 7.44 × 101
25	4.27 × 102 ± 2.05 × 101	3.86 × 102 ± 5.84 × 101	3.22 × 102 ± 1.31 × 102	3.57 × 102 ± 1.03 × 102	4.17 × 102 ± 2.30 × 101	3.97 × 102 ± 1.32 × 10−1
26	3.00 × 102 ± 0.00	2.53 × 102 ± 1.10 × 102	1.17 × 102 ± 1.10 × 102	2.33 × 102 ± 1.09 × 102	3.00 × 102 ± 6.43 × 10−14	3.00 × 102 ± 0.00
27	3.89 × 102 ± 2.17 × 10−1	3.89 × 102 ± 5.59 × 10−1	3.89 × 102 ± 1.73	3.89 × 102 ± 6.64 × 10−1	3.89 × 102 ± 1.70 × 10−1	3.89 × 102 ± 1.98 × 10−1
28	3.12 × 102 ± 3.20 × 101	2.29 × 102 ± 1.29 × 102	2.05 × 102 ± 1.40 × 102	2.18 × 102 ± 1.35 × 102	3.00 × 102 ± 1.37 × 10−13	3.00 × 102 ± 0.00
29	2.48 × 102 ± 4.76	2.43 × 102 ± 5.66	2.44 × 102 ± 1.03 × 101	2.44 × 102 ± 6.14	2.32 × 102 ± 2.88	2.38 × 102 ± 3.33
30	4.56 × 102 ± 3.44 × 101	4.70 × 102 ± 4.92 × 101	5.15 × 102 ± 8.71 × 101	4.97 × 102 ± 8.37 × 101	4.45 × 102 ± 3.58 × 101	4.29 × 102 ± 2.09 × 101

,

Table 10. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with GSK and its modifications over 51 runs for D = 30 on CEC 2017.

Fun.	GSK [16]	AGSK [46]	APGSK [45]	FDBAGSK [36]	eGSK [49]	ATMALS-GSK
1	0.00 ± 0.00	0.00 ± 0.00	1.65 × 10−5 ± 8.63 × 10−5	0.00 ± 0.00	1.75 × 10−1 ± 1.24	0.00 ± 0.00
3	7.07 × 10−7 ± 1.82 × 10−6	8.62 × 10−3 ± 3.06 × 10−2	4.37 × 10−1 ± 8.49 × 10−1	1.75 × 10−1 ± 1.08	0.00 ± 0.00	0.00 ± 0.00
4	1.11 × 101 ± 2.17 × 101	8.87 ± 1.81 × 101	6.98 ± 1.68 × 101	3.07 × 101 ± 3.18 × 101	2.58 ± 1.92	1.99 ± 2.10
5	1.60 × 102 ± 8.87	9.21 × 101 ± 1.27 × 101	1.15 × 102 ± 1.33 × 101	7.12 × 101 ± 1.07 × 101	2.31 × 101 ± 8.24	1.53 × 101 ± 1.90 × 101
6	1.52 × 10−6 ± 2.36 × 10−6	6.98 × 10−6 ± 1.80 × 10−5	1.11 × 10−4 ± 3.68 × 10−4	4.71 × 10−6 ± 8.00 × 10−6	3.78 × 10−6 ± 4.11 × 10−6	6.82 × 10−5 ± 3.90 × 10−5
7	1.87 × 102 ± 8.40	1.17 × 102 ± 9.96	1.53 × 102 ± 1.78 × 101	1.06 × 102 ± 9.57	5.31 × 101 ± 8.29	4.33 × 101 ± 1.15 × 101
8	1.55 × 102 ± 1.11 × 101	9.74 × 101 ± 1.35 × 101	1.13 × 102 ± 1.66 × 101	7.87 × 101 ± 1.19 × 101	2.45 × 101 ± 8.47	1.35 × 101 ± 1.29 × 101
9	0.00 ± 0.00	7.50 ± 7.08	3.65 × 101 ± 3.79 × 101	2.22 ± 1.84	8.91 × 10−3 ± 6.36 × 10−2	1.79 × 10−2 ± 3.77 × 10−2
10	6.69 × 103 ± 3.54 × 102	3.31 × 103 ± 1.86 × 102	2.80 × 103 ± 2.40 × 102	3.24 × 103 ± 2.22 × 102	4.92 × 103 ± 9.42 × 102	2.80 × 103 ± 5.90 × 102
11	3.30 × 101 ± 3.83 × 101	3.65 × 101 ± 1.47 × 101	3.19 × 101 ± 1.32 × 101	1.96 × 101 ± 5.74	1.33 × 101 ± 2.06 × 101	1.46 × 101 ± 1.73 × 101
12	6.62 × 103 ± 4.59 × 103	8.86 × 103 ± 3.97 × 103	1.31 × 104 ± 9.01 × 103	1.04 × 104 ± 5.90 × 103	4.24 × 103 ± 4.97 × 103	3.84 × 103 ± 2.62 × 103
13	9.83 × 101 ± 3.42 × 101	6.17 × 101 ± 4.51 × 101	5.00 × 101 ± 2.35 × 101	6.03 × 101 ± 2.78 × 101	3.17 × 101 ± 2.31 × 101	3.01 × 101 ± 1.12 × 101
14	5.69 × 101 ± 5.49	4.19 × 101 ± 5.98	4.38 × 101 ± 1.02 × 101	3.76 × 101 ± 4.71	2.32 × 101 ± 3.71	1.78 × 101 ± 4.26
15	1.45 × 101 ± 1.49 × 101	1.90 × 101 ± 9.21	2.37 × 101 ± 1.98 × 101	1.42 × 101 ± 4.35	5.28 ± 3.64	1.43 × 101 ± 5.56
16	7.96 × 102 ± 1.94 × 102	7.40 × 102 ± 1.32 × 102	5.45 × 102 ± 1.39 × 102	6.42 × 102 ± 1.30 × 102	2.96 × 102 ± 2.22 × 102	2.53 × 102 ± 2.81 × 102
17	1.89 × 102 ± 9.44 × 101	1.82 × 102 ± 8.02 × 101	1.92 × 102 ± 7.73 × 101	1.65 × 102 ± 7.82 × 101	5.92 × 101 ± 3.89 × 101	4.71 × 101 ± 2.30 × 101
18	3.68 × 101 ± 5.42	5.28 × 102 ± 6.26 × 102	1.31 × 103 ± 1.14 × 103	2.80 × 102 ± 2.52 × 102	3.27 × 101 ± 2.09 × 101	3.02 × 101 ± 6.67
19	1.29 × 101 ± 6.02	1.52 × 101 ± 2.51	1.48 × 101 ± 3.35	1.37 × 101 ± 1.57	6.52 ± 1.88	1.30 × 101 ± 3.34
20	1.08 × 102 ± 1.14 × 102	2.42 × 102 ± 8.46 × 101	2.65 × 102 ± 7.96 × 101	2.04 × 102 ± 7.54 × 101	5.24 × 101 ± 4.31 × 101	4.74 × 101 ± 3.35 × 101
21	3.46 × 102 ± 8.30	2.91 × 102 ± 2.46 × 101	2.75 × 102 ± 7.85 × 101	2.47 × 102 ± 6.00 × 101	2.22 × 102 ± 9.58	2.66 × 102 ± 9.12
22	1.00 × 102 ± 0.00	3.51 × 102 ± 8.70 × 102	1.65 × 102 ± 4.42 × 102	1.00 × 102 ± 1.06	1.00 × 102 ± 7.33 × 10−13	1.00 × 102 ± 0.00
23	4.70 × 102 ± 4.49 × 101	4.39 × 102 ± 1.38 × 101	4.69 × 102 ± 2.84 × 101	4.18 × 102 ± 9.60	3.59 × 102 ± 8.30	3.44 × 102 ± 1.70 × 101
24	5.68 × 102 ± 1.45 × 101	5.06 × 102 ± 1.57 × 101	5.65 × 102 ± 5.90 × 101	4.96 × 102 ± 1.32 × 101	4.33 × 102 ± 7.37	4.75 × 102 ± 1.48 × 101
25	3.87 × 102 ± 2.11 × 10−1	3.87 × 102 ± 1.92	3.87 × 102 ± 1.34	3.87 × 102 ± 6.73 × 10−1	3.87 × 102 ± 1.86	3.86 × 102 ± 1.07
26	9.87 × 102 ± 2.49 × 102	5.85 × 102 ± 6.62 × 102	3.74 × 102 ± 4.90 × 102	7.18 × 102 ± 6.73 × 102	8.55 × 102 ± 3.21 × 102	1.79 × 103 ± 1.93 × 102
27	4.93 × 102 ± 8.02	4.98 × 102 ± 1.04 × 101	5.10 × 102 ± 1.35 × 101	5.01 × 102 ± 1.05 × 101	4.97 × 102 ± 8.59	4.96 × 102 ± 6.31
28	3.21 × 102 ± 4.22 × 101	3.13 × 102 ± 4.15 × 101	3.16 × 102 ± 3.46 × 101	3.32 × 102 ± 5.33 × 101	3.10 × 102 ± 3.10 × 101	3.10 × 102 ± 3.26 × 101
29	5.77 × 102 ± 1.01 × 102	6.41 × 102 ± 8.28 × 101	6.30 × 102 ± 5.35 × 101	5.88 × 102 ± 5.92 × 101	4.35 × 102 ± 4.07 × 101	4.32 × 102 ± 5.95 × 101
30	2.08 × 103 ± 9.27 × 101	2.14 × 103 ± 8.88 × 101	2.25 × 103 ± 1.79 × 102	2.18 × 103 ± 1.95 × 102	2.08 × 103 ± 1.35 × 102	2.06 × 103 ± 5.55 × 101

,

Table 11. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with GSK and its modifications over 51 runs for D = 50 on CEC 2017.

Fun.	GSK [16]	AGSK [46]	APGSK [45]	FDBAGSK [36]	eGSK [49]	ATMALS-GSK
1	1.09 × 103 ± 1.24 × 103	1.59 ± 3.54	9.59 × 102 ± 1.25 × 103	1.86 × 10−1 ± 6.16 × 10−1	2.82 × 10−1 ± 8.75 × 10−1	1.25 × 10−1 ± 2.15 × 10−1
3	3.85 × 103 ± 1.51 × 103	1.87 × 103 ± 2.10 × 103	3.60 × 103 ± 1.66 × 103	6.83 × 102 ± 7.91 × 102	9.42 × 10−7 ± 5.65 × 10−8	7.85 × 102 ± 6.73 × 102
4	8.33 × 101 ± 5.00 × 101	5.80 × 101 ± 4.07 × 101	8.13 × 101 ± 3.96 × 101	5.73 × 101 ± 4.13 × 101	1.02 × 101 ± 2.78 × 101	9.80 ± 1.82 × 101
5	3.20 × 102 ± 1.79 × 101	2.57 × 102 ± 2.06 × 101	2.71 × 102 ± 2.56 × 101	1.98 × 102 ± 2.09 × 101	3.79 × 101 ± 9.72	3.21 × 101 ± 1.27 × 101
6	3.78 × 10−6 ± 3.52 × 10−6	2.48 × 10−4 ± 3.74 × 10−4	1.17 × 10−3 ± 2.24 × 10−3	1.73 × 10−4 ± 2.59 × 10−4	1.18 × 10−5 ± 9.13 × 10−6	7.14 × 10−6 ± 8.31 × 10−6
7	3.70 × 102 ± 1.41 × 101	2.99 × 102 ± 2.85 × 101	3.53 × 102 ± 3.40 × 101	2.54 × 102 ± 2.41 × 101	9.07 × 101 ± 9.86	1.08 × 102 ± 1.12 × 101
8	3.24 × 102 ± 1.36 × 101	2.56 × 102 ± 2.81 × 101	2.75 × 102 ± 2.44 × 101	1.95 × 102 ± 1.91 × 101	4.09 × 101 ± 1.01 × 101	3.85 × 101 ± 9.92
9	1.07 × 10−2 ± 2.79 × 10−2	5.13 × 101 ± 4.26 × 101	1.46 × 103 ± 1.16 × 103	2.34 × 101 ± 2.49 × 101	2.02 × 10−1 ± 4.51 × 10−1	1.91 × 10−1 ± 2.06 × 10−1
10	1.30 × 104 ± 4.50 × 102	7.60 × 103 ± 3.54 × 102	6.06 × 103 ± 2.61 × 102	7.57 × 103 ± 3.37 × 102	1.06 × 104 ± 1.22 × 103	8.80 × 103 ± 9.25 × 102
11	3.45 × 101 ± 2.32 × 101	1.25 × 102 ± 2.74 × 101	9.18 × 101 ± 2.22 × 101	7.83 × 101 ± 2.09 × 101	2.97 × 101 ± 6.59	2.91 × 101 ± 5.17
12	9.46 × 103 ± 7.01 × 103	9.71 × 104 ± 8.84 × 104	1.08 × 105 ± 5.96 × 104	1.22 × 105 ± 8.58 × 104	3.98 × 104 ± 2.56 × 104	3.65 × 104 ± 2.12 × 104
13	1.49 × 103 ± 2.16 × 103	3.50 × 102 ± 1.74 × 102	3.31 × 102 ± 1.81 × 102	3.17 × 102 ± 1.76 × 102	8.40 × 101 ± 7.01 × 101	1.19 × 102 ± 5.17 × 101
14	1.24 × 102 ± 1.87 × 101	1.11 × 102 ± 3.55 × 101	1.58 × 102 ± 1.09 × 102	8.35 × 101 ± 1.15 × 101	4.03 × 101 ± 8.34	3.69 × 101 ± 9.02
15	4.20 × 101 ± 1.68 × 101	8.43 × 101 ± 3.82 × 101	1.06 × 102 ± 3.85 × 101	6.12 × 101 ± 9.46	5.08 × 101 ± 2.03 × 101	4.45 × 101 ± 1.14 × 101
16	1.83 × 103 ± 6.59 × 102	1.47 × 103 ± 1.39 × 102	1.02 × 103 ± 1.83 × 102	1.19 × 103 ± 1.84 × 102	5.01 × 102 ± 4.01 × 102	4.91 × 102 ± 3.35 × 102
17	1.35 × 103 ± 1.90 × 102	1.10 × 103 ± 1.51 × 102	8.70 × 102 ± 1.14 × 102	8.45 × 102 ± 1.71 × 102	7.48 × 102 ± 2.95 × 102	7.08 × 102 ± 2.31 × 102
18	5.98 × 102 ± 3.37 × 102	1.07 × 104 ± 1.23 × 104	9.11 × 103 ± 5.51 × 103	3.72 × 103 ± 2.25 × 103	1.21 × 102 ± 3.42 × 102	1.17 × 102 ± 7.30 × 101
19	3.05 × 101 ± 9.59	4.23 × 101 ± 8.07	4.03 × 101 ± 6.80	3.86 × 101 ± 6.89	3.00 × 101 ± 9.42	2.83 × 101 ± 7.93
20	1.37 × 103 ± 1.28 × 102	8.68 × 102 ± 1.25 × 102	7.39 × 102 ± 1.35 × 102	6.04 × 102 ± 1.68 × 102	7.57 × 102 ± 2.08 × 102	6.91 × 102 ± 1.48 × 102
21	5.21 × 102 ± 1.31 × 101	4.47 × 102 ± 3.33 × 101	4.55 × 102 ± 2.34 × 101	3.89 × 102 ± 2.14 × 101	2.36 × 102 ± 8.86	1.98 × 102 ± 6.81
22	1.10 × 104 ± 4.85 × 103	7.90 × 103 ± 1.64 × 103	6.18 × 103 ± 1.76 × 103	6.79 × 103 ± 2.97 × 103	9.57 × 103 ± 3.65 × 103	8.71 × 103 ± 2.41 × 103
23	5.42 × 102 ± 1.39 × 102	6.65 × 102 ± 2.22 × 101	7.69 × 102 ± 3.06 × 101	6.19 × 102 ± 2.23 × 101	4.51 × 102 ± 1.14 × 101	4.39 × 102 ± 1.27 × 101
24	6.34 × 102 ± 1.39 × 102	7.26 × 102 ± 2.94 × 101	8.40 × 102 ± 4.04 × 101	6.90 × 102 ± 2.31 × 101	5.20 × 102 ± 8.56	5.14 × 102 ± 1.35 × 101
25	5.56 × 102 ± 4.62 × 101	5.52 × 102 ± 3.40 × 101	5.63 × 102 ± 3.10 × 101	5.22 × 102 ± 4.06 × 101	5.63 × 102 ± 3.89 × 101	5.45 × 102 ± 2.74 × 101
26	1.27 × 103 ± 9.18 × 101	3.59 × 103 ± 7.34 × 102	2.80 × 103 ± 1.80 × 103	3.00 × 103 ± 4.95 × 102	1.16 × 103 ± 4.43 × 102	1.35 × 103 ± 3.55 × 102
27	5.92 × 102 ± 8.29 × 101	5.28 × 102 ± 1.63 × 101	5.62 × 102 ± 3.51 × 101	5.24 × 102 ± 1.94 × 101	6.09 × 102 ± 1.26 × 102	5.71 × 102 ± 2.78 × 101
28	4.94 × 102 ± 2.24 × 101	4.99 × 102 ± 3.50 × 101	4.93 × 102 ± 1.91 × 101	4.83 × 102 ± 2.86 × 101	4.93 × 102 ± 3.43 × 101	4.88 × 102 ± 2.39 × 101
29	3.60 × 102 ± 2.23 × 101	1.00 × 103 ± 1.36 × 102	1.01 × 103 ± 1.16 × 102	8.32 × 102 ± 1.26 × 102	4.37 × 102 ± 5.82 × 101	5.01 × 102 ± 4.19 × 101
30	5.96 × 105 ± 2.24 × 104	5.83 × 105 ± 6.20 × 103	5.86 × 105 ± 5.35 × 103	5.86 × 105 ± 1.60 × 104	6.09 × 105 ± 3.27 × 104	5.92 × 105 ± 1.85 × 104

and

Table 12. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with GSK and its modifications over 51 runs for D = 100 on CEC 2017.

Fun.	GSK [16]	AGSK [46]	APGSK [45]	FDBAGSK [36]	eGSK [49]	ATMALS-GSK
1	5.80 × 103 ± 4.63 × 103	5.08 × 101 ± 1.19 × 102	4.38 × 103 ± 2.89 × 103	5.99 ± 9.79	9.58 × 103 ± 4.05 × 104	2.55 ± 4.79
3	1.15 × 105 ± 2.15 × 104	1.08 × 105 ± 2.11 × 104	7.83 × 104 ± 9.64 × 103	5.55 × 104 ± 1.42 × 104	2.00 × 10−3 ± 7.54 × 10−3	4.91 × 104 ± 1.26 × 104
4	2.05 × 102 ± 4.57 × 101	1.68 × 102 ± 4.47 × 101	2.10 × 102 ± 5.52 × 101	2.07 × 102 ± 4.53 × 101	6.30 ± 1.87 × 101	4.90 ± 1.45 × 101
5	5.31 × 102 ± 3.39 × 102	7.28 × 102 ± 5.28 × 101	7.72 × 102 ± 4.46 × 101	6.66 × 102 ± 5.37 × 101	1.12 × 102 ± 1.72 × 101	9.70 × 101 ± 2.17 × 101
6	1.72 × 10−3 ± 6.46 × 10−3	2.01 × 10−2 ± 1.20 × 10−2	3.30 × 10−1 ± 2.97 × 10−1	1.63 × 10−2 ± 1.27 × 10−2	4.01 × 10−2 ± 5.90 × 10−2	1.24 × 10−2 ± 2.73 × 10−2
7	8.75 × 102 ± 1.83 × 101	8.73 × 102 ± 5.20 × 101	1.17 × 103 ± 6.49 × 101	8.03 × 102 ± 5.29 × 101	2.13 × 102 ± 2.45 × 101	1.68 × 102 ± 1.91 × 101
8	4.92 × 102 ± 3.41 × 102	7.15 × 102 ± 4.81 × 101	7.82 × 102 ± 4.48 × 101	6.47 × 102 ± 5.60 × 101	1.12 × 102 ± 1.66 × 101	9.65 × 101 ± 1.12 × 101
9	8.46 ± 3.41	5.50 × 102 ± 2.39 × 102	1.34 × 104 ± 4.65 × 103	7.14 × 102 ± 3.16 × 102	1.98 × 101 ± 1.11 × 101	1.05 × 101 ± 3.14
10	2.95 × 104 ± 4.44 × 102	2.33 × 104 ± 6.30 × 102	1.82 × 104 ± 4.05 × 102	2.34 × 104 ± 6.43 × 102	2.52 × 104 ± 3.03 × 103	2.44 × 104 ± 1.01 × 103
11	2.78 × 102 ± 6.85 × 101	8.81 × 102 ± 1.30 × 102	9.17 × 102 ± 1.39 × 102	9.43 × 102 ± 1.41 × 102	1.86 × 102 ± 7.58 × 101	1.76 × 102 ± 6.28 × 101
12	8.34 × 104 ± 7.52 × 104	2.36 × 105 ± 9.14 × 104	3.86 × 105 ± 1.42 × 105	4.37 × 105 ± 1.86 × 105	4.57 × 105 ± 2.05 × 105	3.78 × 105 ± 1.69 × 105
13	3.20 × 103 ± 2.64 × 103	3.61 × 103 ± 2.30 × 103	3.66 × 103 ± 2.39 × 103	4.10 × 103 ± 3.05 × 103	9.96 × 101 ± 6.48 × 101	1.31 × 102 ± 5.43 × 101
14	4.64 × 103 ± 4.47 × 103	1.57 × 104 ± 1.14 × 104	2.47 × 104 ± 1.82 × 104	2.17 × 104 ± 1.30 × 104	3.12 × 103 ± 2.51 × 103	2.57 × 103 ± 2.31 × 103
15	7.33 × 102 ± 1.09 × 103	4.34 × 102 ± 1.65 × 102	5.57 × 102 ± 2.25 × 102	4.64 × 102 ± 9.10 × 101	8.68 × 101 ± 3.39 × 101	1.04 × 102 ± 4.05 × 101
16	2.27 × 103 ± 2.61 × 103	4.11 × 103 ± 1.62 × 102	2.89 × 103 ± 3.19 × 102	3.68 × 103 ± 2.09 × 102	1.34 × 103 ± 4.90 × 102	1.89 × 103 ± 4.24 × 102
17	3.91 × 103 ± 6.68 × 102	2.87 × 103 ± 1.17 × 102	2.32 × 103 ± 1.77 × 102	2.51 × 103 ± 2.85 × 102	1.68 × 103 ± 7.56 × 102	1.57 × 103 ± 6.92 × 102
18	5.73 × 104 ± 3.63 × 104	6.22 × 104 ± 2.83 × 104	1.12 × 105 ± 4.34 × 104	7.75 × 104 ± 3.64 × 104	9.77 × 103 ± 4.86 × 103	1.14 × 104 ± 4.25 × 103
19	1.00 × 103 ± 8.30 × 102	1.82 × 102 ± 4.94 × 101	2.39 × 102 ± 1.06 × 102	1.93 × 102 ± 5.25 × 101	1.96 × 102 ± 3.37 × 102	1.61 × 102 ± 7.63 × 101
20	4.46 × 103 ± 2.22 × 102	2.76 × 103 ± 1.53 × 102	2.45 × 103 ± 2.07 × 102	2.41 × 103 ± 2.63 × 102	3.02 × 103 ± 6.43 × 102	2.84 × 103 ± 5.21 × 102
21	6.07 × 102 ± 3.37 × 102	9.36 × 102 ± 3.92 × 101	9.75 × 102 ± 5.17 × 101	8.77 × 102 ± 5.08 × 101	3.27 × 102 ± 1.75 × 101	2.89 × 102 ± 2.11 × 101
22	3.00 × 104 ± 4.57 × 102	2.38 × 104 ± 3.51 × 103	1.99 × 104 ± 3.84 × 102	2.41 × 104 ± 7.56 × 102	2.64 × 104 ± 2.41 × 103	2.43 × 104 ± 1.92 × 103
23	6.11 × 102 ± 1.58 × 101	1.19 × 103 ± 2.41 × 101	1.37 × 103 ± 3.88 × 101	1.04 × 103 ± 5.64 × 101	6.26 × 102 ± 1.94 × 101	5.87 × 102 ± 1.45 × 101
24	9.32 × 102 ± 1.70 × 101	1.62 × 103 ± 5.79 × 101	1.65 × 103 ± 5.48 × 101	1.53 × 103 ± 5.13 × 101	9.57 × 102 ± 2.07 × 101	9.11 × 102 ± 1.77 × 101
25	8.21 × 102 ± 4.34 × 101	8.07 × 102 ± 5.07 × 101	8.19 × 102 ± 3.76 × 101	7.71 × 102 ± 5.78 × 101	7.85 × 102 ± 6.35 × 101	7.53 × 102 ± 4.18 × 101
26	3.66 × 103 ± 1.78 × 102	1.09 × 104 ± 1.11 × 103	1.22 × 104 ± 7.50 × 102	1.01 × 104 ± 7.08 × 102	3.06 × 103 ± 1.64 × 103	3.42 × 103 ± 1.28 × 103
27	6.57 × 102 ± 3.07 × 101	6.41 × 102 ± 5.25 × 101	9.63 × 102 ± 5.64 × 101	6.37 × 102 ± 3.41 × 101	7.08 × 102 ± 4.65 × 101	6.87 × 102 ± 3.47 × 101
28	5.53 × 102 ± 3.25 × 101	5.63 × 102 ± 3.15 × 101	5.74 × 102 ± 2.83 × 101	5.72 × 102 ± 2.75 × 101	3.84 × 102 ± 1.17 × 102	3.51 × 102 ± 1.10 × 102
29	1.21 × 103 ± 1.74 × 102	3.69 × 103 ± 1.61 × 102	3.71 × 103 ± 2.41 × 102	3.17 × 103 ± 2.80 × 102	1.55 × 103 ± 2.86 × 102	1.39 × 103 ± 2.14 × 102
30	2.99 × 103 ± 2.73 × 102	5.01 × 103 ± 2.41 × 103	7.42 × 103 ± 2.35 × 103	4.71 × 103 ± 2.90 × 103	3.68 × 103 ± 7.86 × 102	3.23 × 103 ± 8.70 × 102

, respectively.

As seen in Table 9, for functions 1 to 4 and 6, which demonstrate perfect or near-perfect convergence across all algorithms, the basic GSK algorithm alone is sufficient. For functions 5, 7, and 8, we begin to see notable differences. For example, function 5 showcases that ATMALS-GSK (7.57 ± 2.71) falls somewhere between AGSK (6.33 ± 1.89) and eGSK (4.82 ± 1.90). However, for more rugged terrains like functions 7 and 8, ATMALS-GSK performs more competitively, especially when compared to the original GSK, which suffers from high error means (30.7 and 20.2, respectively). Here, ATMALS-GSK manages to cut down the error significantly (to 16.3 and 7.57), underscoring its adaptive capacity. Functions like 10 and 12 really illustrate the leap ATMALS-GSK offers. In function 10, the original GSK’s performance (1060 ± 133) is substantially improved by AGSK and APGSK, but ATMALS-GSK achieves a more favorable trade-off (455 ± 193), suggesting it maintains robustness even in highly multimodal scenarios. Function 12 reveals an even more dramatic advantage—ATMALS-GSK posts a mean of 11.2 with very low deviation, compared to AGSK’s 44.6 and the original’s 89.3.

Although some modifications, particularly AGSK (0.039 ± 0.195), deliver astonishing performance, ATMALS-GSK lags slightly behind (2.78 ± 0.93). ATMALS-GSK continues to impress in functions 13, 15, and 16. Especially in function 13, its low mean (2.65) coupled with a reasonable standard deviation suggests more reliable convergence than many competitors. In function 17, however, ATMALS-GSK shows a significant spike in variance (mean = 12.5, SD = 90.2), possibly revealing sensitivity to initial population or parameter randomness in highly stochastic landscapes. This outlier behavior points to a possible need for parameter fine-tuning or incorporating stability controls. Looking at complex composite functions like 21–28, ATMALS-GSK either performs on par with the best (e.g., functions 26 and 28, both achieving perfect or near-perfect solutions like eGSK) or slightly underperforms in terms of variance but retains competitive means. Notably, in function 24, despite not having the best mean, ATMALS-GSK is very close to eGSK and performs better than all other variants, suggesting good generalization.

The results in Table 9, Table 10, Table 11 and Table 12 show that across most benchmark problems, the STDs of ATMALS-GSK are relatively controlled, signaling dependable convergence behavior. Its hybridization strategy, by embedding adaptive tuning and local search layers, appears to provide resilience against premature convergence—a known weakness in some simpler modifications. The results show that ATMALS-GSK proves itself to be a strong contender among the GSK family. While it does not dominate in every single function, it consistently offers balanced, reliable, and often superior performance, especially for complex or multimodal problems.

4.4. Comparison with Other Metaheuristic Algorithms

In this section, the performance of ATMALS-GSK is compared with recent metaheuristics (i.e., IADE, GJO-JOS, QCSCA, cSM, and FOWFO) in terms of the average and standard deviation results (Mean ± STD) over 51 runs for different dimensions D = 10, 30, 50, and 100 on CEC 2017, as reported in

Table 13. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with existing metaheuristics over 51 runs for D = 10 on CEC 2017.

Fun.	IADE [50]	GJO-JOS [51]	QCSCA [52]	cSM [53]	FOWFO [54]	ATMALS-GSK
1	8.54 × 102 ± 1.51 × 103	2.30 × 104 ± 1.40 × 104	3.09 × 104 ± 1.37 × 105	2.19 × 103 ± 2.39 × 103	3.91 × 10−4 ± 3.18 × 10−4	0.00 ± 0.00
3	3.00 × 102 ± 8.90 × 10−10	3.00 × 102 ± 3.70 × 10−1	3.62 × 102 ± 1.59 × 102	1.38 × 10−12 ± 9.10 × 10−13	4.58 × 10−7 ± 4.09 × 10−7	0.00 ± 0.00
4	3.51 × 103 ± 4.55 × 10−13	4.10 × 102 ± 1.70	4.11 × 102 ± 1.79 × 101	6.32 × 10−1 ± 3.23 × 10−1	5.60 × 10−6 ± 1.71 × 10−5	0.00 ± 0.00
5	6.18 × 102 ± 3.88	5.20 × 102 ± 1.10 × 101	5.22 × 102 ± 7.13	1.46 × 101 ± 4.84	5.70 ± 1.69	7.57 ± 2.71
6	6.43 × 102 ± 6.73	6.00 × 102 ± 9.60 × 10−1	6.01 × 102 ± 2.69 × 10−1	5.44 × 10−4 ± 1.93 × 10−3	1.91 × 10−2 ± 1.04 × 10−2	0.00 ± 0.00
7	7.38 × 102 ± 1.53 × 101	7.30 × 102 ± 1.20 × 101	6.20 × 102 ± 3.72	2.35 × 101 ± 7.14	1.73 × 101 ± 3.03	1.63 × 101 ± 2.21
8	8.13 × 102 ± 4.02	8.20 × 102 ± 8.70	8.20 × 102 ± 5.90	1.57 × 101 ± 7.46	6.65 ± 2.21	7.57 ± 2.11
9	1.54 × 103 ± 5.51 × 101	9.00 × 102 ± 2.90 × 10−1	9.00 × 102 ± 1.31 × 10−1	3.56 × 10−2 ± 2.42 × 10−1	3.68 × 10−4 ± 8.83 × 10−4	0.00 ± 0.00
10	1.77 × 103 ± 2.69 × 102	1.70 × 103 ± 4.00 × 102	1.69 × 103 ± 2.11 × 102	5.92 × 102 ± 2.36 × 102	2.56 × 102 ± 1.24 × 102	4.55 × 102 ± 1.93 × 102
11	1.12 × 103 ± 6.18	1.10 × 103 ± 8.60	1.12 × 103 ± 6.32	1.32 × 101 ± 6.38	1.51 ± 9.92 × 10−1	1.67 ± 7.10 × 10−1
12	8.05 × 103 ± 1.29 × 104	5.60 × 105 ± 6.30 × 105	1.43 × 104 ± 1.39 × 104	5.79 × 104 ± 4.39 × 104	3.42 × 101 ± 4.12 × 101	1.12 × 101 ± 9.45
13	4.96 × 103 ± 7.30 × 103	1.10 × 104 ± 5.70 × 103	6.08 × 103 ± 6.93 × 103	4.92 × 103 ± 6.76 × 103	5.09 ± 2.97	2.65 ± 2.18
14	1.42 × 103 ± 2.02 × 101	1.50 × 103 ± 1.90 × 101	1.43 × 103 ± 2.49 × 101	8.09 × 101 ± 3.66 × 101	2.49 ± 1.78	2.78 ± 9.25 × 10−1
15	1.51 × 103 ± 6.90	1.60 × 103 ± 7.40 × 101	1.65 × 103 ± 4.16 × 102	4.40 × 102 ± 3.34 × 102	5.33 × 10−1 ± 3.54 × 10−1	1.47 × 10−1 ± 8.60 × 10−2
16	1.70 × 103 ± 1.17 × 102	1.80 × 103 ± 1.40 × 102	1.63 × 103 ± 4.81 × 101	5.18 × 101 ± 6.35 × 101	1.86 ± 6.26 × 10−1	1.12 ± 4.70 × 10−1
17	1.76 × 103 ± 6.04 × 101	1.70 × 103 ± 1.20 × 101	1.74 × 103 ± 2.17 × 101	3.76 × 101 ± 2.94 × 101	5.21 ± 2.30	1.25 × 101 ± 9.02 × 101
18	1.83 × 103 ± 9.11	3.00 × 104 ± 1.30 × 104	7.20 × 103 ± 7.51 × 103	1.02 × 104 ± 9.11 × 103	1.88 ± 1.10	3.04 × 10−1 ± 1.85 × 10−1
19	3.45 × 103 ± 8.35	2.00 × 103 ± 5.60 × 102	2.10 × 103 ± 7.50 × 102	4.68 × 102 ± 9.55 × 102	7.92 × 10−1 ± 3.88 × 10−1	1.39 × 10−1 ± 5.66 × 10−2
20	2.06 × 103 ± 5.02 × 101	2.10 × 103 ± 6.00 × 101	2.03 × 103 ± 3.05 × 101	1.00 × 101 ± 1.07 × 101	4.21 ± 2.44	2.51 ± 1.14
21	2.23 × 103 ± 6.78 × 101	2.20 × 103 ± 4.00 × 101	2.30 × 103 ± 4.95 × 101	1.67 × 102 ± 5.80 × 101	9.80 × 101 ± 1.40 × 101	1.00 × 102 ± 0.00
22	2.30 × 103 ± 1.00	2.30 × 103 ± 3.00	2.30 × 103 ± 2.02 × 101	8.66 × 101 ± 3.13 × 101	1.42 × 101 ± 3.23 × 101	6.38 × 101 ± 4.73 × 101
23	2.62 × 103 ± 1.43 × 101	2.60 × 103 ± 1.00 × 101	2.62 × 103 ± 5.07	3.15 × 102 ± 6.13	2.75 × 102 ± 9.27 × 101	2.76 × 102 ± 9.72 × 101
24	2.75 × 103 ± 1.04 × 101	2.60 × 103 ± 1.00 × 102	2.75 × 103 ± 3.58 × 101	3.22 × 102 ± 6.89 × 101	8.88 × 101 ± 3.13 × 101	3.11 × 102 ± 7.44 × 101
25	2.91 × 103 ± 2.35 × 101	2.90 × 103 ± 2.30 × 101	2.93 × 103 ± 5.10 × 101	4.03 × 102 ± 4.73 × 101	1.94 × 102 ± 1.12 × 102	3.97 × 102 ± 1.32 × 10−1
26	2.94 × 103 ± 4.97 × 101	2.90 × 103 ± 7.70 × 101	2.98 × 103 ± 2.48 × 102	3.20 × 102 ± 1.91 × 102	8.73 × 101 ± 1.01 × 102	3.00 × 102 ± 0.00
27	3.10 × 103 ± 5.62	3.10 × 103 ± 1.70 × 101	3.10 × 103 ± 7.53	3.92 × 102 ± 5.24	3.82 × 102 ± 4.52 × 101	3.89 × 102 ± 1.98 × 10−1
28	3.17 × 103 ± 6.95 × 10−13	3.30 × 103 ± 1.20 × 102	3.29 × 103 ± 1.20 × 102	3.70 × 102 ± 1.15 × 102	1.76 × 102 ± 1.42 × 102	3.00 × 102 ± 0.00
29	3.68 × 103 ± 3.40 × 101	3.20 × 103 ± 2.50 × 101	3.20 × 103 ± 2.53 × 101	2.68 × 102 ± 2.57 × 101	2.45 × 102 ± 1.27 × 101	2.38 × 102 ± 3.33
30	5.08 × 107 ± 6.10 × 10−8	5.10 × 105 ± 1.60 × 105	3.32 × 105 ± 3.68 × 105	1.26 × 105 ± 3.51 × 105	3.97 × 102 ± 6.67	4.29 × 102 ± 2.09 × 101

,

Table 14. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with existing metaheuristics over 51 runs for D = 30 on CEC 2017.

Fun.	IADE [50]	GJO-JOS [51]	QCSCA [52]	cSM [53]	FOWFO [54]	ATMALS-GSK
1	8.49 × 103 ± 6.38 × 103	2.10 × 105 ± 1.10 × 105	3.39 × 103 ± 3.66 × 103	2.30 × 103 ± 3.01 × 103	8.75 × 10−4 ± 3.89 × 10−4	0.00 ± 0.00
3	3.00 × 102 ± 7.95 × 10−1	6.40 × 102 ± 2.50 × 102	3.09 × 102 ± 2.95 × 101	9.89 × 10−10 ± 1.17× 10−9	4.53 × 10−6 ± 3.42 × 10−6	0.00 ± 0.00
4	1.94 × 104 ± 0.00	5.00 × 102 ± 1.70 × 101	4.07 × 102 ± 9.72	8.07 × 101 ± 1.02 × 101	5.43 ± 1.52 × 101	1.99 ± 2.10
5	7.93 × 102 ± 1.47 × 101	6.00 × 102 ± 5.60 × 101	5.17 × 102 ± 4.78	8.61 × 101 ± 2.30 × 101	5.49 × 101 ± 1.39 × 101	1.53 × 101 ± 1.90 × 101
6	6.55 × 102 ± 4.63	6.10 × 102 ± 5.90	6.00 × 102 ± 3.52 × 10−2	4.13 × 10−4 ± 8.21 × 10−4	4.61 × 10−2 ± 3.07 × 10−2	6.82 × 10−5 ± 3.90 × 10−5
7	9.04 × 102 ± 7.38 × 101	8.30 × 102 ± 6.20 × 101	7.27 × 102 ± 6.06	1.33 × 102 ± 2.61 × 101	7.91 × 101 ± 1.13 × 101	4.33 × 101 ± 1.15 × 101
8	9.24 × 102 ± 3.24 × 101	8.90 × 102 ± 4.60 × 101	8.14 × 102 ± 4.32	1.01 × 102 ± 2.82 × 101	5.60 × 101 ± 1.19 × 101	1.35 × 101 ± 1.29 × 101
9	5.44 × 103 ± 4.41 × 102	1.20 × 103 ± 4.40 × 102	9.00 × 102 ± 4.28 × 10−3	9.32 × 102 ± 8.65 × 102	1.90 × 101 ± 2.15 × 101	1.79 × 10−2 ± 3.77 × 10−2
10	4.75 × 103 ± 1.12 × 103	5.00 × 103 ± 1.70 × 103	1.57 × 103 ± 2.19 × 102	2.85 × 103 ± 4.80 × 102	2.02 × 103 ± 3.11 × 102	2.80 × 103 ± 5.90 × 102
11	1.22 × 103 ± 4.95 × 101	1.20 × 103 ± 4.20 × 101	1.11 × 103 ± 1.86	1.61 × 102 ± 5.33 × 101	1.26 × 101 ± 4.65	1.46 × 101 ± 1.73 × 101
12	5.57 × 104 ± 4.87 × 104	1.20 × 107 ± 1.10 × 107	1.80 × 104 ± 1.44 × 104	8.87 × 105 ± 8.57 × 105	2.92 × 102 ± 1.78 × 102	3.84 × 103 ± 2.62 × 103
13	1.57 × 104 ± 1.84 × 104	1.00 × 105 ± 7.50 × 104	4.94 × 103 ± 6.23 × 103	2.17 × 104 ± 2.35 × 104	4.43 × 101 ± 2.89 × 101	3.01 × 101 ± 1.12 × 101
14	1.64 × 103 ± 8.18 × 101	3.80 × 104 ± 3.00 × 104	1.42 × 103 ± 2.20 × 101	1.01 × 104 ± 7.92 × 103	2.91 × 101 ± 7.91	1.78 × 101 ± 4.26
15	6.84 × 103 ± 1.01 × 104	3.40 × 104 ± 2.20 × 104	1.53 × 103 ± 3.46 × 101	1.79 × 104 ± 1.39 × 104	1.27 × 101 ± 4.03	1.43 × 101 ± 5.56
16	2.37 × 103 ± 2.31 × 102	2.50 × 103 ± 3.60 × 102	1.61 × 103 ± 3.89	6.87 × 102 ± 2.37 × 102	4.08 × 102 ± 1.68 × 102	2.53 × 102 ± 2.81 × 102
17	2.08 × 103 ± 1.11 × 102	2.00 × 103 ± 1.40 × 102	1.73 × 103 ± 1.99 × 101	2.83 × 102 ± 1.70 × 102	8.79 × 101 ± 4.18 × 101	4.71 × 101 ± 2.30 × 101
18	1.47 × 104 ± 1.93 × 104	3.20 × 105 ± 3.30 × 105	6.49 × 103 ± 9.86 × 103	2.02 × 105 ± 1.89 × 105	2.67 × 101 ± 4.82	3.02 × 101 ± 6.67
19	7.20 × 103 ± 4.67 × 103	3.40 × 105 ± 3.20 × 105	1.91 × 103 ± 9.97	2.30 × 104 ± 2.07 × 104	1.36 × 101 ± 3.53	1.30 × 101 ± 3.34
20	2.53 × 103 ± 2.27 × 102	2.30 × 103 ± 1.10 × 102	2.01 × 103 ± 9.03	3.75 × 102 ± 1.45 × 102	1.12 × 102 ± 6.64 × 101	4.74 × 101 ± 3.35 × 101
21	2.38 × 103 ± 2.88 × 101	2.40 × 103 ± 5.70 × 101	2.30 × 103 ± 4.74 × 101	2.86 × 102 ± 2.26 × 101	1.96 × 102 ± 7.84 × 101	2.66 × 102 ± 9.12
22	4.08 × 103 ± 2.49 × 103	2.30 × 103 ± 1.30	2.30 × 103 ± 1.89 × 101	2.61 × 103 ± 1.27 × 103	9.93 × 101 ± 5.19	1.00 × 102 ± 0.00
23	2.73 × 103 ± 2.00 × 101	2.80 × 103 ± 5.30 × 101	2.62 × 103 ± 2.93	4.35 × 102 ± 2.61 × 101	2.64 × 102 ± 1.47 × 102	3.44 × 102 ± 1.70 × 101
24	2.94 × 103 ± 5.02 × 101	2.90 × 103 ± 5.10 × 101	2.72 × 103 ± 8.70 × 101	4.96 × 102 ± 6.26 × 101	3.64 × 102 ± 1.77 × 102	4.75 × 102 ± 1.48 × 101
25	2.90 × 103 ± 9.48	2.90 × 103 ± 1.20 × 101	2.93 × 103 ± 2.24 × 101	3.86 × 102 ± 1.21	3.85 × 102 ± 1.64	3.86 × 102 ± 1.07
26	5.56 × 103 ± 6.13 × 102	3.40 × 103 ± 8.50 × 102	2.90 × 103 ± 1.53 × 102	1.95 × 103 ± 4.32 × 102	2.55 × 102 ± 4.88 × 101	1.79 × 103 ± 1.93 × 102
27	3.24 × 103 ± 1.34 × 101	3.20 × 103 ± 1.60 × 101	3.09 × 103 ± 7.50	5.17 × 102 ± 9.77	4.97 × 102 ± 1.22 × 101	4.96 × 102 ± 6.31
28	3.54 × 103 ± 1.51 × 10−12	3.20 × 103 ± 2.70 × 101	3.23 × 103 ± 1.26 × 102	3.63 × 102 ± 5.49 × 101	3.00 × 102 ± 2.34 × 10−2	3.10 × 102 ± 3.26 × 101
29	7.34 × 103 ± 2.95 × 102	3.70 × 103 ± 1.60 × 102	3.18 × 103 ± 1.95 × 101	7.81 × 102 ± 1.72 × 102	4.97 × 102 ± 5.92 × 101	4.32 × 102 ± 5.95 × 101
30	2.65 × 109 ± 4.42 × 10−6	2.00 × 106 ± 9.60 × 105	1.18 × 105 ± 2.66 × 105	2.26 × 104 ± 1.53 × 104	1.98 × 103 ± 5.31 × 101	2.06 × 103 ± 5.55 × 101

,

Table 15. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with existing metaheuristics over 51 runs for D = 50 on CEC 2017.

Fun.	IADE [50]	GJO-JOS [51]	QCSCA [52]	cSM [53]	FOWFO [54]	ATMALS-GSK
1	3.25 × 103 ± 4.46 × 103	7.80 × 105 ± 4.50 × 105	3.02 × 103 ± 3.12 × 103	5.99 × 103 ± 6.65 × 103	1.78 × 10−2 ± 1.37 × 10−2	1.25 × 10−1 ± 2.15 × 10−1
3	2.46 × 103 ± 3.21 × 103	3.90 × 103 ± 1.90 × 103	3.02 × 102 ± 3.33	1.76 × 10−7 ± 1.72 × 10−7	2.24 ± 3.72	7.85 × 102 ± 6.73 × 102
4	3.12 × 104 ± 1.24 × 10−11	5.80 × 102 ± 5.10 × 101	4.05 × 102 ± 2.28	1.13 × 102 ± 3.89 × 101	2.15 × 101 ± 1.73 × 101	9.80 ± 1.82 × 101
5	8.88 × 102 ± 3.44 × 101	6.90 × 102 ± 8.60 × 101	5.15 × 102 ± 3.68	1.89 × 102 ± 4.15 × 101	1.17 × 102 ± 1.94 × 101	3.21 × 101 ± 1.27 × 101
6	6.62 × 102 ± 3.92	6.20 × 102 ± 1.20 × 101	6.00 × 102 ± 1.40 × 10−2	1.75 × 10−4 ± 4.74 × 10−4	1.01 × 10−1 ± 4.49 × 10−2	7.14 × 10−6 ± 8.31 × 10−6
7	1.23 × 103 ± 7.64 × 101	9.80 × 102 ± 1.20 × 102	7.25 × 102 ± 4.68	2.48 × 102 ± 4.20 × 101	1.45 × 102 ± 1.80 × 101	1.08 × 102 ± 1.12 × 101
8	1.09 × 103 ± 4.66 × 101	1.00 × 103 ± 9.00 × 101	8.12 × 102 ± 3.46	1.85 × 102 ± 3.01 × 101	1.18 × 102 ± 2.37 × 101	3.85 × 101 ± 9.92
9	1.31 × 104 ± 5.76 × 102	9.00 × 103 ± 7.00 × 103	9.00 × 102 ± 3.31 × 10−4	4.67 × 103 ± 2.57 × 103	3.95 × 102 ± 2.83 × 102	1.91 × 10−1 ± 2.06 × 10−1
10	7.98 × 103 ± 1.36 × 103	8.20 × 103 ± 3.00 × 103	1.46 × 103 ± 1.49 × 102	5.05 × 103 ± 6.73 × 102	3.84 × 103 ± 4.36 × 102	8.80 × 103 ± 9.25 × 102
11	1.32 × 103 ± 6.22 × 101	1.40 × 103 ± 6.80 × 101	1.11 × 103 ± 1.31	2.26 × 102 ± 5.19 × 101	5.21 × 101 ± 1.41 × 101	2.91 × 101 ± 5.17
12	9.99 × 105 ± 5.62 × 105	4.80 × 107 ± 2.50 × 107	2.00 × 104 ± 1.69 × 104	3.89 × 106 ± 2.38 × 106	2.83 × 103 ± 2.42 × 103	3.65 × 104 ± 2.12 × 104
13	8.85 × 103 ± 8.49 × 103	1.30 × 105 ± 6.50 × 104	3.48 × 103 ± 3.45 × 103	1.62 × 104 ± 1.37 × 104	1.04 × 102 ± 4.79 × 101	1.19 × 102 ± 5.17 × 101
14	5.27 × 103 ± 2.38 × 103	1.90 × 105 ± 1.00 × 105	1.43 × 103 ± 2.19 × 101	5.51 × 104 ± 4.25 × 104	5.83 × 101 ± 1.03 × 101	3.69 × 101 ± 9.02
15	1.00 × 104 ± 9.72 × 103	5.00 × 104 ± 2.60 × 104	1.52 × 103 ± 2.87 × 101	1.44 × 104 ± 9.47 × 103	4.33 × 101 ± 9.08	4.45 × 101 ± 1.14 × 101
16	3.07 × 103 ± 4.27 × 102	2.90 × 103 ± 4.50 × 102	1.61 × 103 ± 4.96 × 101	1.44 × 104 ± 3.69 × 102	9.73 × 102 ± 2.25 × 102	4.91 × 102 ± 3.35 × 102
17	3.27 × 103 ± 3.10 × 102	2.80 × 103 ± 4.30 × 102	1.72 × 103 ± 1.87 × 101	1.44 × 104 ± 2.87 × 102	6.46 × 102 ± 1.76 × 102	7.08 × 102 ± 2.31 × 102
18	1.12 × 105 ± 8.98 × 104	1.50 × 106 ± 7.80 × 105	3.60 × 103 ± 3.85 × 103	1.44 × 104 ± 1.11 × 105	4.18 × 101 ± 5.98	1.17 × 102 ± 7.30 × 101
19	1.67 × 108 ± 4.01 × 101	5.00 × 105 ± 3.40 × 105	1.90 × 103 ± 6.17	1.44 × 104 ± 1.71 × 104	3.05 × 101 ± 7.38	2.83 × 101 ± 7.93
20	3.39 × 103 ± 2.65 × 102	2.80 × 103 ± 3.60 × 102	2.00 × 103 ± 3.25	1.44 × 104 ± 1.80 × 102	5.07 × 102 ± 1.35 × 102	6.91 × 102 ± 1.48 × 102
21	2.52 × 103 ± 4.60 × 101	2.50 × 103 ± 8.70 × 101	2.29 × 103 ± 4.82 × 101	1.44 × 104 ± 4.00 × 101	3.15 × 102 ± 3.84 × 101	1.98 × 102 ± 6.81
22	1.01 × 104 ± 7.53 × 102	5.80 × 103 ± 4.20 × 103	2.30 × 103 ± 1.97 × 101	1.44 × 104 ± 7.06 × 102	3.82 × 103 ± 1.67 × 103	8.71 × 103 ± 2.41 × 103
23	2.97 × 103 ± 3.90 × 101	2.90 × 103 ± 5.40 × 101	2.61 × 103 ± 2.88	1.44 × 104 ± 3.65 × 101	5.40 × 102 ± 8.26 × 101	4.39 × 102 ± 1.27 × 101
24	3.13 × 103 ± 3.48 × 101	3.10 × 103 ± 6.90 × 101	2.72 × 103 ± 8.02 × 101	1.44 × 104 ± 4.28 × 101	7.02 × 102 ± 1.12 × 102	5.14 × 102 ± 1.35 × 101
25	3.09 × 103 ± 1.91 × 101	3.10 × 103 ± 3.60 × 101	2.92 × 103 ± 2.34 × 101	1.44 × 104 ± 3.34 × 101	4.89 × 102 ± 3.03 × 101	5.45 × 102 ± 2.74 × 101
26	8.18 × 103 ± 1.53 × 103	2.90 × 103 ± 3.30 × 102	2.90 × 103 ± 1.41 × 102	1.44 × 104 ± 3.58 × 102	6.14 × 102 ± 8.00 × 102	1.35 × 103 ± 3.55 × 102
27	3.62 × 103 ± 1.34 × 102	3.40 × 103 ± 6.30 × 101	3.09 × 103 ± 7.06	1.44 × 104 ± 4.77 × 101	5.50 × 102 ± 3.24 × 101	5.71 × 102 ± 2.78 × 101
28	3.88 × 103 ± 2.03 × 101	3.30 × 103 ± 4.10 × 101	3.23 × 103 ± 1.48 × 102	1.44 × 104 ± 1.70 × 101	4.60 × 102 ± 4.29	4.88 × 102 ± 2.39 × 101
29	1.55 × 105 ± 6.22 × 102	4.20 × 103 ± 2.90 × 102	3.17 × 103 ± 1.66 × 101	1.44 × 104 ± 2.58 × 102	6.76 × 102 ± 1.60 × 102	5.01 × 102 ± 4.19 × 101
30	6.21 × 109 ± 7.78 × 10−5	2.00 × 107 ± 5.20 × 106	1.79 × 105 ± 3.34 × 105	1.44 × 104 ± 3.35 × 106	5.97 × 105 ± 3.14 × 104	5.92 × 105 ± 1.85 × 104

and

Table 16. Comparison of the average and standard deviation results (Mean ± STD) of ATMALS-GSK with existing metaheuristics over 51 runs for D = 100 on CEC 2017.

Fun.	IADE [50]	GJO-JOS [51]	QCSCA [52]	cSM [53]	FOWFO [54]	ATMALS-GSK
1	9.46 × 104 ± 1.85 × 105	3.90 × 106 ± 1.10 × 106	3.27 × 103 ± 3.72 × 103	8.14 × 103 ± 1.28 × 104	1.88 × 103 ± 2.03 × 103	2.55 ± 4.79
3	3.68 × 104 ± 8.20 × 103	5.20 × 104 ± 1.10 × 104	3.00 × 102 ± 3.85 × 10−1	4.83 × 103 ± 2.26 × 103	4.23 × 104 ± 9.62 × 103	4.91 × 104 ± 1.26 × 104
4	7.98 × 104 ± 1.23 × 10−10	7.70 × 102 ± 5.20 × 101	4.04 × 102 ± 2.27	2.31 × 102 ± 2.72 × 101	1.52 × 102 ± 4.63 × 101	4.90 ± 1.45 × 101
5	1.41 × 103 ± 3.44 × 101	1.00 × 103 ± 1.70 × 102	5.11 × 102 ± 2.32	5.00 × 102 ± 6.65 × 101	3.11 × 102 ± 5.11 × 101	9.70 × 101 ± 2.17 × 101
6	6.62 × 102 ± 1.18	6.40 × 102 ± 9.70	6.00 × 102 ± 3.58 × 10−3	6.53 × 10−5 ± 1.06 × 10−4	2.96 × 10−1 ± 8.85 × 10−2	1.24 × 10−2 ± 2.73 × 10−2
7	2.59 × 103 ± 2.92 × 102	1.50 × 103 ± 2.30 × 102	7.23 × 102 ± 3.62	7.07 × 102 ± 7.90 × 101	3.43 × 102 ± 3.11 × 101	1.68 × 102 ± 1.91 × 101
8	1.66 × 103 ± 1.28 × 102	1.30 × 103 ± 1.80 × 102	8.09 × 102 ± 2.70	5.19 × 102 ± 7.77 × 101	3.28 × 102 ± 4.38 × 101	9.65 × 101 ± 1.12 × 101
9	2.39 × 104 ± 7.15 × 102	4.50 × 104 ± 1.60 × 104	9.00 × 102 ± 1.88 × 10−5	1.84 × 104 ± 5.08 × 103	7.71 × 103 ± 2.00 × 103	1.05 × 101 ± 3.14
10	1.57 × 104 ± 1.19 × 103	1.60 × 104 ± 5.70 × 103	1.28 × 103 ± 1.05 × 102	1.18 × 104 ± 1.00 × 103	1.05 × 104 ± 7.56 × 102	2.44 × 104 ± 1.01 × 103
11	3.40 × 103 ± 1.16 × 103	2.80 × 103 ± 2.90 × 102	1.10 × 103 ± 8.65 × 10−1	1.29 × 103 ± 2.17 × 102	2.70 × 102 ± 5.01 × 101	1.76 × 102 ± 6.28 × 101
12	1.07 × 107 ± 2.86 × 106	2.70 × 108 ± 1.20 × 108	1.55 × 104 ± 1.43 × 104	1.27 × 107 ± 4.80 × 106	1.36 × 105 ± 6.66 × 104	3.78 × 105 ± 1.69 × 105
13	1.02 × 104 ± 3.42 × 103	4.00 × 105 ± 8.70 × 105	2.96 × 103 ± 2.86 × 103	1.29 × 104 ± 1.32 × 104	3.49 × 102 ± 1.10 × 102	1.31 × 102 ± 5.43 × 101
14	2.31 × 105 ± 1.40 × 105	1.00 × 106 ± 4.50 × 105	1.42 × 103 ± 2.04 × 101	1.67 × 105 ± 7.05 × 104	2.10 × 102 ± 3.31 × 101	2.57 × 103 ± 2.31 × 103
15	4.79 × 103 ± 2.66 × 103	6.40 × 104 ± 2.90 × 104	1.42 × 103 ± 2.04 × 101	8.26 × 103 ± 8.01 × 103	1.92 × 102 ± 6.45 × 101	1.04 × 102 ± 4.05 × 101
16	5.59 × 103 ± 6.96 × 102	5.40 × 103 ± 5.50 × 102	1.61 × 103 ± 4.81	3.39 × 103 ± 6.43 × 102	2.87 × 103 ± 3.97 × 102	1.89 × 103 ± 4.24 × 102
17	4.86 × 103 ± 7.32 × 102	4.60 × 103 ± 5.00 × 102	1.72 × 103 ± 2.08 × 101	2.81 × 103 ± 4.78 × 102	2.07 × 103 ± 3.26 × 102	1.57 × 103 ± 6.92 × 102
18	5.51 × 105 ± 2.28 × 105	2.10 × 106 ± 9.40 × 105	2.02 × 103 ± 8.45 × 102	3.66 × 105 ± 1.29 × 105	3.90 × 102 ± 2.46 × 102	1.14 × 104 ± 4.25 × 103
19	2.82 × 109 ± 1.06 × 102	1.70 × 106 ± 9.60 × 105	1.90 × 103 ± 9.14	1.17 × 104 ± 1.16 × 104	1.10 × 102 ± 2.40 × 101	1.61 × 102 ± 7.63 × 101
20	5.27 × 103 ± 6.49 × 102	4.60 × 103 ± 1.00 × 103	2.00 × 103 ± 2.19 × 10−1	2.66 × 103 ± 4.39 × 102	1.97 × 103 ± 2.92 × 102	2.84 × 103 ± 5.21 × 102
21	2.95 × 103 ± 1.46 × 102	2.80 × 103 ± 1.50 × 102	2.29 × 103 ± 4.70 × 101	7.33 × 102 ± 7.54 × 101	5.46 × 102 ± 4.89 × 101	2.89 × 102 ± 2.11 × 101
22	1.83 × 104 ± 1.13 × 103	1.30 × 104 ± 9.10 × 103	2.30 × 103 ± 8.75 × 10−3	1.32 × 104 ± 1.24 × 103	1.18 × 104 ± 8.06 × 102	2.43 × 104 ± 1.92 × 103
23	3.47 × 103 ± 9.11 × 101	3.30 × 103 ± 1.40 × 102	2.61 × 103 ± 2.65	9.18 × 102 ± 6.16 × 101	7.72 × 102 ± 3.13 × 101	5.87 × 102 ± 1.45 × 101
24	4.07 × 103 ± 1.09 × 102	3.80 × 103 ± 1.90 × 102	2.71 × 103 ± 7.38 × 101	1.40 × 103 ± 7.62 × 101	1.25 × 103 ± 5.48 × 101	9.11 × 102 ± 1.77 × 101
25	3.38 × 103 ± 6.40 × 101	3.50 × 103 ± 4.90 × 101	2.92 × 103 ± 2.34 × 101	7.09 × 102 ± 6.05 × 101	7.30 × 102 ± 5.31 × 101	7.53 × 102 ± 4.18 × 101
26	1.87 × 104 ± 3.28 × 103	6.20 × 103 ± 4.20 × 103	2.92 × 103 ± 2.34 × 101	8.62 × 103 ± 8.60 × 102	6.37 × 103 ± 1.61 × 103	3.42 × 103 ± 1.28 × 103
27	3.92 × 103 ± 2.80 × 102	3.50 × 103 ± 5.90 × 101	3.10 × 103 ± 1.19 × 101	7.47 × 102 ± 4.96 × 101	6.85 × 102 ± 3.88 × 101	6.87 × 102 ± 3.47 × 101
28	5.46 × 103 ± 1.54 × 102	3.50 × 103 ± 5.10 × 101	3.20 × 103 ± 1.26 × 102	5.53 × 102 ± 3.30 × 101	5.55 × 102 ± 2.73 × 101	3.51 × 102 ± 1.10 × 102
29	5.20 × 104 ± 9.53 × 102	6.90 × 103 ± 8.90 × 102	3.16 × 103 ± 1.40 × 101	3.95 × 103 ± 5.28 × 102	2.67 × 103 ± 2.49 × 102	1.39 × 103 ± 2.14 × 102
30	2.06 × 1010 ± 1.98 × 10−1	2.10 × 107 ± 6.10 × 106	1.70 × 105 ± 3.26 × 105	1.78 × 104 ± 6.67 × 103	2.85 × 103 ± 2.31 × 102	3.23 × 103 ± 8.70 × 102

, respectively.

Table 13 highlights the strong performance of the ATMALS-GSK algorithm against other metaheuristics across a range of 10-dimensional benchmark functions. It consistently outperforms or matches competing methods, showing exceptional accuracy and robustness—often achieving exact solutions (e.g., 0.00 with zero variance) for simpler functions like F1, F3, and F4, while remaining highly competitive on more complex or multimodal ones like F5 and F10. Even on the most challenging hybrid and composite functions (F14–F29), ATMALS-GSK frequently delivers near-optimal results with low variability, outperforming methods like IADE, GJO-JOS, and QCSCA in many cases. Its ability to maintain low standard deviations throughout confirms its reliability and stable search behavior, making it a dependable choice for diverse and complex optimization tasks.

The results in Table 13, Table 14, Table 15 and Table 16 for different dimensions demonstrate the superior performance of ATMALS-GSK for solving high-dimensional problems. Its hybrid mechanism—blending adaptive learning and guided search via GSK—allows it to effectively balance exploration and exploitation. This advantage is evident not only in reaching optimal or near-optimal solutions but also in doing so reliably across a broad set of benchmark problems, outperforming recent metaheuristic algorithms.

4.5. Statistical Analysis

The statistical analysis of the CEC 2017 benchmark functions using the Wilcoxon and Friedman tests (shown in

Table 17. Wilcoxon signed-rank test of the ATMALS-GSK algorithm with the GSK variants.

D	Methods		R+	R−	p Value	+	=	−	Dec.
10	ATMALS-GSK vs	GSK	238	15	0.0003	19	7	3	+
		AGSK	134	191	0.4432	11	4	14	≈
		APGSK	138	187	0.5098	13	4	12	≈
		FDBAGSK	97	203	0.1300	8	5	16	≈
		eGSK	152	79	0.2045	11	8	10	≈
30	ATMALS-GSK vs	GSK	337	41	0.0003	22	2	5	+
		AGSK	378	28	0.0001	26	1	2	+
		APGSK	379	27	0.0000	27	1	1	+
		FDBAGSK	340	38	0.0002	23	2	4	+
		eGSK	244	107	0.0818	18	3	8	≈
50	ATMALS-GSK vs	GSK	375	60	0.0003	23	0	6	+
		AGSK	354	81	0.0023	25	0	4	+
		APGSK	356	79	0.0019	25	0	4	+
		FDBAGSK	314	121	0.0365	21	0	8	+
		eGSK	330	105	0.0138	24	0	5	+
100	ATMALS-GSK vs	GSK	378.5	56	0.0002	23	0	6	+
		AGSK	368	67	0.0006	24	0	5	+
		APGSK	383	52	0.0001	26	0	3	+
		FDBAGSK	394	41	0.0000	25	0	4	+
		eGSK	323.5	111.5	0.0215	23	0	6	+

,

Table 18. Friedman test of the ATMALS-GSK algorithm with the GSK variants.

Method	Ranks				Mean Rank	Overall Rank
	D = 10	D = 30	D = 50	D = 100
ATMALS-GSK	3.37	1.84	1.93	1.82	2.24	1
eGSK	3.53	2.24	2.76	2.86	2.84	2
FDBAGSK	2.81	3.66	3.09	3.82	3.34	3
AGSK	2.87	4.36	4.41	3.93	3.89	4
GSK	4.86	4.16	4.28	3.58	4.22	5
APGSK	3.53	4.74	4.53	4.96	4.44	6
p-value	0.0003	0.0001	0.0000	0.0000	0.0000

,

Table 19. Wilcoxon signed-rank test of the ATMALS-GSK algorithm with existing metaheuristics.

D	Methods		R+	R−	p Value	+	=	−	Dec.
10	ATMALS-GSK vs.	IADE	435	0	0.0000	29	0	0	+
		GJO-JOS	435	0	0.0000	29	0	0	+
		QCSCA	435	0	0.0000	29	0	0	+
		cSM	435	0	0.0000	29	0	0	+
		FOWFO	143	292	0.1104	14	0	15	≈
30	ATMALS-GSK vs.	IADE	435	0	0.0000	29	0	0	+
		GJO-JOS	435	0	0.0000	29	0	0	+
		QCSCA	425	10	0.0000	28	0	1	+
		cSM	406	0	0.0000	28	1	0	+
		FOWFO	205.5	229.5	0.7983	16	0	13	≈
50	ATMALS-GSK vs.	IADE	433	2	0.0000	28	0	1	+
		GJO-JOS	415	20	0.0000	27	0	2	+
		QCSCA	322	113	0.0229	24	0	5	+
		cSM	392	43	0.0000	26	0	3	+
		FOWFO	217	218	1.00	15	0	14	≈
100	ATMALS-GSK vs.	IADE	383	52	0.0001	26	0	3	+
		GJO-JOS	396	39	0.0000	27	0	2	+
		QCSCA	276	159	0.2132	20	0	9	≈
		cSM	360	75	0.0013	23	0	6	+
		FOWFO	236	199	0.7013	18	0	11	≈

and

Table 20. Friedman test of the ATMALS-GSK algorithm with existing metaheuristics.

Method	Ranks				Mean Rank	Overall Rank
	D = 10	D = 30	D = 50	D = 100
ATMALS-GSK	1.52	1.50	1.86	2.03	1.72	1
FOWFO	1.55	1.65	1.72	2.07	1.75	2
cSM	3.10	3.47	4.48	3.41	3.61	3
QCSCA	5.00	3.78	3.02	2.86	3.66	4
GJO-JOS	4.8.3	5.31	4.84	5.21	5.04	5
IADE	5.00	5.29	5.07	5.41	5.19	6
p-value	0.0000	0.0000	0.0000	0.0000	0.0000

and Figure 13 and Figure 14) supports the strong performance of ATMALS-GSK.

The Wilcoxon test results in Table 17 demonstrate that the proposed ATMALS-GSK algorithm surpasses GSK, AGSK, APGSK, FDBAGSK, and eGSK algorithms in 87, 86, 91, 77, and 76 cases out of all 29 × 4 = 116 cases (29 functions over four dimensions). This results in success ratios of ATMALS-GSK against GSK, AGSK, APGSK, FDBAGSK, and eGSK of 75%, 74.1%, 78.4%, 66.4%, and 65.5%, respectively. Furthermore, the results in Table 19 indicate that ATMALS-GSK exceeds IADE, GJO-JOS, QCSCA, cSM, and FOWFO, recording success ratios of 96.6%, 96.6%, 87.1%, 91.4%, and 54.3%, respectively, across all functions in overall dimensions.

The Friedman test results in Table 18 and Table 20 further showcase the effectiveness of the proposed ATMALS-GSK algorithm. In these tables, “Ranks” provide the average Friedman test rankings of the different algorithms over 51 runs for each dimension (D = 10, 30, 50, and 100). Furthermore, “Mean Rank” calculates the average rankings for all dimensions. Finally, “Overall Rank” provides the sorted rank of the different algorithms based on their Mean Rank.

Furthermore, Figure 13 and Figure 14 visually reinforce the outcomes derived from the Friedman test shown in Table 18 and Table 20, in terms of the “Mean Rank” and “Overall Rank” of the different algorithms. In Figure 13, based on the “Overall Rank” results, ATMALS-GSK secures the top rank when compared with its GSK-based variants across all dimensions. Similarly, Figure 14 presents a comparative ranking against other metaheuristics, where ATMALS-GSK again outperforms existing metaheuristics by a significant margin. The consistent first-place ranking across all dimensions highlights its competitiveness and dominance in achieving optimal solutions.

When compared to its GSK-based variants (Table 17 and Table 18 and Figure 13) and other well-known metaheuristics (Table 19 and Table 20 and Figure 14), ATMALS-GSK consistently performs better. The Wilcoxon test demonstrates that it significantly outperforms most of the competing methods across all problem dimensions (D = 10 to 100), with p-values often much lower than 0.05, especially when compared to the original GSK and its variants. Although ATMALS-GSK performs similarly to improved variants like AGSK and APGSK at lower dimensions (e.g., for D = 10), it becomes noticeably superior as the problem size increases. It also clearly outperforms popular metaheuristics like IADE, GJO-JOS, QCSCA, and cSM in nearly all cases. These results show that ATMALS-GSK is not only effective but also consistently reliable across a wide range of benchmark problems and dimensions.

4.6. Parametric Study of ATMALS-GSK

This section provides a parametric study of the proposed ATMALS-GSK algorithm to evaluate the individual effect of each contribution to the overall performance. To achieve this purpose, we compare ATMALS-GSK with its five components (ATMALS-GSK-1 to ATMALS-GSK-5). For each algorithm, one change among the five adaptive parameters (K, K_r, K_f, P, and P_LS) is enabled, while keeping the other four parameters fixed as the basic GSK algorithm. The aim is to show that our algorithm ATMALS-GSK (with five adaptive parameters) is superior to ATMALS-GSK-1, ATMALS-GSK-2, ATMALS-GSK-3, ATMALS-GSK-4, and ATMALS-GSK-5 (each considering just one parameter as adaptive and fixing the other parameters).

In the following, we test the individual components of the ATMALS-GSK algorithm to assess the impact and efficiency of the modifications made to the GSK algorithm and to verify that ATMALS-GSK performs as expected. The ATMALS-GSK algorithm includes five main modifications compared to the original GSK algorithm: GSK with adaptive K (ATMALS-GSK-1), GSK with adaptive K_r (ATMALS-GSK-2), GSK with adaptive K_f (ATMALS-GSK-3), GSK with adaptive P (ATMALS-GSK-4), and GSK with adaptive local search mechanism P_LS (ATMALS-GSK-5). These modifications influence the convergence and balance the exploration and exploitation aspects of the ATMALS-GSK algorithm.

To assess the effectiveness of the five ATMALS-GSK components, these versions are compared with ATMALS-GSK and GSK on CEC 2017 with D = 30. The average results of ATMALS-GSK, ATMALS-GSK-1, ATMALS-GSK-2, ATMALS-GSK-3, ATMALS-GSK-4, ATMALS-GSK-5, and GSK over 51 runs on different problems are reported in

Table 21. Comparison of the average results of ATMALS-GSK with original GSK and five ATMALS-GSK versions over 51 runs for D = 30 on CEC 2017.

Fun.	GSK	ATMALS-GSK-1	ATMALS-GSK-2	ATMALS-GSK-3	ATMALS-GSK-4	ATMALS-GSK-5	ATMALS-GSK
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00
3	7.07 × 10−7	0.00	5.03 × 10−7	4.28 × 10−7	1.75 × 10−7	0.00	0.00
4	1.11 × 101	4.32	9.32	1.18 × 101	7.93	3.27	1.99
5	1.60 × 102	6.42 × 101	4.29 × 101	1.13 × 101	6.11 × 101	2.82 × 101	1.53 × 101
6	1.52 × 10−6	1.37 × 10−6	9.86 × 10−5	4.80 × 10−5	7.12 × 10−5	1.12 × 10−6	6.82 × 10−5
7	1.87 × 102	8.44 × 101	6.93 × 101	8.33 × 101	1.63 × 102	8.27 × 101	4.33 × 101
8	1.55 × 102	1.34 × 102	4.28 × 101	7.30 × 101	5.25 × 101	1.35 × 101	1.35 × 101
9	0.00	6.52 × 10−1	0.00	2.58 × 10−3	5.22 × 10−3	0.00	1.79 × 10−2
10	6.69 × 103	5.23 × 103	4.22 × 103	6.88 × 103	6.48 × 103	5.40 × 103	2.80 × 103
11	3.30 × 101	2.91 × 101	3.55 × 101	3.08 × 101	2.01 × 101	2.32 × 101	1.46 × 101
12	6.62 × 103	6.82 × 103	6.04 × 103	6.22 × 103	5.86 × 103	5.14 × 103	3.84 × 103
13	9.83 × 101	7.23 × 101	6.45 × 101	6.38 × 101	8.91 × 101	7.40 × 101	3.01 × 101
14	5.69 × 101	3.13 × 101	7.48 × 101	6.11 × 101	5.22 × 101	4.17 × 101	1.78 × 101
15	1.45 × 101	1.41 × 101	1.48 × 101	1.52 × 101	1.50 × 101	1.42 × 101	1.43 × 101
16	7.96 × 102	4.33 × 102	7.12 × 102	6.53 × 102	5.44 × 102	4.67 × 102	2.53 × 102
17	1.89 × 102	1.05 × 102	1.78 × 102	8.31 × 101	2.22 × 102	7.99 × 101	4.71 × 101
18	3.68 × 101	3.55 × 102	3.81 × 101	3.39 × 101	3.97 × 101	3.45 × 101	3.02 × 101
19	1.29 × 101	1.28 × 101	1.45 × 101	1.34 × 101	1.38 × 101	1.27 × 101	1.30 × 101
20	1.08 × 102	9.62 × 101	8.77 × 101	7.30 × 101	7.75 × 101	8.91 × 101	4.74 × 101
21	3.46 × 102	3.88 × 102	3.29 × 102	3.14 × 102	3.01 × 102	2.98 × 102	2.66 × 102
22	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102	1.00 × 102
23	4.70 × 102	4.48 × 102	4.96 × 102	4.50 × 102	4.04 × 102	4.10 × 102	3.44 × 102
24	5.68 × 102	5.24 × 102	5.75 × 102	5.33 × 102	5.59 × 102	5.21 × 102	4.75 × 102
25	3.87 × 102	3.88 × 102	3.86 × 102	3.87 × 102	3.89 × 102	3.86 × 102	3.86 × 102
26	9.87 × 102	9.77 × 102	1.23 × 103	1.55 × 103	2.26 × 103	9.54 × 102	1.79 × 103
27	4.93 × 102	5.03 × 102	5.38 × 102	6.06 × 102	4.91 × 102	4.97 × 102	4.96 × 102
28	3.21 × 102	3.17 × 102	3.24 × 102	3.15 × 102	3.20 × 102	3.16 × 102	3.10 × 102
29	5.77 × 102	5.42 × 102	6.37 × 102	5.58 × 102	5.25 × 102	5.12 × 102	4.32 × 102
30	2.08 × 103	2.07 × 103	2.08 × 103	2.15 × 103	2.12 × 103	2.07 × 103	2.06 × 103

. The results show that all components impact the efficiency of the combined modifications and confirm that the ATMALS-GSK algorithm functions as intended. It is evident that the ATMALS-GSK algorithm significantly outperforms others in F1, F3, F4, F7, F8, F10, F11, F12, F13, F14, F16, F17, F18, F20, F21, F22, F23, F24, F25, F28, F29, and F30.

The statistical analysis results are provided in

Table 22. Wilcoxon signed-rank test of the ATMALS-GSK algorithm with the original GSK and five ATMALS-GSK versions over 51 runs for D = 30 on CEC 2017.

Methods		R+	R−	p Value	+	=	−	Dec.
ATMALS-GSK vs.	GSK	337	41	0.0003	22	2	5	+
	ATMALS-GSK-1	321	30	0.0002	22	3	4	+
	ATMALS-GSK-2	324	27	0.0002	24	3	2	+
	ATMALS-GSK-3	341	37	0.0003	23	2	4	+
	ATMALS-GSK-4	363	15	0.0000	24	2	3	+
	ATMALS-GSK-5	268	32	0.0007	19	5	5	+

and Figure 15. The Wilcoxon test results in Table 22 indicate that the proposed ATMALS-GSK algorithm surpasses its five versions in 22, 24, 23, 22, and 19 functions, which results in success ratios of ATMALS-GSK against ATMALS-GSK-1 to ATMALS-GSK-5 by 75.9%, 82.7%, 79.3%, 82.7%, and 65.5%, respectively. The rank analysis using the Friedman test in Figure 15 demonstrates the effectiveness of applying all modifications of the ATMALS-GSK algorithm together. According to the mean and overall ranks in Figure 15, ATMALS-GSK secures the top rank when compared with GSK and the ATMALS-GSK components. The Friedman test yields a p-value of 4.11 × 10⁻¹⁰, indicating a significant difference in ATMALS-GSK performance compared to GSK and the components of the ATMALS-GSK algorithm.

5. Conclusions

This paper introduced ATMALS-GSK, an enhanced version of the GSK algorithm, empowering auto-tuning memory-based historical information and adaptive local search mechanisms. Unlike the original GSK algorithm, which relies on fixed control parameters and often struggles with premature convergence, the proposed ATMALS-GSK algorithm integrates memory-based auto-tuning and an adaptive local search strategy. The key advantage of the proposed ATMALS-GSK algorithm lies in the adaptive Gaussian-driven adjustment of its controllable parameters using a historical memory mechanism that tracks their effectiveness over time. A historical memory-powered RWS, informed by a weighted moving average, ensures that the algorithm dynamically prioritizes more successful configurations. Furthermore, the adaptive local search component focuses on fine-tuning senior solutions, enabling better exploration of the search space and avoiding getting stuck in local optima.

Extensive experiments conducted on the CEC 2011 and CEC 2017 benchmark suites have demonstrated the superiority of the proposed ATMALS-GSK algorithm over the original GSK and its modified variants, as well as other recent metaheuristic algorithms. The Wilcoxon test results indicated that ATMALS-GSK surpasses the original GSK and its variants by obtaining success ratios of 75%, 74.1%, 78.4%, 66.4%, and 65.5% against GSK, AGSK, APGSK, FDBAGSK, and eGSK, respectively. Furthermore, ATMALS-GSK outperforms other metaheuristic algorithms—IADE, GJO-JOS, QCSCA, cSM, and FOWFO—achieving success ratios of 96.6%, 96.6%, 87.1%, 91.4%, and 54.3%, respectively, across all functions in overall dimensions. The Friedman test results also indicate that ATMALS-GSK ranks first across all dimensions in terms of average performance. The p-values from both tests confirm that these improvements are statistically significant. These results consistently showed improved robustness and higher solution quality across a variety of complex optimization problems.

Despite these achievements, the proposed algorithm is not without limitations. The introduction of memory-based mechanisms and local search strategies inevitably adds computational overhead, which might become significant in high-dimensional or real-time applications. Additionally, while the Gaussian-based parameter tuning works well in the current setup, it might need recalibration or refinement for highly irregular or domain-specific fitness landscapes. Furthermore, the performance of ATMALS-GSK may vary based on the characteristics of the problem being solved, suggesting that further customization or hybridization with problem-specific heuristics might be beneficial.

Therefore, there are several promising directions for future research. One main area involves exploring alternative memory management techniques, such as adaptive forgetting factors or reinforcement learning-inspired models, to further improve the parameter selection strategy. Integrating multi-objective capabilities into ATMALS-GSK would also broaden its applicability to problems with competing criteria. Moreover, the use of parallel and distributed computing could help mitigate the added computational complexity and make the algorithm more suitable for large-scale real-time problems. Finally, applying the ATMALS-GSK algorithm to real-world engineering optimization problems (especially those with discrete and integer variables) and exploring hybridization with problem-specific heuristics would be a promising direction for future work.