
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71828
10.1038/s41598-024-71828-y
Article
PMSOMA: optical microscope algorithm based on piecewise linear chaotic mapping and sparse adaptive exploration
Guo Linyi
Gu Wei 20171054@hbut.edu.cn

https://ror.org/02d3fj342 grid.411410.1 0000 0000 8822 034X School of Computer Science, Hubei University of Technology, Wuhan, 430074 China
6 9 2024
6 9 2024
2024
14 2084926 3 2024
30 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
The optical microscope algorithm (OMA) is a metaheuristic algorithm that draws inspiration from the magnifying functionality of optical microscopes. This study introduces an enhanced variant of OMA, termed PMSOMA, designed to mitigate the original version's limitations, notably its slow convergence rates and vulnerability to local optima. PMSOMA integrates a piecewise linear chaotic map to refine population initialization and augment diversity, alongside a sparse adaptive exploration mechanism to bolster search efficacy. The performance of PMSOMA was rigorously tested using a suite of 50 benchmark functions, the CEC2017 test suite, feature selection datasets, and three classical engineering challenges. The empirical findings confirm that PMSOMA surpasses both the original OMA and competing algorithms by delivering superior solutions, accelerating convergence, and demonstrating enhanced robustness in convergence.

Keywords

Optical microscope algorithm
Piecewise linear chaotic mapping
Sparse adaptive exploration mechanism
Engineering problem
Subject terms

Engineering
Mathematics and computing
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Metaheuristic algorithms have gained prominence in solving complex problems across a range of disciplines, thanks to their ease of implementation, robust capabilities, and clear underlying principles. These algorithms excel in global exploration and optimization, demonstrating their effectiveness across various domains.

Agrawal et al1. have provided a classification of metaheuristic algorithms based on their inspiration sources, dividing them into four main categories: swarm intelligence, evolution, physics, and human behaviour. Swarm intelligence algorithms draw inspiration from the collective behaviours of insects, animals, fish, and birds, aiming to emulate their social dynamics. Notable examples include particle swarm optimization (PSO)2, differential evolution (DE)3, the firefly algorithm (FA)4, and ant colony optimization (ACO)5. Evolution-based algorithms, on the other hand, are influenced by genetic evolution principles, embodying the concept of natural selection where the most adaptable individuals are more likely to survive. This category includes genetic programming (GP)6 and the evolution strategy algorithm (ESA)7. Physics-based algorithms derive their mechanisms from physical laws, with examples such as the gravitational search algorithm (GSA)8, electromagnetic field optimization (EFO)9, and simulated annealing (SA)10. Lastly, algorithms inspired by human behaviour aim to mimic human societal and cognitive processes, including the cultural algorithm (CA)11, social learning optimization (SLO)12, and forensic investigation (FI)13.

Diversification (exploration) and intensification (exploitation) are pivotal phases in the functioning of metaheuristic algorithms, with the primary challenge being the effective balance between these two to efficiently arrive at near-optimal solutions14,15. Diversification focuses on the exploration of new solutions across various regions, while intensification is dedicated to refining and enhancing solutions within a specific area16. Achieving a harmonious balance and synergistic interplay between these phases is essential for the successful pursuit of global optimal solutions.

The optical microscope algorithm (OMA)17, introduced in 2023 as an intelligent optimization algorithm, draws inspiration from the magnification process of optical microscopes targeting specific objects. OMA employs two key phases for target acquisition: exploration, facilitated by the objective lens, and exploitation, conducted through the eyepiece. Despite its robustness, simplicity in implementation, and minimal control parameters, OMA faces challenges such as slow convergence rates and a tendency to get trapped in local optima, which complicates its application to complex, high-dimensional problems.

To address the challenges faced by the OMA, we introduce a novel algorithm that integrates piecewise linear chaotic mapping (PWLCM) and sparse adaptive exploration into the OMA framework. This enhanced algorithm is named PMSOMA. In the initial phase of population setup, PMSOMA employs PWLCM to rectify the issue of uneven population distribution encountered in OMA. Furthermore, it adopts a sparse adaptive exploration mechanism, enabling optimal individuals to conduct piecewise adaptive random searches across each dimension. This approach facilitates the entire population's escape from local optima. Additionally, PMSOMA incorporates a dynamic range adjustment strategy to bolster the algorithm's overall performance, significantly improving its exploration capabilities both before and after the primary search phase.

The performance of PMSOMA was evaluated on various datasets, including benchmark functions, standard testing sets, classification datasets, function approximation datasets, and three engineering problems. The experimental results demonstrate the superior performance of PMSOMA compared with the original OMA.

The main contributions of this paper are as follows:Enhancement of the original algorithm through the adoption of a piecewise linear chaotic map, which addresses the challenge of uneven population distribution and enriches population diversity.

Integration of a sparse adaptive exploration mechanism to mitigate the original OMA’s propensity for local optima entrapment, thereby augmenting its solution-searching efficiency.

Rigorous evaluation of PMSOMA across 50 benchmark functions, the CEC2017 test suite, a feature selection dataset, and three classical engineering scenarios, demonstrating its robustness and effectiveness.

Utilization of Friedman and Wilcoxon signed-rank tests to conduct a comparative analysis of PMSOMA against eight other heuristic algorithms, with the findings presented through statistical interpretation.

The remainder of this paper is organized as follows: In Section "Related works", the inspiration, the implementation and existing research background of the original OMA algorithm are described in detail. Section "Methodology" presents the implementation idea, mechanism analysis and detailed procedure of the improved PMSOMA algorithm. In Section "Experimental results and discussion", the performance of PMSOMA is verified on a wide range of datasets. Our conclusions are summarized in Section "Conclusion".

Related works

Optical microscope algorithm

The OMA is inspired by the physical principles of microscopy, simulating the zooming process on a target object by an observer. In OMA, “magnification” refers to the process of enlarging the target object, achieved through adjustments of the objective lens and eyepiece to reach optimal magnification and focus.

The overall magnification of the target in OMA is derived from the magnification principle used in compound microscopes, as shown in Eq. (1) 18.1 Mtotal=Mo∗ME

here, Mtotal is the total visual magnification of the microscope, Mo is the magnification of the objective lens, and ME is the magnification of the eyepiece. The magnification equation for the objective lens is expressed by Eq. (2) 18.2 Mo=Lfo

here, L is the tube length of the microscope, and fo is the focal length of the objective lens.

The modified target object (Minew) in the objective lens phase is expressed mathematically by Eq. (3).3 Minew=Mi+MT∗1.40∗Mbest

here, Mi denotes the initial target object, and the magnification scale (MT) is used for the initial magnification, representing random magnification values ranging from zero to one. The value of 1.40 is the numerical aperture corresponding to the maximum magnification of the objective lens. The modified target object (Mi new) is then compared with the current object, and the better of the two is selected as the best magnification (Mbest).

The second lens of the microscope is the eyepiece, and the magnification equation for the eyepiece is expressed according to Eq. (4).4 Me=Dfe

here, D is the minimum viewing distance, and fe is the focal length of the eyepiece.

The eyepiece phase, which is a specific phase for advanced magnification, uses Eqs. (5) and (6) to simulate the magnification and modification patterns of the target object, respectively 17.5 space=Mj-MiiffMi≥fMjMi-MjiffMi<fMj

6 Minew=Mi+Mr∗0.55∗space

here, Mi and Mj denote different initial target objects, and “space” refers to the local search space for magnification. Mr represents random magnification values ranging from zero to one. The coefficient 0.55 corresponds to the numerical aperture associated with the eyepiece’s maximum magnification capacity. The modified target object (Minew) is compared with the current object, and the object that produces the superior outcome is selected as the optimal magnification (Mbest). The pseudocode for the OMA is presented in Algorithm 1.

Algorithm 1 Pseudocode for the OMA.

Algorithm analysis

Fusing Eqs. 5 and 6 leads to Eq. 7, which shows that the search for the target object takes 0.55∗Mr as coefficients, where the setting of the parameter 0.55 affects the search of the algorithm, and that the search space is drastically reduced when the target objects Mj and Mi are approximated, causing the algorithm to fall into a locally optimal solution. Therefore, the dispersion of the initialization population, and the setting of the algorithm parameter 0.55, can seriously affect the search performance of the algorithm.7 Minew=(1-0.55∗Mr)Mi+0.55∗Mr∗Mj,iffMi≥fMj(1+0.55∗Mr)Mi-0.55∗Mr∗Mj,iffMi<fMj

In this paper, we propose a fusion strategy to improve the defects of the OMA, select the chaos mapping algorithm to improve the initialization of the algorithm population, and propose the sparse adaptive exploration to enhance the ability of population individuals to jump out of the local optimum, in order to solve the algorithm's problems and improve the performance of the algorithm.

Related research

Despite the benefits of intelligent optimization algorithms, such as their low computational complexity, straightforward parameters, and ease of implementation, they encounter challenges including slow convergence rates and reduced accuracy of solutions. To mitigate these issues, significant research and development efforts have been made. Rong introduced reverse learning and solutions to augment population diversity19. Singh enhanced exploration capabilities by integrating a sine–cosine operator for position updates in the sea squirt algorithm20. Li et al21. used a logistic map to improve the population initialization of the dung beetle optimizer algorithm, while Kohli et al22. employed various one-dimensional chaotic maps to adjust critical parameters and improve the performance of the grey wolf optimizer. Song et al23. using chaotic interference factor and elementary mathematical to improve the pelican optimization algorithm. Considering ant colony optimization, Cui et al24. introduced a direction-guidance mechanism and three other strategies to optimize parameters and achieve efficient convergence. Çelik utilized information exchange and quasi-opposition learning to achieve a desirable balance between exploration and exploitation25. Moreover, Wang et al26. using a chaotic mapping strategy involving dynamic inertia weighting based on cosine variation and a position update strategy to enhance the golden jackal optimization algorithm. Furthermore, Fan et al27. developed an advanced version of the African vulture optimization algorithm, utilizing time-varying techniques and tent map chaos to boost performance.

The OMA is a new algorithm initially proposed in September, 202317, although the OMA algorithm has good performance, it still has some defects, such as low quality of initial population, easy to fall into local optimal solution, low accuracy of obtaining optimal solution, and it has not yet been improved upon. Currently, no scholars have made defective improvements and performance enhancements to OMA, and the proposed PMSOMA system can fill this gap.

Methodology

The original OMA is recognized for its robust search capabilities but is hindered by slow convergence rates and a tendency to fall into local optima. These drawbacks challenge its effectiveness in tackling complex, high-dimensional problems. PMSOMA design integrates PWLCM and sparse adaptive exploration to enhance performance compared with the original OMA.

Population initialisation utilising PWLCM

The population initialization of the OMA is performed according to Eq. (8), where i corresponds to NP and k corresponds to dim. The calculation of x(i, j) is defined in Eq. (9), where Ub and Lb represent the upper and lower bounds of the given search space, respectively.8 populationi,j=Lbj+xi,j∗Ubj-Lbj

9 xi+1,:=x(i,:).∗1-xi,:

With NP set to 100 and dim set to 30, and with Ub set to 1 and Lb set to − 1, the scatter plot and histogram in Fig. 1 represent the initialization of the population. It can be observed that the majority of individuals in the population initialized by the OMA are concentrated in the range of [− 1, − 0.5]. The distribution is not uniform, which can contribute to premature convergence and the algorithm becoming trapped in local optima.Fig. 1 Scatter plot and histogram of a population initialized by the OMA.

Chaos mapping, a complex dynamical method present in nonlinear systems, possesses a set of deterministic equations governing its underlying mechanisms. This enables the method to manifest stochastic behavior externally while maintaining superior dynamical and statistical properties. The traversal and non-repeating nature of chaotic mapping facilitate a more comprehensive exploration of the solution space by members of the initial population28,29. Extensive research suggests that the integration of chaos mapping into meta-heuristic search algorithms enhances the preservation of population diversity30–36.

There are 10 commonly used chaotic mapping algorithms: Tent, Logistic, Cubic, Chebyshev, Piecewise, Sinusoidal, Sine, ICMIC, Circle, and Bernoulli. In this study, these 10 chaotic mapping methods combined with subsequent improvements were individually applied to the OMA and tested on the CEC2017 standard testing set. The number of dimensions was set at 10, and 3000 iterations were executed. The mean values were calculated after 30 independent runs. The results are presented in Table 1. The mean and Friedman test results indicate that the piecewise mapping algorithm outperformed the other nine chaotic mapping algorithms when applied to the OMA. The descriptions of the CEC2017 testing set and the Friedman assay are given in the "Experiments" section. Table 1 Testing results for the 10 chaotic mapping algorithms applied to the OMA.

	Tent	Logistic	Cubic	Chebyshev	Piecewise	Sinusoidal	Sine	ICMIC	Circle	Bernoulli	
Frideman	6.0167	8.5000	5.6167	5.0500	3.0333	5.7500	5.5500	5.2500	5.4500	4.7833	
rank	9	10	7	3	1	8	6	4	5	2	
F1	1.84E + 03	3.73E + 03	1.28E + 03	9.11E + 02	1.10E + 03	1.29E + 03	1.33E + 03	9.36E + 02	1.08E + 03	1.08E + 03	
F2	2.00E + 02	2.15E + 12	2.00E + 02	2.00E + 02	2.00E + 02	2.04E + 02	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	
F3	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	
F4	4.04E + 02	4.07E + 02	4.04E + 02	4.04E + 02	4.05E + 02	4.09E + 02	4.04E + 02	4.04E + 02	4.04E + 02	4.08E + 02	
F5	5.21E + 02	5.85E + 02	5.19E + 02	5.12E + 02	5.10E + 02	5.13E + 02	5.11E + 02	5.11E + 02	5.10E + 02	5.14E + 02	
F6	6.00E + 02	6.30E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	
F7	7.48E + 02	7.92E + 02	7.42E + 02	7.35E + 02	7.28E + 02	7.46E + 02	7.32E + 02	7.32E + 02	7.36E + 02	7.31E + 02	
F8	8.21E + 02	8.24E + 02	8.17E + 02	8.11E + 02	8.09E + 02	8.13E + 02	8.11E + 02	8.10E + 02	8.13E + 02	8.11E + 02	
F9	9.01E + 02	1.60E + 03	9.01E + 02	9.01E + 02	9.00E + 02	9.01E + 02	9.01E + 02	9.02E + 02	9.00E + 02	9.01E + 02	
F10	1.86E + 03	1.99E + 03	1.77E + 03	1.83E + 03	1.58E + 03	1.75E + 03	1.78E + 03	1.81E + 03	1.79E + 03	1.71E + 03	
F11	1.12E + 03	1.12E + 03	1.12E + 03	1.12E + 03	1.11E + 03	1.12E + 03	1.12E + 03	1.12E + 03	1.12E + 03	1.12E + 03	
F12	1.26E + 04	2.22E + 07	1.14E + 04	9.87E + 03	9.35E + 03	8.06E + 03	1.26E + 04	1.06E + 04	1.07E + 04	1.16E + 04	
F13	3.36E + 03	1.56E + 08	3.42E + 03	2.33E + 03	2.12E + 03	2.44E + 03	2.17E + 03	2.38E + 03	3.44E + 03	2.45E + 03	
F14	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	1.43E + 03	
F15	1.53E + 03	1.75E + 03	1.53E + 03	1.53E + 03	1.52E + 03	1.53E + 03	1.53E + 03	1.53E + 03	1.54E + 03	1.53E + 03	
F16	1.62E + 03	2.12E + 03	1.62E + 03	1.62E + 03	1.61E + 03	1.62E + 03	1.61E + 03	1.61E + 03	1.61E + 03	1.62E + 03	
F17	1.74E + 03	1.87E + 03	1.74E + 03	1.74E + 03	1.73E + 03	1.75E + 03	1.74E + 03	1.74E + 03	1.74E + 03	1.74E + 03	
F18	2.27E + 03	2.05E + 08	1.96E + 03	3.25E + 03	3.14E + 03	2.46E + 03	3.96E + 03	3.77E + 03	3.38E + 03	3.99E + 03	
F19	1.91E + 03	1.91E + 03	1.91E + 03	1.92E + 03	1.91E + 03	1.92E + 03	1.92E + 03	1.92E + 03	1.91E + 03	1.92E + 03	
F20	2.03E + 03	2.39E + 03	2.03E + 03	2.02E + 03	2.02E + 03	2.03E + 03	2.03E + 03	2.03E + 03	2.03E + 03	2.02E + 03	
F21	2.27E + 03	2.20E + 03	2.26E + 03	2.27E + 03	2.27E + 03	2.28E + 03	2.28E + 03	2.28E + 03	2.26E + 03	2.25E + 03	
F22	2.30E + 03	2.39E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	
F23	2.62E + 03	2.69E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.63E + 03	2.62E + 03	
F24	2.76E + 03	2.67E + 03	2.72E + 03	2.72E + 03	2.73E + 03	2.72E + 03	2.73E + 03	2.73E + 03	2.74E + 03	2.71E + 03	
F25	2.92E + 03	3.00E + 03	2.92E + 03	2.92E + 03	2.92E + 03	2.91E + 03	2.93E + 03	2.94E + 03	2.93E + 03	2.92E + 03	
F26	3.05E + 03	3.57E + 03	2.99E + 03	2.97E + 03	2.99E + 03	2.97E + 03	2.95E + 03	2.94E + 03	2.94E + 03	2.93E + 03	
F27	3.10E + 03	3.17E + 03	3.10E + 03	3.14E + 03	3.10E + 03	3.10E + 03	3.19E + 03	3.11E + 03	3.10E + 03	3.10E + 03	
F28	3.07E + 03	3.14E + 03	3.20E + 03	3.30E + 03	3.22E + 03	3.37E + 03	3.30E + 03	3.28E + 03	3.16E + 03	3.21E + 03	
F29	3.26E + 03	3.32E + 03	3.18E + 03	3.18E + 03	3.18E + 03	3.18E + 03	3.18E + 03	3.18E + 03	3.18E + 03	3.18E + 03	
F30	7.33E + 03	6.28E + 03	8.29E + 04	2.03E + 04	1.24E + 05	5.61E + 03	4.67E + 03	6.24E + 03	7.83E + 03	2.13E + 05	
Significant values are in [bold].

Previously proposed population initialization methods for meta-heuristic algorithms include pseudo- and quasi-random number methods, chaos theory, and probability distributions37. To further validate the suitability of chaotic mapping in OMA systems, this study compares the currently employed and prevalent methods of good point set mapping, Halton mapping, and Sobol mapping with the PWLCM mapping chosen from the aforementioned experiments. The findings are presented in Table 2. It can be seen that the chaotic mapping initialization performs better than the other methods and is more suitable for the OMA, which may stem from its traversal and non-repetitive nature. Table 2 Testing results of different population initialization methods applied to the OMA.

	Piecewise	Good point	Halton	Sobol	
 + /–/ = 	NAN	26/4/0	26/4/0	23/7/0	
Frideman	1.5333	3.3000	2.8667	2.3000	
rank	1	4	3	2	
F1	1.10E + 03	3.05E + 03	1.21E + 03	1.15E + 03	
F2	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	
F3	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	
F4	4.05E + 02	4.02E + 02	4.04E + 02	4.04E + 02	
F5	5.10E + 02	6.17E + 02	5.10E + 02	5.11E + 02	
F6	6.00E + 02	6.55E + 02	6.00E + 02	6.00E + 02	
F7	7.28E + 02	8.09E + 02	7.30E + 02	7.29E + 02	
F8	8.09E + 02	8.35E + 02	8.11E + 02	8.10E + 02	
F9	9.00E + 02	1.54E + 03	9.01E + 02	9.02E + 02	
F10	1.58E + 03	1.82E + 03	1.77E + 03	1.73E + 03	
F11	1.11E + 03	1.12E + 03	1.12E + 03	1.12E + 03	
F12	9.35E + 03	5.77E + 03	9.21E + 03	1.01E + 04	
F13	2.12E + 03	3.67E + 03	2.26E + 03	2.38E + 03	
F14	1.43E + 03	1.44E + 03	1.43E + 03	1.43E + 03	
F15	1.52E + 03	1.56E + 03	1.56E + 03	1.53E + 03	
F16	1.61E + 03	2.08E + 03	2.05E + 03	1.61E + 03	
F17	1.73E + 03	1.80E + 03	1.80E + 03	1.74E + 03	
F18	3.14E + 03	1.38E + 04	1.42E + 04	3.79E + 03	
F19	1.91E + 03	1.97E + 03	1.97E + 03	1.92E + 03	
F20	2.02E + 03	2.37E + 03	2.40E + 03	2.03E + 03	
F21	2.27E + 03	2.34E + 03	2.33E + 03	2.27E + 03	
F22	2.30E + 03	2.38E + 03	2.43E + 03	2.30E + 03	
F23	2.62E + 03	3.09E + 03	3.00E + 03	2.62E + 03	
F24	2.73E + 03	2.53E + 03	2.51E + 03	2.72E + 03	
F25	2.92E + 03	2.94E + 03	2.94E + 03	2.92E + 03	
F26	2.99E + 03	3.90E + 03	4.01E + 03	2.96E + 03	
F27	3.10E + 03	3.21E + 03	3.20E + 03	3.10E + 03	
F28	3.22E + 03	3.34E + 03	3.34E + 03	3.27E + 03	
F29	3.18E + 03	3.23E + 03	3.23E + 03	3.17E + 03	
F30	1.24E + 05	6.09E + 03	5.76E + 03	3.37E + 04	
Significant values are in [bold].

PWLCM is a segmented mapping function that uses an initial position value as a segmentation indicator and adopts different calculation formulas for individuals at different positions. The formula for the PWLCM is defined as follows38:10 x(i+1)=x(i)d,0≤x(i)≤dx(i)-d0.5-d,d≤x(i)<0.51-d-x(i)0.5-d,0.5≤x(i)<1-d1-x(i)d,1-d≤x(i)<1.

here, d is a control parameter used to determine the segmentation range of the four-segment formula. Wang et al39. proved that the optimal results are achieved when d = 0.3, and this value was therefore used in all experiments in this study. We used the results of Eq. (10) to initialize the positions of the individuals in the initial population as follows:11 Mi=Ub-Lbxi+Lb.

here, M(i) initializes the location of each target object.

The population for PMSOMA was initialized using the PWLCM with NP set to 100, dim set to 30, Ub set to 1, and Lb set to −1. Fig. 2 shows the corresponding scatter plot and frequency distribution histogram. Comparing Figs. 1 and 2 shows that within the given parameter range, the chaotic sequence generated by PWLCM is more evenly distributed within the search space than the initial sequence generated by the original algorithm.Fig. 2 Scatter plot and histogram of a population initialized using PWLCM.

Sparse adaptive exploration

The original OMA was enhanced by incorporating sparse adaptive exploration and integrating a random search in each dimension to ensure the diversity of individual positions and prevent the algorithm from becoming trapped in local optima. Additionally, the scope of the search is dynamically adjusted based on the movement of individuals. If the result of the current iteration is inferior to that of the previous iteration, then the previous step is repeated until a superior solution is discovered. The probability of obtaining a better solution is effectively enhanced by this fusion method.

Additionally, the utilization of a sampling control mechanism determines the execution of the stepwise search mechanism. The stepwise mechanism does not take effect initially but begins to be used at a specific stage, thereby effectively delaying algorithm convergence and preventing entrapment in local optima. Sparse adaptive exploration is performed based on the actual results obtained during the search process, as shown in Eqs. (12) and (13).12 ift>M&&t%N1==0

13 Xbt=Xbt±C·Rand·U-L·e-k·tTmax

Here, t represents the number of iterations, and control intervention is activated only when the search has reached the M-th generation. N1 refers to the number of iterations in the control interval. C is a dynamic control parameter, and its value is related to the success rate of the sparse search mechanism. Rand denotes a random number between zero and one. Additionally, e-k∙tTmax is an adaptive control parameter that is subject to variation based on the position of the target object, and k is the convergence rate control parameter.

Mechanism analysis

The analysis presented in Section "Algorithm analysis" reveals that the original OMA algorithm's search space is significantly diminished when target objects are approximated. This results in the emergence of super-individuals and rapid convergence, which also leads to a reduction in population diversity and a subsequent tendency to converge to local optima. The fusion mechanism introduced in this study addresses these issues. Initially, the application of a chaotic mapping strategy boosts the diversity of the initial population, thereby mitigating the impact of super-individuals. Furthermore, in the original algorithmic framework, the search scope for new individuals is contingent upon the existing population. In scenarios where there is a high degree of similarity among the population's members, the algorithm struggles to escape the local optimal solution. In contrast, the sparse adaptive exploration mechanism proposed here features a search scope that is not contingent upon the aggregation state of the group but is instead dynamically modified based on individual positions and iteration counts. This mechanism, when integrated with the chaotic mapping strategy, effectively counters the premature convergence issue arising from the approximation of group individuals. Additionally, the sparse adaptive exploration mechanism is introduced into the algorithm after an initial phase of conventional searching, thereby preventing the premature emergence of super-individuals, this approach also ensures that the sparse exploration mechanism functions effectively. Moreover, interspersing a few generations between policy searches enhances the efficiency of the algorithm's search process.

PMSOMA process

The pseudocode for PMSOMA, which incorporates the aforementioned improvement strategies, is presented in Algorithm 2. A flowchart of PMSOMA is shown in Fig. 3.Fig. 3 Flowchart of the PMSOMA.

Algorithm 2 Pseudocode for PMSOMA

Experimental results and discussion

In this section, 50 benchmark functions15,18,40–55, the CEC2017 testing set, feature selection testing sets, and four classical engineering problems are used to evaluate the performance of the proposed algorithm. Notably, the functions in the 50 benchmark functions and CEC2017 test sets do not overlap. All experiments were conducted using a PC running Microsoft Windows 11. The system was equipped with an Intel(R) Core(TM) i5-1135G7 (2.40 GHz) CPU and 16.0 GB of RAM. The MATLAB R2020a software was used for algorithm implementation.

In each trial, the iteration count was fixed at 3000, with the mean and standard deviation of the fitness function values computed from 30 independent runs. To highlight the advantages of PMSOMA comprehensively, we considered the classical WOA56 and GWO57 algorithms, as well as the newly developed APO58, FVIM59, COA60, SHIO61, and WSO62 algorithms for comparison.here, “ + ” and “ − ” indicate that PMSOMA results are substantially better or worse than those of the compared algorithms, respectively, whereas “ = ” signifies a non-significant outcome.

Benchmark function

The 50 benchmark test functions used in this study are classified into unimodal–separable functions (F1–F5), unimodal–non-separable functions (F6–F17), multimodal–separable functions (F18 to F26), and multimodal–non-separable functions (F27 to F50), covering a dimensional range from 2 to 30. Detailed information on these functions can be found in Table 317. Table 3 Benchmark functions considered for the experiment.

No	Function	Range	D	Opt	Type	
F1	Stepint	[–5.12,5.12]	5.00	0	US	
F2	Step	[–100,100]	30	0	US	
F3	Sphere	[–100,100]	30	0	US	
F4	SumSquares	[–10,10]	30	0	US	
F5	Quartic	[–1.28,1.28]	30	0	US	
F6	Beale	[–4.5,4.5]	2	0	UN	
F7	Easom	[–100,100]	2	–1	UN	
F8	Matyas	[–10,10]	2	0	UN	
F9	Colville	[–10,10]	4	0	UN	
F10	Trid 6	[–D2,D]	6	–50	UN	
F11	Trid 10	[–D2,D]	10	–210	UN	
F12	Zakharov	[–5,10]	10	0	UN	
F13	Powell	[–4,5]	24	0	UN	
F14	Schwefel 2.22	[–10,10]	30	0	UN	
F15	Schwefel 1.2	[–100,100]	30	0	UN	
F16	Rosenbrock	[–30,30]	30	0	UN	
F17	Dixon–Price	[–10,10]	30	0	UN	
F18	Foxholes	[–65.536,65.536]	2	0.998	MS	
F19	Branin	[–5,10]x[0,15]	2	0.398	MS	
F20	Bohachevsky 1	[–100,100]	2	0	MS	
F21	Booth	[–10,10]	2	0	MS	
F22	Rastrigin	[–5.12,5.12]	30	0	MS	
F23	Schwefel	[–500,500]	30	–12,569.50	MS	
F24	Michalewicz 2	[0,π]	2	–1.8013	MS	
F25	Michalewicz 5	[0,π]	5	–4.6877	MS	
F26	Michalewicz 10	[0,π]	10	–9.6602	MS	
F27	Schaffer	[–100,100]	2	0	MN	
F28	Six-Hump Camel	[–5,5]	2	–1.0316	MN	
F29	Bohachevsky 2	[–100,100]	2	0	MN	
F30	Bohachevsky 3	[–100,100]	2	0	MN	
F31	Shubert	[–10,10]	2	–186.7309	MN	
F32	GoldStein–Price	[–2,2]	2	3	MN	
F33	Kowalik	[–5,5]	4	0.000307	MN	
F34	Shekel 5	[0,10]	4	–10.1532	MN	
F35	Shekel 7	[0,10]	4	–10.4029	MN	
F36	Shekel 10	[0,10]	4	–10.5364	MN	
F37	Perm	[–D,D]	4	0	MN	
F38	PowerSum	[0,1]	4	0	MN	
F39	Hartman 3	[0,D]	3	–3.8628	MN	
F40	Hartman 6	[0,1]	6	–3.3224	MN	
F41	Griewank	[–600,600]	30	0	MN	
F42	Ackley	[–32,32]	30	0	MN	
F43	Penalized	[–50,50]	30	0	MN	
F44	Penalized 2	[–50,50]	30	0	MN	
F45	Langermann 2	[0,10]	2	–1.0809	MN	
F46	Langermann 5	[0,10]	5	–1.5	MN	
F47	Langermann 10	[0,10]	10	–1.34	MN	
F48	Fletcher–Powell 2	[–π,π]	2	0	MN	
F49	Fletcher–Powell 5	[–π,π]	5	0	MN	
F50	Fletcher–Powell 10	[–π,π]	10	0	MN	
D = dimensions (D), opt. = optimal value, U = unimodal, M = multimodal, S = separable, N = non-separable	
Significant values are in [bold].

The CEC2017 testing function is a set of functions utilized for evaluating optimization algorithm performances. In the existing CEC2017 test set, unimodal functions (F1–F3) have a global minimum, while multimodal functions (F4–F10) contain local extreme points. Mixed functions (F11–F20) comprise three or more CEC2017 benchmark functions with displacement or rotation, assigning each sub-function a specific weight. Composite functions (F21 to F30) consist of at least three mixed or benchmark functions with additional bias values and weights. The CEC2017 test function is distinguished by characteristics and complexities including non-linearity, multi-mode, and non-convexity, posing challenges such as local optimal solutions and dimension disasters for algorithms attempting to solve them.

Parameter setting

PMSOMA incorporates two innovative mechanisms (adaptive and sparse exploration mechanisms), as well as several control parameters. These parameters were subjected to numerical analysis to determine the optimal parameter combination. This process was performed passively, and the results highlighted the role of each mechanism in the performance of the algorithm.

Fes is a parameter representative of the progressive intervention parameters, whereas the MaxFEs values of all algorithms are assigned to a 20,000*dimensional space. FC, as the frequency control parameter, was optimally set to 20. C serves as the sparse control parameter, and the impact of different parameter combinations on algorithm performance is summarized in Table 4. Table 4 Combination of control parameters.

	iter	mod	k	
PMSOMA1	300	30	5	
PMSOMA2	300	30	6	
PMSOMA3	300	80	5	
PMSOMA4	300	80	6	
PMSOMA5	1000	30	5	
PMSOMA6	1000	30	6	
PMSOMA7	1000	80	5	
PMSOMA8	1000	80	6	

In each trial, the iteration count was fixed at 3000, with the mean and standard deviation of the fitness function values computed from 30 independent runs. The findings are summarized in Table 5. The CEC2017 testing set was employed. The combination of parameters under PMSOMA2 demonstrated superior performance, identifying 300 as the optimal number of iterations for initiating the local search strategy. The mod parameter was optimally set at 30 to manage the frequency of local searches and the convergence step control parameter k was optimally set to 6, which was used in further tests of the algorithm. Table 5 Test results for different dynamic control parameters.

	PMSOMA1	PMSOMA2	PMSOMA3	PMSOMA4	PMSOMA5	PMSOMA6	PMSOMA7	PMSOMA8	
Frideman	3.7667	2.3000	4.3667	5.0667	4.7333	4.5000	5.6667	5.6000	
rank	2	1	3	6	5	4	8	7	
F1	1.27E + 03	8.57E + 02	1.67E + 03	9.72E + 02	8.05E + 02	1.11E + 03	9.79E + 02	9.48E + 02	
F2	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	2.00E + 02	
F3	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	3.00E + 02	
F4	4.00E + 02	4.00E + 02	4.00E + 02	4.00E + 02	4.00E + 02	4.00E + 02	4.00E + 02	4.00E + 02	
F5	5.10E + 02	5.07E + 02	5.09E + 02	5.09E + 02	5.09E + 02	5.08E + 02	5.09E + 02	5.09E + 02	
F6	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	6.00E + 02	
F7	7.23E + 02	7.24E + 02	7.24E + 02	7.25E + 02	7.25E + 02	7.26E + 02	7.24E + 02	7.26E + 02	
F8	8.07E + 02	8.07E + 02	8.07E + 02	8.09E + 02	8.07E + 02	8.07E + 02	8.08E + 02	8.07E + 02	
F9	9.00E + 02	9.00E + 02	9.00E + 02	9.01E + 02	9.00E + 02	9.00E + 02	9.00E + 02	9.00E + 02	
F10	1.43E + 03	1.29E + 03	1.45E + 03	1.48E + 03	1.38E + 03	1.39E + 03	1.41E + 03	1.40E + 03	
F11	1.11E + 03	1.11E + 03	1.12E + 03	1.12E + 03	1.11E + 03	1.11E + 03	1.11E + 03	1.12E + 03	
F12	9.21E + 03	9.13E + 03	8.98E + 03	1.22E + 04	1.09E + 04	1.09E + 04	1.17E + 04	1.10E + 04	
F13	2.23E + 03	1.78E + 03	1.98E + 03	2.10E + 03	2.06E + 03	2.10E + 03	2.20E + 03	1.86E + 03	
F14	1.42E + 03	1.42E + 03	1.42E + 03	1.42E + 03	1.43E + 03	1.43E + 03	1.42E + 03	1.43E + 03	
F15	1.51E + 03	1.51E + 03	1.51E + 03	1.51E + 03	1.51E + 03	1.52E + 03	1.52E + 03	1.52E + 03	
F16	1.60E + 03	1.60E + 03	1.61E + 03	1.60E + 03	1.61E + 03	1.61E + 03	1.61E + 03	1.62E + 03	
F17	1.72E + 03	1.72E + 03	1.72E + 03	1.73E + 03	1.73E + 03	1.73E + 03	1.73E + 03	1.73E + 03	
F18	3.12E + 03	3.05E + 03	3.64E + 03	3.69E + 03	3.61E + 03	3.37E + 03	3.65E + 03	3.68E + 03	
F19	1.91E + 03	1.91E + 03	1.91E + 03	1.91E + 03	1.91E + 03	1.91E + 03	1.92E + 03	1.91E + 03	
F20	2.01E + 03	2.01E + 03	2.01E + 03	2.02E + 03	2.02E + 03	2.02E + 03	2.02E + 03	2.02E + 03	
F21	2.25E + 03	2.25E + 03	2.27E + 03	2.24E + 03	2.24E + 03	2.24E + 03	2.26E + 03	2.25E + 03	
F22	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	2.30E + 03	
F23	2.61E + 03	2.61E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	2.62E + 03	
F24	2.68E + 03	2.69E + 03	2.71E + 03	2.67E + 03	2.70E + 03	2.70E + 03	2.71E + 03	2.70E + 03	
F25	2.93E + 03	2.92E + 03	2.92E + 03	2.93E + 03	2.92E + 03	2.93E + 03	2.93E + 03	2.93E + 03	
F26	2.92E + 03	2.96E + 03	2.93E + 03	2.94E + 03	2.95E + 03	2.94E + 03	2.96E + 03	2.94E + 03	
F27	3.10E + 03	3.10E + 03	3.10E + 03	3.10E + 03	3.10E + 03	3.10E + 03	3.10E + 03	3.10E + 03	
F28	3.22E + 03	3.20E + 03	3.17E + 03	3.23E + 03	3.20E + 03	3.21E + 03	3.16E + 03	3.18E + 03	
F29	3.17E + 03	3.16E + 03	3.17E + 03	3.17E + 03	3.17E + 03	3.17E + 03	3.17E + 03	3.16E + 03	
F30	3.40E + 04	4.81E + 03	1.69E + 05	5.70E + 03	5.98E + 04	3.26E + 04	5.81E + 03	3.29E + 04	
Significant values are in [bold].

The parameters of the comparison algorithms refer to the original algorithm criteria, as shown in Table 6. Table 6 Parameter settings of the comparison algorithms.

Algorithm	Parameter settings	Year	
WOA	a = [2,0]	2016	
GWO	a = [2,0]	2014	
APO	np = 1, pfmax = 0.1	2024	
FVIM	alpha = 1.5	2024	
COA	no special parameters	2023	
SHIO	a = 1.5	2022	
WSO	fmax = 0.75, fmin = 0.07, tau = 4.11, pmin = 0.5,

pmax = 1.5, a0 = 6.25, a1 = 100, a2 = 0.0005

	2022	

Evaluation on the 50 benchmark functions

PMSOMA was evaluated using the specified benchmark functions, with the Rosenbrock function serving as a notable example. This non-convex function is commonly used to assess the efficacy of optimization algorithms63. As depicted on the left side of Fig. 4, the global minimum of the Rosenbrock function forms a parabolic shape, making it challenging to pinpoint the exact global minimum due to minor value fluctuations within the valley. The convergence trajectories of both OMA and PMSOMA, when applied to this function, are illustrated on the right side of Fig. 4. It is evident that PMSOMA achieves solutions of greater precision compared to the original OMA, successfully overcoming issues related to accuracy and the tendency to fall into local optima.Fig. 4 Example of visualization and convergence (Rosenbrock function).

As presented in Tables 7, 8, 9, 10 and 11, the PMSOMA algorithm was compared with the original OMA, two classical algorithms (WOA and GWO), and five recently proposed algorithms (APO, FVIM, COA, SHIO and WSO). PMSOMA demonstrates superiority over the original OMA, with its Friedman test outcomes positioning it at the forefront in a global comparison against other algorithms. It can be seen that PMSOAM performs more prominently on the multimodal function, which means that it can better jump out of the local optimum to find the global optimal solution. Table 7 Results of PMSOMA and other algorithms for 50 benchmark functions.

		PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
 + /–/ = 	NAN	30/2/17	35/9/5	33/8/5	25/10/14	39/5/5	28/13/8	40/4/5	29/5/15	
Frideman	2.9400	5.0900	5.4700	5.3000	3.8300	6.6400	4.1700	6.6200	4.9400	
rank	1	5	7	6	2	9	3	8	4	

Table 8 Results of PMSOMA and other algorithms for 50 benchmark unimodal separable functions (F1 to F5).

Fun		PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
F1	Mean	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	–6.00E–01	
Std	0.00E + 00	1.45E–15	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F2	Mean	0.00E + 00	1.15E–03	4.44E–04	4.82E–01	2.78E–15	3.68E + 00	1.21E–06	1.79E + 00	3.80E–01	
Std	0.00E + 00	2.05E–03	2.02E–04	3.13E–01	1.51E–14	7.61E–01	3.71E–06	6.30E–01	2.32E–01	
F3	Mean	2.05E–223	5.97E–01	0.00E + 00	2.00E–184	1.57E–44	5.75E–76	0.00E + 00	1.81E–76	3.17E–04	
Std	0.00E + 00	3.24E + 00	0.00E + 00	0.00E + 00	1.95E–44	1.64E–75	0.00E + 00	6.76E–76	3.75E–04	
F4	Mean	1.83E–218	1.17E–03	0.00E + 00	6.21E–186	7.58E–45	6.94E–76	0.00E + 00	1.91E–76	2.63E–05	
Std	0.00E + 00	4.11E–03	0.00E + 00	0.00E + 00	7.74E–45	2.76E–75	0.00E + 00	6.46E–76	3.24E–05	
F5	Mean	3.79E–02	1.10E–01	6.94E–04	2.31E–04	1.79E–03	6.25E–03	1.35E–05	2.70E–03	7.09E–02	
Std	1.93E–02	3.54E–02	7.51E–04	1.35E–04	5.72E–04	2.73E–03	1.43E–05	1.48E–03	2.37E–02	
Significant values are in [bold].

Table 9 Results of PMSOMA and other algorithms for 50 benchmark unimodal non-separable functions (F6 to F17).

Fun		PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
F6	Mean	0.00E + 00	0.00E + 00	1.97E–13	3.96E–09	0.00E + 00	3.59E–05	1.39E–14	4.31E–05	0.00E + 00	
Std	0.00E + 00	0.00E + 00	4.63E–13	3.95E–09	0.00E + 00	9.52E–05	2.04E–14	8.67E–05	0.00E + 00	
F7	Mean	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	–1.00E + 00	
Std	2.06E–17	0.00E + 00	2.97E–09	2.60E–08	0.00E + 00	2.69E–06	8.99E–17	3.56E–05	0.00E + 00	
F8	Mean	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	5.02E–213	0.00E + 00	0.00E + 00	0.00E + 00	9.38E–208	
Std	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F9	Mean	0.00E + 00	3.98E–16	5.30E–01	4.63E–01	0.00E + 00	1.07E + 00	1.96E–03	1.09E + 00	0.00E + 00	
Std	0.00E + 00	2.00E–15	8.39E–01	7.19E–01	0.00E + 00	6.41E–01	2.65E–03	6.12E–01	0.00E + 00	
F10	Mean	–5.00E + 01	–5.00E + 01	–5.00E + 01	–5.00E + 01	–5.00E + 01	–4.52E + 01	–5.00E + 01	–4.70E + 01	–5.00E + 01	
Std	1.27E–13	6.80E–14	2.07E–06	5.60E–06	2.98E–14	9.24E + 00	1.17E–10	7.88E + 00	3.23E–14	
F11	Mean	–2.10E + 02	–2.10E + 02	–2.10E + 02	–1.84E + 02	–2.10E + 02	–6.63E + 01	–2.10E + 02	–1.08E + 02	–2.10E + 02	
Std	5.89E–12	2.43E–02	1.26E–03	4.78E + 01	3.99E–06	6.47E + 01	5.32E–07	6.58E + 01	9.56E–04	
F12	Mean	9.03E–85	7.66E–21	1.69E–07	7.45E–216	7.63E–46	3.69E–97	0.00E + 00	1.10E–106	1.29E–04	
Std	4.78E–84	2.54E–20	5.36E–07	0.00E + 00	1.15E–45	9.60E–97	0.00E + 00	2.82E–106	4.71E–04	
F13	Mean	2.12E–07	2.26E–02	8.03E–07	1.20E–06	2.22E–05	9.21E–05	0.00E + 00	2.54E–05	6.36E–03	
Std	7.00E–08	1.82E–02	1.53E–06	1.81E–06	4.55E–06	8.15E–05	0.00E + 00	2.19E–05	3.90E–03	
F14	Mean	5.76E–123	1.10E–03	0.00E + 00	7.84E–107	5.19E–28	1.51E–44	0.00E + 00	2.70E–46	2.29E–04	
Std	3.01E–122	3.43E–03	0.00E + 00	1.27E–106	2.23E–28	2.29E–44	0.00E + 00	3.53E–46	1.58E–04	
F15	Mean	9.16E–36	5.86E + 01	3.20E + 03	1.17E–53	4.22E–14	5.97E–08	0.00E + 00	1.13E–16	9.36E + 01	
Std	3.82E–35	4.92E + 01	3.32E + 03	4.98E–53	1.38E–13	2.04E–07	0.00E + 00	4.36E–16	4.78E + 01	
F16	Mean	4.14E–01	1.39E + 02	2.59E + 01	2.66E + 01	3.91E + 01	2.83E + 01	2.30E + 01	2.76E + 01	1.02E + 02	
Std	1.13E + 00	8.24E + 01	3.07E–01	6.03E–01	3.07E + 01	6.91E–01	3.46E–01	9.98E–01	5.88E + 01	
F17	Mean	5.33E–01	3.67E + 00	6.67E–01	6.67E–01	7.40E–01	6.99E–01	6.67E–01	6.69E–01	1.90E + 00	
Std	2.71E–01	1.98E + 00	7.39E–06	1.37E–07	4.01E–01	8.66E–02	1.52E–09	7.27E–03	1.62E + 00	
Significant values are in [bold].

Table 10 Results of PMSOMA and other algorithms for 50 benchmark multimodal separable functions (F18 to F26).

Fun		PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
F18	Mean	9.98E–01	9.98E–01	1.20E + 00	4.13E + 00	9.98E–01	1.01E + 00	1.39E + 00	1.01E + 00	9.98E–01	
Std	4.12E–17	7.14E–17	6.05E–01	4.14E + 00	0.00E + 00	2.99E–02	1.81E + 00	3.34E–02	0.00E + 00	
F19	Mean	3.98E–01	3.98E–01	3.98E–01	3.98E–01	3.98E–01	3.98E–01	3.98E–01	3.98E–01	3.98E–01	
Std	0.00E + 00	0.00E + 00	5.52E–09	1.38E–07	0.00E + 00	4.60E–07	1.22E–13	8.21E–07	0.00E + 00	
F20	Mean	0.00E + 00	2.36E–03	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
Std	0.00E + 00	1.29E–02	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F21	Mean	0.00E + 00	0.00E + 00	5.50E–06	2.18E–08	0.00E + 00	3.97E–08	1.25E–14	5.94E–08	0.00E + 00	
Std	0.00E + 00	0.00E + 00	6.39E–06	1.52E–08	0.00E + 00	4.67E–08	2.49E–14	4.77E–08	0.00E + 00	
F22	Mean	6.63E–01	4.60E + 01	0.00E + 00	0.00E + 00	6.97E + 00	3.83E + 01	0.00E + 00	3.42E + 01	6.43E + 00	
Std	7.08E–01	2.05E + 01	0.00E + 00	0.00E + 00	2.76E + 00	1.26E + 01	0.00E + 00	2.22E + 01	2.37E + 00	
F23	Mean	–1.15E + 04	–5.82E + 03	–1.19E + 04	–5.91E + 03	–1.13E + 04	–5.47E + 03	–9.30E + 03	–5.79E + 03	–7.53E + 03	
Std	3.01E + 02	5.17E + 02	1.24E + 03	7.49E + 02	3.67E + 02	7.65E + 02	7.43E + 02	6.49E + 02	5.37E + 02	
F24	Mean	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	–1.80E + 00	
Std	9.03E–16	9.03E–16	3.90E–09	1.17E–07	9.03E–16	3.90E–04	1.78E–15	1.87E–03	9.03E–16	
F25	Mean	–4.69E + 00	–4.68E + 00	–4.07E + 00	–4.44E + 00	–4.69E + 00	–4.40E + 00	–4.53E + 00	–4.29E + 00	–4.65E + 00	
Std	3.84E–15	2.52E–02	4.60E–01	3.17E–01	7.62E–03	3.90E–01	3.00E–01	4.13E–01	4.76E–02	
F26	Mean	–9.61E + 00	–9.48E + 00	–6.55E + 00	–7.76E + 00	–9.66E + 00	–8.25E + 00	–9.30E + 00	–7.28E + 00	–9.51E + 00	
Std	6.09E–02	2.00E–01	1.01E + 00	1.05E + 00	1.22E–02	7.89E–01	4.62E–01	1.00E + 00	1.26E–01	
Significant values are in [bold].

Table 11 Results of PMSOMA and other algorithms for 50 benchmark multimodal non-separable functions (F27 to F50).

Fun		PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
F27	Mean	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
Std	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F28	Mean	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	–1.03E + 00	
Std	6.78E–16	6.78E–16	2.08E–12	1.45E–09	6.78E–16	1.80E–07	5.22E–16	3.13E–09	6.78E–16	
F29	Mean	7.28E–03	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
Std	3.99E–02	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F30	Mean	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
Std	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F31	Mean	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	–1.87E + 02	
Std	4.35E–14	1.90E–14	1.47E–06	2.43E–02	5.63E–10	9.47E–02	7.68E–10	8.69E–03	1.53E–01	
F32	Mean	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	
Std	1.89E–15	1.19E–15	1.70E–06	1.24E–06	1.84E–15	2.48E–05	6.40E–15	1.04E–05	1.22E–15	
F33	Mean	3.07E–04	3.07E–04	5.50E–04	3.75E–03	3.81E–04	5.22E–04	1.68E–03	6.60E–04	3.38E–04	
Std	1.96E–19	8.52E–19	3.06E–04	7.56E–03	2.39E–04	3.14E–04	5.08E–03	3.46E–04	1.67E–04	
F34	Mean	–1.02E + 01	–1.00E + 01	–1.02E + 01	–9.81E + 00	–9.65E + 00	–8.14E + 00	–8.11E + 00	–7.63E + 00	–9.41E + 00	
Std	6.22E–15	7.18E–01	2.46E–03	1.29E + 00	1.54E + 00	2.88E + 00	2.54E + 00	3.10E + 00	2.28E + 00	
F35	Mean	–1.04E + 01	–1.04E + 01	–9.93E + 00	–1.04E + 01	–9.76E + 00	–9.70E + 00	–8.28E + 00	–8.62E + 00	–9.96E + 00	
Std	1.09E–15	8.65E–11	1.82E + 00	2.94E–05	1.96E + 00	1.70E + 00	2.65E + 00	2.68E + 00	1.69E + 00	
F36	Mean	–1.05E + 01	–1.05E + 01	–1.05E + 01	–1.05E + 01	–9.87E + 00	–9.04E + 00	–9.23E + 00	–9.18E + 00	–1.05E + 01	
Std	7.44E–15	1.14E–02	2.85E–03	2.83E–05	2.04E + 00	2.69E + 00	2.42E + 00	2.67E + 00	1.81E–15	
F37	Mean	1.42E–03	3.16E–03	5.40E–01	3.83E–03	9.98E–04	5.07E–01	3.56E–03	4.21E–01	1.36E–03	
Std	3.23E–03	4.59E–03	4.31E–01	5.11E–03	2.72E–03	7.77E–01	4.88E–03	7.51E–01	3.25E–03	
F38	Mean	5.54E–05	7.51E–05	9.78E + 00	1.76E–01	6.02E–04	6.88E–02	6.58E–02	3.52E–01	4.31E–05	
Std	7.65E–05	1.14E–04	2.82E + 01	3.46E–01	5.52E–04	1.83E–01	2.22E–01	4.17E–01	1.63E–04	
F39	Mean	–3.86E + 00	–3.86E + 00	–3.37E + 00	–3.86E + 00	–3.86E + 00	–3.86E + 00	–1.67E + 00	–3.86E + 00	–3.86E + 00	
Std	2.67E–15	2.56E–15	1.26E + 00	3.64E–03	2.71E–15	1.29E–03	1.23E + 00	3.53E–03	2.71E–15	
F40	Mean	–3.32E + 00	–3.32E + 00	–3.24E + 00	–3.26E + 00	–3.31E + 00	–3.30E + 00	–3.28E + 00	–3.29E + 00	–3.30E + 00	
Std	1.82E–15	4.51E–07	1.17E–01	7.39E–02	3.63E–02	4.88E–02	5.70E–02	6.27E–02	4.51E–02	
F41	Mean	4.76E–03	1.72E–01	3.21E–03	1.26E–03	2.13E–03	6.38E–03	0.00E + 00	5.76E–03	3.24E–02	
Std	6.07E–03	3.68E–01	1.23E–02	5.30E–03	5.44E–03	6.54E–03	0.00E + 00	6.53E–03	2.98E–02	
F42	Mean	4.78E–14	7.11E + 00	3.73E–15	8.11E–15	4.44E–15	7.99E–15	8.88E–16	1.04E–14	6.25E–02	
Std	1.13E–14	1.54E + 00	2.17E–15	6.49E–16	0.00E + 00	1.62E–15	0.00E + 00	3.41E–15	2.38E–01	
F43	Mean	3.66E–04	9.40E + 01	7.35E + 01	7.51E + 05	1.83E–03	9.53E + 06	3.80E + 01	3.32E + 06	2.27E + 03	
Std	2.01E–03	1.23E + 02	1.67E + 01	1.20E + 06	4.16E–03	1.18E + 07	2.00E + 01	6.57E + 06	5.92E + 03	
F44	Mean	1.35E–32	3.32E + 01	9.17E–03	4.49E–01	3.66E–04	2.27E + 00	1.49E + 00	1.54E + 00	6.22E–01	
Std	5.57E–48	1.43E + 01	2.12E–02	1.80E–01	2.01E–03	4.98E–01	5.67E–01	2.71E–01	2.49E–01	
F45	Mean	–4.16E + 00	–4.16E + 00	–4.09E + 00	–4.05E + 00	–4.15E + 00	–4.14E + 00	–4.01E + 00	–4.09E + 00	–4.15E + 00	
Std	9.47E–16	9.03E–16	1.43E–01	1.98E–01	9.76E–03	8.70E–02	2.03E–01	1.64E–01	5.15E–03	
F46	Mean	–3.89E + 00	–3.89E + 00	–3.35E + 00	–3.66E + 00	–3.84E + 00	–3.80E + 00	–3.71E + 00	–3.44E + 00	–3.81E + 00	
Std	6.28E–03	5.98E–03	8.30E–01	5.84E–01	2.80E–01	3.09E–01	5.16E–01	7.71E–01	4.43E–01	
F47	Mean	–2.75E + 00	–2.06E + 00	–1.51E + 00	–3.42E + 00	–3.56E + 00	–3.31E + 00	–2.67E + 00	–2.36E + 00	–2.25E + 00	
Std	9.35E–01	9.41E–01	9.60E–01	9.75E–01	8.36E–01	1.04E + 00	1.19E + 00	1.23E + 00	1.07E + 00	
F48	Mean	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
Std	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	
F49	Mean	2.29E–11	4.54E–10	1.40E–06	5.05E–10	1.70E–59	3.31E–08	0.00E + 00	2.15E–08	5.17E–71	
Std	1.51E–11	3.52E–10	1.06E–06	5.02E–10	9.23E–59	2.64E–08	0.00E + 00	2.11E–08	2.61E–70	
F50	Mean	8.51E–09	2.47E–07	2.75E–06	1.02E–07	3.05E–10	6.16E–06	0.00E + 00	2.28E–06	4.46E–06	
Std	4.08E–09	1.19E–07	1.98E–06	1.62E–07	8.29E–10	5.70E–06	0.00E + 00	3.03E–06	1.20E–05	
Significant values are in [bold].

Figure 5 shows the particle search history, average fitness curve, trajectory of the first particle, and convergence curve for PMSOMA applied to some of the 50 benchmark functions. The results clearly show PMSOMA’s superior performance across all scenarios. For unimodal functions (F2, F4, F12, and F17), PMSOMA’s characteristics enable the rapid and undisturbed exploration of promising regions. Similarly, for multimodal functions (F22, F23, and F27), PMSOMA’s performance is noteworthy, showcasing its proficiency in swiftly escaping local optima.Fig. 5 Fitness landscape, search history, average fitness, trajectory of the first particle, and convergence curve of unimodal (F2, F4, F12 and F17), multimodal (F22, F23 and F27) functions.

Evaluation on the CEC2017 testing set

In this subsection, the effectiveness of PMSOMA for solving the complex IEEE CEC2017 benchmark functions is evaluated. The function dimension was set to 10, and Table 12 presents the results of PMSOMA and eight other algorithms on objective functions F1 to F30. The Friedman test outcomes for PMSOMA and other algorithms are presented at the top of Table 12. The corresponding box plot is shown in Fig. 6. It is shown that PMSOMA ranked first overall, exhibiting superior robustness compared with the original OMA and other algorithms. Table 12 Comparison of results on the CEC2017 testing set.

	PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
 + /–/ = 	NAN	28/1/1	30/0/0	29/1/0	17/13/0	30/0/0	30/0/0	30/0/0	23/7/0	
Frideman	1.7333	3.6	8.0667	5.9667	2.4	7.3	5.3	7.3333	3.3	
Rank	1	4	9	6	2	7	5	8	3	
F1	8.57E + 02	1.12E + 03	2.47E + 05	3.14E + 07	5.74E + 03	3.44E + 08	3.05E + 03	1.43E + 08	1.75E + 03	
F2	2.00E + 02	2.00E + 02	8.23E + 03	4.10E + 06	4.68E + 03	4.97E + 08	2.00E + 02	1.57E + 08	5.74E + 02	
F3	3.00E + 02	3.00E + 02	1.12E + 03	1.46E + 03	1.09E + 03	4.87E + 03	3.00E + 02	5.33E + 03	3.00E + 02	
F4	4.00E + 02	4.04E + 02	4.31E + 02	4.11E + 02	4.07E + 02	4.24E + 02	4.03E + 02	4.37E + 02	4.02E + 02	
F5	5.07E + 02	5.11E + 02	5.47E + 02	5.13E + 02	5.08E + 02	5.27E + 02	5.15E + 02	5.28E + 02	5.15E + 02	
F6	6.00E + 02	6.00E + 02	6.35E + 02	6.01E + 02	6.00E + 02	6.10E + 02	6.06E + 02	6.10E + 02	6.01E + 02	
F7	7.24E + 02	7.31E + 02	7.78E + 02	7.28E + 02	7.18E + 02	7.42E + 02	7.72E + 02	7.46E + 02	7.18E + 02	
F8	8.07E + 02	8.11E + 02	8.43E + 02	8.13E + 02	8.07E + 02	8.20E + 02	8.27E + 02	8.24E + 02	8.09E + 02	
F9	9.00E + 02	9.01E + 02	1.36E + 03	9.17E + 02	9.00E + 02	9.99E + 02	1.05E + 03	1.02E + 03	9.10E + 02	
F10	1.29E + 03	1.80E + 03	2.01E + 03	1.51E + 03	1.43E + 03	2.11E + 03	1.75E + 03	1.94E + 03	1.41E + 03	
F11	1.11E + 03	1.12E + 03	1.21E + 03	1.13E + 03	1.10E + 03	1.23E + 03	1.14E + 03	1.19E + 03	1.12E + 03	
F12	9.13E + 03	9.92E + 03	3.36E + 06	4.10E + 05	1.64E + 04	1.25E + 06	1.58E + 04	1.33E + 06	3.06E + 03	
F13	1.78E + 03	2.34E + 03	1.27E + 04	1.08E + 04	1.44E + 03	1.03E + 04	3.28E + 03	1.40E + 04	1.35E + 03	
F14	1.42E + 03	1.43E + 03	1.67E + 03	2.43E + 03	1.41E + 03	1.83E + 03	1.48E + 03	1.94E + 03	1.43E + 03	
F15	1.51E + 03	1.53E + 03	4.72E + 03	3.16E + 03	1.50E + 03	3.13E + 03	1.71E + 03	3.21E + 03	1.51E + 03	
F16	1.60E + 03	1.61E + 03	1.85E + 03	1.76E + 03	1.61E + 03	1.78E + 03	1.66E + 03	1.78E + 03	1.65E + 03	
F17	1.72E + 03	1.74E + 03	1.81E + 03	1.75E + 03	1.72E + 03	1.78E + 03	1.74E + 03	1.78E + 03	1.72E + 03	
F18	3.05E + 03	4.16E + 03	1.35E + 04	2.86E + 04	1.94E + 03	1.36E + 04	1.28E + 04	1.84E + 04	1.82E + 03	
F19	1.91E + 03	1.92E + 03	2.70E + 04	6.77E + 03	1.90E + 03	8.06E + 03	1.98E + 03	6.50E + 03	1.90E + 03	
F20	2.01E + 03	2.02E + 03	2.15E + 03	2.07E + 03	2.02E + 03	2.13E + 03	2.02E + 03	2.12E + 03	2.03E + 03	
F21	2.25E + 03	2.28E + 03	2.32E + 03	2.31E + 03	2.22E + 03	2.30E + 03	2.31E + 03	2.28E + 03	2.30E + 03	
F22	2.30E + 03	2.30E + 03	2.31E + 03	2.31E + 03	2.30E + 03	2.35E + 03	2.36E + 03	2.33E + 03	2.31E + 03	
F23	2.61E + 03	2.62E + 03	2.64E + 03	2.62E + 03	2.61E + 03	2.64E + 03	2.62E + 03	2.64E + 03	2.63E + 03	
F24	2.69E + 03	2.73E + 03	2.76E + 03	2.75E + 03	2.69E + 03	2.76E + 03	2.74E + 03	2.76E + 03	2.71E + 03	
F25	2.92E + 03	2.93E + 03	2.95E + 03	2.93E + 03	2.93E + 03	2.95E + 03	2.94E + 03	2.95E + 03	2.92E + 03	
F26	2.96E + 03	2.95E + 03	3.56E + 03	3.14E + 03	2.93E + 03	3.16E + 03	3.18E + 03	3.20E + 03	2.95E + 03	
F27	3.10E + 03	3.10E + 03	3.13E + 03	3.09E + 03	3.09E + 03	3.10E + 03	3.11E + 03	3.10E + 03	3.11E + 03	
F28	3.20E + 03	3.23E + 03	3.38E + 03	3.38E + 03	3.26E + 03	3.31E + 03	3.28E + 03	3.30E + 03	3.25E + 03	
F29	3.16E + 03	3.18E + 03	3.35E + 03	3.18E + 03	3.17E + 03	3.24E + 03	3.18E + 03	3.26E + 03	3.17E + 03	
F30	4.81E + 03	1.18E + 05	3.26E + 05	6.08E + 05	1.31E + 05	5.72E + 05	4.04E + 05	4.63E + 05	3.18E + 04	
Significant values are in [bold].

Fig. 6 Boxplots of results for some example functions in the CEC2017 testing set.

This study conducted a comparative analysis of the convergence capabilities among various algorithms. Figure 7 presents the convergence curves for PMSOMA alongside OMA, WOA, GWO, APO, FVIM, COA, SHIO, and WSO across both single- and multi-peak benchmark functions. It can be seen that PMSOMA shows superior performance in terms of convergence speed and accuracy.Fig. 7 Convergence curves for the selected functions.

The Wilcoxon’s rank-sum test was conducted with a 5% significance threshold. Wilcoxon rank-sum test can determine whether there is a statistical difference in the results between the proposed PMSOMA algorithm and other algorithms. The calculated p values are presented in Table 13. P values greater than 0.05 are marked in bold in the table, which shows that PMSOMA provides significant advantages over the other algorithms in most cases. Table 13 P values of the Wilcoxon rank-sum test for the CEC2017 testing set functions.

Fun	PMSOMA	OMA	WOA	GWO	APO	FVIM	COA	SHIO	WSO	
1	NaN	0.089718	0.0000017	0.000002	0.000003	0.000002	0.000894	0.000002	0.271155	
2	NaN	0.000004	0.0000017	0.000002	0.000002	0.000002	0.000002	0.000002	0.000005	
3	NaN	0.002930	0.0000017	0.000002	0.000002	0.000002	0.000002	0.000002	0.000002	
4	NaN	0.000002	0.0000017	0.000002	0.000002	0.000002	0.000002	0.000002	0.000004	
5	NaN	0.000174	0.0000017	0.000283	0.517048	0.000002	0.000125	0.000002	0.000028	
6	NaN	0.000003	0.0000017	0.000002	0.000002	0.000002	0.000002	0.000002	0.000024	
7	NaN	0.003379	0.0000017	0.057096	0.000026	0.000022	0.000002	0.000004	0.001484	
8	NaN	0.000664	0.0000017	0.000028	0.975387	0.000003	0.000002	0.000002	0.164432	
9	NaN	0.001652	0.0000017	0.000006	0.102011	0.000003	0.000006	0.000002	0.000024	
10	NaN	0.000003	0.0000029	0.005320	0.005320	0.000002	0.000012	0.000003	0.042767	
11	NaN	0.009842	0.0000017	0.000359	0.000034	0.000002	0.001833	0.000002	0.044919	
12	NaN	0.765519	0.0000017	0.000002	0.025637	0.000002	0.085896	0.000002	0.000306	
13	NaN	0.007731	0.0000017	0.000002	0.000261	0.000002	0.001382	0.000002	0.000008	
14	NaN	0.000014	0.0000017	0.000002	0.065641	0.000002	0.000002	0.000002	0.003609	
15	NaN	0.000002	0.0000017	0.000002	0.001287	0.000002	0.000002	0.000002	0.000963	
16	NaN	0.000002	0.0000017	0.000002	0.000005	0.000002	0.000003	0.000002	0.000283	
17	NaN	0.000003	0.0000019	0.000006	0.085896	0.000002	0.000616	0.000002	0.036826	
18	NaN	0.044919	0.0000026	0.000002	0.000018	0.000002	0.000005	0.000003	0.000002	
19	NaN	0.000003	0.0000017	0.000002	0.000009	0.000002	0.000002	0.000002	0.028486	
20	NaN	0.000044	0.0000017	0.000002	0.001287	0.000002	0.280214	0.000002	0.000616	
21	NaN	0.021827	0.0000486	0.000010	0.047162	0.000332	0.000005	0.057096	0.000115	
22	NaN	0.262299	0.0000260	0.000082	0.006424	0.000002	0.688359	0.000004	0.000005	
23	NaN	0.001833	0.0000024	0.059836	0.000148	0.000002	0.017518	0.000002	0.000005	
24	NaN	0.003609	0.0001359	0.000004	0.658331	0.000026	0.000003	0.000020	0.009271	
25	NaN	0.428430	0.0003589	0.082206	0.033269	0.000894	0.097772	0.001593	0.360039	
26	NaN	0.736534	0.0000058	0.177907	0.490798	0.000037	0.047162	0.000041	0.781264	
27	NaN	0.909931	0.0000486	0.010444	0.000002	0.004682	0.036826	0.044919	0.000420	
28	NaN	0.077803	0.0000689	0.000006	0.033269	0.000571	0.019569	0.000894	0.004114	
29	NaN	0.000089	0.0000017	0.002585	0.147040	0.000002	0.057096	0.000002	0.198610	
30	NaN	0.004682	0.0000017	0.000002	0.000006	0.000002	0.000002	0.000002	0.221022	
Significant values are in [bold].

Feature selection

In this section, PMSOMA’s performance is evaluated using three classification datasets and three function approximation datasets. The classification datasets, namely XOR and Cancer, were sourced from the University of California, Irvine (UCI) Machine Learning Repository64. For function approximation, the Sigmoid and Cosine datasets were utilized.

For the experiments, populations were initialized randomly within the range of [− 10, 10]. The division of these datasets into training and testing sets is detailed in Tables 14 and 1565. Table 14 Classification dataset splits.

Classification datasets	Number of attributes	Number of training samples	Number of testing samples	Number of classes	
XOR	3	8	8	2	
Cancer	9	599	100	2	

Table 15 Function approximation dataset splits.

Function	Training samples	Test samples	
Sigmoid:y=1/(1+e-x)	61: x in [− 3:0.1:3]	121: x in [− 3:0.05:3]	
Cosine:y=(cos(xπ/2))7	31: x in [1.25:0.05:2.75]	38: x in [1.25:0.04:2.75]	

Classification problems

XOR classification problem

The N-bit XOR problem is a well-known nonlinear benchmark problem. The goal is to recognise the number of “1”s in an input vector. The XOR result of the input vector should be returned.

The experimental results for this problem are presented in Table 16. It can be observed that the classification rates of PMSOMA, GWO, APO and SHIO are 100%; however, the mean and standard deviation results of PMSOMA are better than those of the other training algorithms. Overall, PMSOMA exhibited excellent performance on this classification problem. Table 16 Experimental results for XOR dataset.

Algorithm	Mean	Std	Final_rate	
PMSOMA	1.68E-09	9.93E-10	100	
OMA	2.32E-03	3.29E-03	25	
WOA	1.92E-01	5.60E-02	50	
GWO	1.68E-02	3.13E-02	100	
APO	1.11E-06	6.06E-06	100	
FVIM	1.01E-02	2.63E-02	62.5	
COA	1.17E-01	1.27E-01	0	
SHIO	1.48E-02	3.97E-02	100	
WSO	1.15E-03	4.22E-02	87.5	
Significant values are in [bold].

Cancer classification problem

The Cancer dataset, created by William H. Wolberg from the University of Wisconsin Hospital in Madison, comprises 699 instances. It features nine attributes, such as clump thickness, uniformity of cell size, uniformity of cell shape, and marginal adhesion 66.

The experimental results for the different algorithms are presented in Table 17. The mean and standard deviation values indicate that PMSOMA avoided the local minima in the dataset the best. However, in terms of classification rate, the classification rates of APO,SHIO and WSO reached 100%, whereas those of PMSOMA and WOA were 97%. Table 17 Experimental results for Cancer dataset.

Algorithm	Mean	Std	Final_rate	
PMSOMA	5.65E-04	8.57E-05	97	
OMA	3.20E-03	1.36E-03	93	
WOA	2.82E-03	8.96E-04	97	
GWO	2.23E-03	5.37E-04	95	
APO	1.34E-03	7.00E-05	99	
FVIM	1.37E-03	5.62E-05	96	
COA	1.37E-03	5.62E-05	96	
SHIO	1.31E-03	6.06E-05	99	
WSO	1.45E-03	2.09E-04	99	
Significant values are in [bold].

Function approximation problems

Sigmoid function

The Sigmoid dataset contains data points in the interval [− 3, 3] in increments of 0.1, and the number of training data points is 61. The number of testing data points is 121, and these points exist within the same range. The results of the different algorithms for this dataset are presented in Table 18, test _ error represents the sum of the absolute error between the predicted value and the true value in 30 runs. Table 18 Experimental results for sigmoid dataset (one dimensional).

Algorithm	Mean	Std	test_error	
PMSOMA	2.46E-01	8.10E-06	17.5118	
OMA	2.47E-01	2.90E-04	17.9436	
WOA	2.49E-01	4.26E-03	17.9541	
GWO	2.47E-01	1.35E-04	17.8958	
APO	2.46E-01	3.27E-05	17.6318	
FVIM	2.46E-01	3.04E-05	17.5399	
COA	2.46E-01	4.61E-05	17.5430	
SHIO	2.46E-01	1.21E-04	17.6735	
WSO	2.46E-01	2.24E-05	17.5143	
Significant values are in [bold].

The results for mean, std, and test_error indicate that PMSOMA is much better at avoiding local minima than the other algorithms.

Cosine function

The Cosine dataset contains 31 training and 38 testing samples. The results for this dataset are presented in Table 19. It can be observed that the test _ error of PMSOMA is the lowest at 4.6509. Additionally, the mean and standard deviation values of PMSOMA are lower than those of the other seven algorithms. Table 19 Experimental results for Cosine dataset.

Algorithm	Mean	Std	test_error	
PMSOMA	1.75E-01	8.68E-05	4.6509	
OMA	1.85E-01	3.05E-03	5.1075	
WOA	2.59E-01	7.67E-02	6.3443	
GWO	1.76E-01	3.85E-04	4.7771	
APO	1.76E-01	2.55E-04	4.7383	
FVIM	2.58E-01	1.04E-01	4.8317	
COA	1.76E-01	3.77E-04	4.6791	
SHIO	1.77E-01	1.11E-03	4.9828	
WSO	1.76E-01	2.35E-04	4.6846	
Significant values are in [bold].

Engineering design problems

In this subsection, PMSOMA is applied to three engineering problems, namely the Step-cone pulley problem, Hydro-static thrust bearing design problem, and Welded beam design problem. These engineering problems feature complex constraints that render them difficult to solve. In this study, PMSOMA is utilized alongside other algorithms to address these three problems, thereby validating the efficacy of PMSOMA.

Step-cone pulley problem

The primary goal of the problem is to reduce the weight of a four-step cone pulley. This is to be achieved by manipulating five variables. Four of these variables pertain to the diameters of each step of the pulley, while the fifth variable is related to the width of the pulley. Additionally, the problem is subject to 11 nonlinear constraints. These constraints serve to guarantee that the transmitted power is maintained at 0.75 horsepower. The mathematical formulation of the problem is as follows:

Minimize:14 fx¯=ρωd121+N1N2+d221+N2N2+d321+N3N2+d421+N4N2

Subject to:15 h1x¯=C1-C2=0,

16 h2x¯=C1-C3=0,

17 h3x¯=C1-C4=0,

18 gi=1,2,3,4x¯=-Ri≤2,

19 gi=5,6,7,8x¯=0.75×745.6998-Pi≤0.

where20 Ci=πdi21+NiN+NiN-124a+2a,i=1,2,3,4,

21 Ri=expμπ-2sin-1NiN-1di2a,i=1,2,3,4,

22 Pi=stω1-RiπdiNi60,i=1,2,3,4,

23 t=8mm,s=1.75MPa,μ=0.35,ρ=7200Kgm3,a=3mm.

As presented in Table 20, the solutions obtained by each algorithm reveal that PMSOMA derived optimal results for this problem with a Best Cost of 16.4804, representing a significant improvement over the results produced by the other algorithms. Table 20 Step-cone pulley problem results.

	d1	d2	d3	d4	ω	Best cost	
PMSOMA	39.3472	54.1438	72.1860	86.5482	87.8597	16.4804	
OMA	40.1464	55.2444	73.6532	88.3058	86.4293	17.0919	
WOA	40.9123	56.2991	75.0593	89.9901	85.0532	20.5308	
GWO	40.9058	56.2930	75.0486	89.9774	87.4897	128,253.7924	
APO	39.8224	54.7981	73.0583	87.5931	86.8087	16.6790	
FVIM	40.7067	56.0254	74.6749	89.5354	87.0651	1,528,116.4612	
COA	40.8538	56.2184	74.9521	89.8613	87.8057	884.3344	
SHIO	40.5628	55.8054	74.4378	89.1828	88.7564	12,468,942.6624	
WSO	38.4140	52.8586	70.4727	84.4957	90.0000	16.6586	
Significant values are in [bold].

Hydro-static thrust bearing design problem

The main objective of the design problem is to optimize the bearing power loss using four design variables, namely oil viscosity μ, bearing radius R, flow rate Q and groove radius R0. The problem is subject to seven nonlinear constraints related to inlet oil pressure, load capacity, oil film thickness and inlet oil pressure, as stated in the following mathematical formulation:

Minimize:24 fx¯=QP00.7+Ef.

Subject to:25 g1x¯=1000-P0≤0,

26 g2x¯=W-101000≤0,

27 g3x¯=5000-WπR2-R02≤0,

28 g4x¯=50-P0≤0,

29 g5x¯=0.001-0.0307386.4P0Q2πRh≤0,

30 g6x¯=R-R0≤0,

31 g7x¯=h-0.001≤0.

where:32 W=πP02R2-R02lnRR0,

33 P0=6μQπh3lnRR0,

34 Ef=9336Q×0.0307×0.5ΔT,ΔT=210P-559.7,

35 P=log10log108.122×106μ+0.8+3.5510.04,

36 h=2π×7506022πμEfR44-R044.

With bounds:37 1≤R≤16,

38 1≤R0≤16,

39 1×10-6≤μ≤16×10-6,

40 1≤Q≤16.

PMSOMA was applied to the engineering optimization problem of Hydro-static thrust bearing design. As presented in Table 21, the Best Cost of PMSOAM is 1616.2952, which is closer to the optimal solution than the other algorithms, indicating good robustness. Table 21 Hydro-static thrust bearing design problem results.

	R	R0	μ	Q	Best cost	
PMSOMA	5.9559	5.3891	0.0000	2.2570	1616.2952	
OMA	5.9555	5.3887	0.0000	2.2566	1625.6366	
WOA	15.9913	15.7890	0.0000	15.9913	6559.8587	
GWO	5.9851	5.4199	0.0000	2.4205	1648.3189	
APO	6.2054	5.6600	0.0000	4.0635	1879.7351	
FVIM	7.1379	6.6639	0.0000	10.4363	2495.3938	
COA	7.0891	6.6136	0.0000	14.2621	2686.9437	
SHIO	6.0152	5.4490	0.0000	2.9367	1727.9274	
WSO	5.9556	5.3887	0.0000	4.0541	1815.7673	
Significant values are in [bold].

Welded beam design problem

The welded beam design problem aims to minimize the design cost, denoted as Func(X→) subject to a set of constraints. This optimization problem involves four design variables: the weld thickness (x1), the length of the clamped reinforcement (x2), the height of the reinforcement (x3), and the thickness of the reinforcement (x4). These variables can be mathematically described as follows:

Minimize:41 fx¯=1.10471x12x2+0.04811x3x4x2+14.

Subject to:42 g1(x¯)=τ(x¯)-13600≤0,

43 g2(x¯)=σ(x¯)-30000≤0,

44 g3(x¯)=x1-x4≤0,

45 g4(x¯)=0.10471x12+0.04811x3x4(14+x2)-5≤0,

46 g5(x¯)=0.125-x1≤0,

47 g6(x¯)=δ(x¯)-0.25≤0,

48 g7(x¯)=6000-Pa(x¯)≤0,

where49 τx¯=τ′+2τ′τ′′x22R+τ′′2,

50 τ′=60002x1x2,

51 τ′′=TKL,

52 T=600014+x22,

53 K=x1+x322+x224,

54 L=2x1x22x2212+x1+x322,

55 σx¯=504000x32x4,

56 Pax¯=4.01330×106x32x4636196×1-x330×106412×10628.

With bounds:57 0.1≤x1,

58 0.1≤x2,

59 x3≤10,

60 x4≤2.

As shown in Table 22, the optimal cost obtained by PMSOMA, APO and WSO is 1.6702, which is the closest to the optimal solution among all the algorithm results. Overall, PMSOMA shows superior performance. Table 22 Welded beam design problem.

	X1	X2	X3	X4	Best cost	
PMSOMA	9.1920	3.3374	0.1988	0.1988	1.6702	
OMA	9.3167	3.9112	0.1842	0.2064	1.8033	
WOA	9.0986	4.1984	0.1732	0.2029	1.7557	
GWO	9.1935	3.4069	0.1954	0.1989	1.6748	
APO	9.1920	3.3374	0.1988	0.1988	1.6702	
FVIM	9.1910	3.4013	0.1958	0.1989	1.6744	
COA	9.1935	3.3405	0.1987	0.1989	1.6712	
SHIO	9.1897	3.3515	0.1983	0.1989	1.6718	
WSO	9.1920	3.3374	0.1988	0.1988	1.6702	
Significant values are in [bold].

Conclusion

This study presents PMSOMA, an innovative adaptation of the OMA, devised to overcome issues encountered by the original OMA including its slow convergence rate and propensity for becoming trapped in local optima. PMSOMA integrates PWLCM with a sparse adaptive exploration mechanism. The chaotic map is specifically employed to rectify the issue of uneven population distribution observed during the initial stages of OMA. The sparse adaptive exploration mechanism enables optimal individuals to execute adaptive searches across each dimension, thus facilitating their escape from local optima.

PMSOMA was benchmarked against the original OMA, two classical algorithms (WOA and GWO), and five recently proposed algorithms (APO, FVIM, COA, SHIO and WSO). This assessment was conducted using 50 benchmark functions, the CEC2017 test suite, feature selection datasets, and three classic engineering challenges. The results unequivocally demonstrate PMSOMA’s superiority in identifying high-quality solutions and enhancing convergence speed.

The study deliberately refrains from altering the magnification stages associated with the objective lens and eyepiece in OMA, which presents a potential avenue for future inquiry. The current algorithm assesses each candidate solution independently at each iteration, potentially constraining its capacity for parallelization. Exploiting the potential of modern multi-core processors, it would be beneficial to develop parallel variants of the algorithm. Notably, the algorithm lacks an explicit exception handling mechanism. In real-world scenarios, it is crucial to incorporate error detection and exception handling to enhance the robustness of the algorithm. Furthermore, the efficacy of the algorithm in handling high-dimensional and multi-modal problems requires comprehensive investigation and enhancement. By combining PMSOMA with machine learning algorithms, it may be possible to further enhance its functionality and thus extend its applicability to a wider range of practical problems.

Author contributions

L.G. proposed the idea of algorithm improvement, wrote the main manuscript, built 50 benchmark test functions, tested the performance of the algorithm before and after improvement, and analysed the test results. W.G. debugged the optimisation algorithm, collected materials related to CEC2017, feature selection, engineering application problems, and checked and modified manuscripts. All authors read and approved the final manuscripts.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
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