Individual different state-based multi-swarm particle swarm optimization

Standard PSO

The standard PSO simulates cooperation and competition within a group, leveraging information sharing among individuals to search for the optimal solution. In each iteration, PSO first evaluates the objective function value at the current position of all particles, determines both the historical best position of each particle (i.e., the best solution found so far) and the best position of the entire group (i.e., the best solution found among all particles). Based on this information, the velocity and position of each particle are then updated. In this way, particles gradually approach the global optimal solution in the search space. PSO operates according to simple rules, and three factors are considered before updating velocity: its own inertial forward direction, the direction it considers best for itself and the globally best direction. PSO achieves a balance under these three choices. The formula can be described by Eqs. (1) and (2):

(1) $$v_{i,j}(t+1)=w\cdot v_{i,j}(t)+c_1{r}_{1,j}\cdot (p{b}_{i,j}(t)-x_{i,j}(t))+c_2{r}_{2,j}\cdot (g{b}_j(t)-x_{i,j}(t))$$

(2) $$x_{i,j}(t+1)=x_{i,j}(t)+v_{i,j}(t+1)$$where w is the inertia weight, c₁ and c₂ represent the cognitive and social learning factors respectively. r₁ and r₂ are random numbers between 0 and 1. pb_i,j (t) and gb_j (t) represent the particle’s personal best position and global best position respectively. x_i,j (t) and v_i,j (t) denote the velocity and position of the i-th particle in the j-th dimension at the t-th iteration.

PSO variants

PSO is known as a valid optimization algorithm. However, when solving complex problems, it often faces issues such as premature convergence, insufficient search precision, and difficulty in maintaining trade-off between global and local searches. To deal with these issues, researchers have developed many effective PSO variants, and they can be roughly divided into three categories: parameter adjustment, topological structure, and multi-swarm technique.

(1) Parameter tuning is a simple yet effective strategy to enhance the search ability of PSO. As for the inertia weight, it determines the extent to which a particle retains its movement velocity from the previous iteration. In Liu, Zhang & Tu (2020), a PSO with chaotic inertia weight (MPSO) is introduced to trade-off the global and local searches of particles. However, it suffers from the problem of premature convergence when applied to the optimization of high-dimensional complex functions. Note that in Zhan & Zhang (2008), APSO is formulated based on the evolutionary state to dynamically adjust the inertia weight. Zhang & Ding (2011) outline a MSCPSO, which adaptively adjusts the inertia weight based on particle fitness and uses adaptive strategies to regulate the influence of historical information. In addition, a new adaptive inertia weight is explored in Jiyue, Liu & Wan (2023) based on evolutionary differences among particles, with the goal of integrating adaptive inertia weights and multiple operators into the PSO. In Nagra, Han & Ling (2019), an improved self-adaptive PSO that incorporates gradient-based local search strategies is proposed, where the enhanced inertia weight significantly boosts the particle’s search performance. Alternatively, a PSO variant combining adaptive dynamic inertia weight and adaptive dynamic acceleration coefficients is introduced in Sekyere, Effah & Okyere (2024) in order to further enhance the particle’s exploration and exploitation capabilities.

Additionally, learning factors are another critical parameter in PSO. They are typically divided into the cognitive learning factor (c₁) and the social learning factor (c₂). The cognitive learning factor represents a particle’s ability to learn from its own historical best position, while the social learning factor represents its ability to learn from the swarm’s historical best position. Ratnaweera, Halgamuge & Watson (2004) discuss an approach where c₁ decreases over time while c₂ increases to balance global and local search capabilities. In Chen et al. (2018b), a hybrid particle swarm optimizer with sine cosine acceleration coefficients (H-PSO-SCAC) is proposed, which effectively controls local search and global convergence by introducing sine and cosine acceleration methods. In Tian, Zhao & Shi (2019), a chaotic particle swarm optimization based on an s-shaped acceleration coefficient (CPSOS) is proposed, where the sigmoid acceleration coefficient is designed to balance early-stage exploration with late-stage exploitation. An improved particle swarm optimization (A-PSO) is proposed in Chen et al. (2018a), where nonlinear dynamic acceleration coefficients are introduced to enhance solution quality and accelerate global convergence. However, most of these PSO methods only focus on adjusting inertia weights or acceleration coefficients independently, lacking a unified multi-strategy framework. As a result, the performance of PSOs can be undoubtedly impaired.

(2) Topology adjustment is widely applied in various PSO variants. Since the topology determines how each particle exchanges information with others, it directly affects the convergence rate and global search capability of the PSO. In Mendes, Kennedy & Neves (2004), the static topology is classified into fully connected, ring, four-cluster, pyramid and square structures. In the fully connected topology, each particle communicates with every other particle in the swarm, which leads to fast convergence but with the risk of getting trapped in local optima. In the ring topology, each particle only exchanges information with a few neighboring particles, which helps maintain diversity and prevents premature convergence. In Sun & Li (2014), a two-swarm cooperative PSO uses a ring topology for neighbor-based learning, while a four-cluster topology divides the swarm into subgroups to balance local and global information exchange, maintaining diversity and avoiding local optima. In literature Blackwell & Kennedy (2018), it is shown that fully connected topologies exhibit outstanding advantages in unimodal functions while ring topologies are better for multimodal functions.

Dynamic topologies change during the optimization process and can enhance the search capability of the PSO variant by adjusting the connections between particles. In Wang, Yang & Orchard (2016), a dynamic tournament topology strategy is introduced to strengthen PSO, where a few good positions are randomly selected from the entire swarm to guide each particle. In order to balance exploration in the early stages and exploitation in the later stages, a multi-swarm PSO based on dynamic topology and purposeful detection is proposed in Xia, Gui & Zhan (2018). In Zhang et al. (2022), KGPSO is presented based on a self-organizing topology and adaptive parameters. During the evolutionary process, the K-means clustering method is periodically used to divide the swarm into multiple distance-based sub-swarms to enhance the global search ability. Besides, Ni & Deng (2013) employ a random topology structure and explore the relationship between swarm topology and PSO’s performance. In Lim et al. (2018), an adaptive topology connection is utilized, where the connections are adaptively modified during different search stages to facilitate particle’s exploration. In summary, although the static topological structure is relatively easy to implement, it may trap the PSO into local optima due to their fixed connections. In contrast, despite the dynamic topologies can boost the search performance via adjustable connections, it will increase the computational complexity of the algorithm.

(3) Multi-swarm techniques have gained significant attention from researchers in recent years. The core idea is to divide the entire particle swarm into multiple sub-swarms, each of which conducts independent searches and shares information through predefined communication mechanisms.

To increase swarm diversity, a dynamic multi-swarm PSO (DMS-PSO) is proposed (Liang & Suganthan, 2005), which divides the population into small sub-swarms with periodic regrouping (R) to enhance diversity, though at the cost of slower convergence. To improve local search in specific regions, the PSO described in Zhao et al. (2008) integrates the quasi-Newton operator into DMS-PSO to enhance local search and accelerate convergence. Besides, in literature Nasir et al. (2012), a dynamic neighborhood learning PSO (DNLPSO) is structured based on the learning mechanism from CLPSO. Another approach, known as the multi-adaptive strategy PSO (MAPSO), is discussed in Wei et al. (2020). This method divides the population into multiple sub-swarms that can be recombined during evolution process, with particles in each generation adaptively selecting learning exemplars based on their performance. In Jiyue, Liu & Wan (2023), the population is dynamically divided into four sub-swarms according to their fitness values, with each sub-swarm adopting a different velocity update mechanism to improve algorithm performance. Similarly, in Gou et al. (2017), the population is divided into three sub-swarms based on emotional states: happy, normal, and unhappy. Particles within each sub-swarm use different velocity update mechanisms to obtain the global optimum. In Yang, Li & Huang (2023), a PSO variant based on a dynamic multi-swarm framework is proposed, combined with a stagnation detection mechanism (SDM) and a spatial exclusion strategy (SES). Alternatively, the ESD-PSO is developed in Yang & Li (2023a) to improve the information exchange efficiency among sub-swarms through an adaptive cooperative mechanism. To balance exploration and exploitation, the fuzzy C-means (FPC)-based partitioning method is used to divide the initial population into multiple sub-swarms in Tao et al. (2022). Although these studies have enhanced the diversity of swarm and local search ability via dynamic multi-swarm structures, they often suffer from slower convergence rate and increased complexity.

Dynamic sub-swarm division

It is well known that particle swarm division and the size of each sub-swarm play an important role in maintaining swarm diversity. In Ye, Feng & Fan (2017), the division of sub-swarms is based on the total number of particles. For instance, if the number of particles is M * N, the population is divided into M sub-swarms, each containing N particles. While this approach accelerates convergence rate, it can be difficult to maintain diversity under certain conditions. In literature Yang et al. (2018), a fixed number of particles are also employed for sub-swarm division, which may hinder the swarm diversity. Besides, in Kong, Jiang & Huang (2019), a parameter-based method for population division is introduced, where ps represents the swarm size and m is the division parameter. The swarm is divided by taking the quotient of ps divided by m, and any remainder is discarded. As iterations progress, the size of sub-swarms declines. In Yang, Li & Yu (2022), sub-swarms are divided into three categories and they are responsible for different search tasks. However, most researchers divide sub-swarms based on the number of particles or custom parameters, allocating a fixed number of particles to each sub-swarm rather than adaptively adjusting based on the dynamic evolution of all particles. In view of this, the swarm is dynamically divided into four sub-swarms based on particle’s fitness values. The formula is defined by Eq. (3):

(3) $$\{\begin{array}{ccc}{f}_1& =& f_{min}\hfill \\ f_2& =& {\displaystyle \frac{(m_1-1)(f_{max}-f_{min})}{4}+f_{min}}\\ f_3& =& {\displaystyle \frac{(m_2-1)(f_{max}-f_{min})}{4}+f_{min}}\\ f_4& =& {\displaystyle \frac{(m_3-1)(f_{max}-f_{min})}{4}+f_{min}}\\ f_5& =& f_{max}\hfill \end{array}$$

$$S_{\mathrm{k}}^{\mathrm{t}}=\{x_i^t|f_{\mathrm{k}}\le fitness(x_i^t)<f_{\mathrm{k}+1},i=1,2\dots N,\mathrm{k}=1,2,3,4\}$$where, f_min represents the minimum fitness value of all particles in the swarm, which is also the optimal fitness value. f_max is the maximum fitness value of all particles in the population, which is also the worst fitness value. $x_i^t$ denotes the position of the i-th particle during the t-th iteration, where the subscript i refers to the particle within the population. The values for m₁, m₂, and m₃ are set to 2, 3, and 4, respectively. $S_{\mathrm{k}}^{\mathrm{t}}$ represents the k-th sub-swarm in the t-th iteration. Since the range of k is (1,4), the swarm is divided into four sub-swarms, denoted as $S_1^t$, $S_2^t$, $S_3^t$ and $S_4^t$. When a particle’s fitness value falls between f₁ and f₂, the particle belongs to S₁ (the best sub-swarm). If the fitness value is between f₂ and f₃, the particle is assigned to S₂ (the better sub-swarm). Similarly, if the fitness value is between f₃ and f₄, the particle belongs to S₃ (the worse sub-swarm), and if the fitness value is between f₄ and f₅, the particle is categorized into S₄ (the worst sub-swarm).

Assuming there are 20 particles in the population, the division of particles after two iterations is shown in the diagram. The orange circle represents the entire population, and we divide the swarm into four sub-swarms based on the division method. As shown in Fig. 1, the number of particles in each sub-swarm varies randomly as the number of iterations increases, which can effectively enhance swarm diversity and accelerate the convergence rate of IDSMPSO.

Adaptive inertia weight

Inertia weight plays a crucial role in the PSO, as it significantly influences particle’s search behavior and the convergence of the swarm. The introduction of inertia weight is intended to balance the particle’s global exploration capability with its local exploitation ability. Specifically, inertia weight controls the extent to which the current velocity affects the velocity in the next iteration. A larger inertia weight encourages particles to maintain higher velocities, enhancing global exploration. Conversely, a smaller inertia weight reduces the velocity, making particles rely more on local neighborhood searches, thus improving local exploitation. However, if the inertia weight is too large, particles may miss the optimal solution, while an excessively small inertia weight could cause particles to prematurely fall into local optima. Therefore, selecting an appropriate inertia weight strategy is key to maintaining a trade-off between global and local searches, as well as ensuring convergence in various complex optimization problems. In Eberhart & Shi (2001), a stochastic inertia weight factor for tracking dynamic systems is introduced, where inertia weight is dynamically adjusted using random numbers between 0 and 1, rather than following a linear decreasing pattern. Inspired by cognitive psychology, a self-regulating inertia weight is proposed in Tanweer, Suresh & Sundararajan (2015), where the best particles use self-adjusting inertia weights for better exploration. To address the limitations of linear decreasing inertia weight, a PSO algorithm with dynamic nonlinear adaptive inertia weight is proposed in Chatterjee & Siarry (2006). Generally, inertia weight decreases linearly, but in this case, it is dynamically modified based on fitness information. In view of this, this article adopts different inertia weights for particles in different subgroups according to particle fitness values. To improve the dynamic particle search capability, the adaptive inertia weights are described by Eq. (4):

(4) $$w=\{\begin{array}{ll}{w}_{max}\hfill & f_i\ge f_4\hfill \\ {\displaystyle \frac{\left(f_4-f_i\right)\left(w_{max}-w_{min}\right)}{f_4-f_3}+w_{min}}& f_3\le f_i<f_4\hfill \\ {\displaystyle \frac{(f_3-f_i)(w_{max}-w_{min})}{f_3-f_2}+{\mathrm{w}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}& f_2\le f_i<f_3\hfill \\ w_{\mathrm{m}\mathrm{a}\mathrm{x}}-\mathrm{t}{\displaystyle \frac{w_{\mathrm{m}\mathrm{a}\mathrm{x}}-w_{\mathrm{m}\mathrm{i}\mathrm{n}}}{T}}\hfill & f_i<f_2\hfill \end{array}$$where, w_max is set to 0.9 and w_min is set to 0.4. t represents the current iteration, and T denotes the maximum number of iterations. f_i stands for the fitness value of the current particle. When $f_i\ge f_4$, the particle belongs to the worst sub-swarm, where a larger inertia weight favors exploration of promising solutions. When $f_3\le f_i<f_4$, the particle is classified into the worse sub-swarm, and when $f_2\le f_i<f_3$, the particle is part of the better sub-swarm. In both worse and better sub-swarms, particles select inertia weight based on the specific formula of their sub-swarm. If f_i < f₂, the particle belongs to the best sub-swarm, where the velocity is updated using a linearly decreasing inertia weight. Our proposed method updates the inertia weight by leveraging each particle’s fitness value and the evolutionary differences between particles. As a result, this method is more effective in improving the performance of PSO.

By observing the average inertia weight of all particles, it is evident that the average inertia weight adaptively decreases with the increasing number of iterations, as shown in Fig. 2.

Different learning strategy for different sub-swarm

Worst sub-swarm

It is well known that the “cask effect” implies that the amount of water a cask can hold depends on its shortest plank rather than its longest. By extending the shortest plank and removing the limitation it creates, the cask’s water capacity can be increased. Hence, to strengthen the performance of the particles in the worst sub-swarm is particularly important for PSO. Particles in the worst sub-swarm are more likely to move in the wrong direction or get trapped in local optima. To deal with this issue, many strategies have been proposed to assist these particles evade local optima and find more promising solutions, such as Gaussian mutation, elite learning and random combination strategy, etc. In Yang & Li (2023b), Gaussian mutation is introduced to further update particles’ positions for the purpose of stopping them from falling into local optima. An elite learning strategy is proposed in Wang et al. (2021), and Xia et al. (2019) that allows particles to study from the best individuals in the best sub-swarm, thereby improving the convergence speed of the swarm. In addition, a random combination strategy is applied in Tang et al. (2021) to guide the velocity of particles in the “isolated domain” so as to increase the swarm diversity. As shown in Fig. 3, the random combination strategy can reconstruct each dimension of the particle X_mix, where each dimension is randomly selected from the best historical positions of different particles in order to ensure multiple dimensions are not selected from the same particle.

According to the operator described in Eq. (3), particles with fitness values greater than or equal to f₄ are assigned to S₄ (the worst sub-swarm). In the worst sub-swarm, particles update their velocities using elite learning and random combination strategies. Additionally, Gaussian mutation is adaptively introduced to further update particle’s position and avoid local optima. The learning strategy can be defined by Eqs. (11)–(13):

(11) $$\begin{array}{rl}& v_{i,j}\left(t+1\right)=w\cdot v_{i,j}\left(t\right)+c_1\cdot {\mathrm{r}}_{1,j}\cdot (p{b}_{i,j}(t)-x_{i,j}(t))+c_2\cdot r_{2,j}\cdot (g{b}_j^{s_1}(t)-x_{i,j}(t))\\ & +c_3\cdot r_{3,j}\cdot (X_{mix,j}(t)-x_{i,j}(t))\end{array}$$

(12) $$r>{\displaystyle \frac{1}{2}[1+\arctan \left({\displaystyle \frac{t}{T_{max}}}\right)\times {\displaystyle \frac{4}{\pi }}]}$$

(13) $$GV(x)=x\cdot (1+randn\left(1,D\right))$$where X_mix,j represents the particle reconstructed in the j-th dimension, and w is the adaptive inertia weight. Since the particle is in the worst sub-swarm, w_max is selected according to Eq. (2) to update the particle’s velocity. r is a random number between 0 and 1 that is used to determine whether a Gaussian mutation should be triggered, x stands for the current particle’s position, and randn denotes a Gaussian-distributed random number with mean of 0 and variance of 1. The new position calculated by GV(x) replaces or influences the particle’s current position x to increase population diversity and avoid local optima. When the condition in Eq. (12) is met, Gaussian mutation will be trigged to help the particles search for better solutions.

BFGS quasi-newton method

The Quasi-Newton method is a classic numerical technique used to solve unconstrained optimization problems, which is developed to avoid directly calculating the Hessian matrix, making it an efficient method for solving large-scale optimization problems. The BFGS (Broyden, 1970; Fletcher, 1970; Goldfarb, 1970; Shanno, 1970) method is one of the most widely used Quasi-Newton methods. It begins by constructing an inverse matrix approximating the Hessian, and then updates this matrix using gradient information. This update is performed according to the BFGS formula, progressively approximating the true Hessian matrix, thus obtaining a better search direction in each iteration.

To address the slow convergence problem of DMS-PSO, a local search phase using the BFGS Quasi-Newton method is added to the algorithm in Zhao et al. (2008), enhancing the local search capability of particles. The Quasi-Newton Method constructs a local quadratic model to approximate the objective function by updating the Hessian matrix approximation. The quadratic model generally takes the form of Eq. (14):

(14) $$m_k\left(p\right)=f\left(x_k\right)+\mathrm{\nabla }f{\left(x_k\right)}^{T}p+{\displaystyle \frac{1}{2}{p}^T{B}_{k}p}$$where m_k(p) is the quadratic approximation model of the objective function at $x_k$, p is the search direction vector, and B_k is the Hessian matrix at x_k that is a positive definite and symmetric matrix.

Based on this quadratic model, the Quasi-Newton method can determine the search direction $p_k$ at each step, which is calculated by Eq. (15):

(15) $$p_k=-B_k^{-1}\mathrm{\nabla }f(x_k).$$

This direction ensures the fastest decrease in the quadratic model, thus accelerating convergence to the minimum of the objective function. If x* is the optimal solution of the quadratic model, then x* must satisfy: $\mathrm{\nabla }f\left(x^{\ast }\right)=0$.

The Quasi-Newton method updates the Hessian matrix approximation at each iteration. The most commonly used update formula is the BFGS formula, which ensures that the matrix remains symmetric and positive definite, preserving the validity of the quadratic model. The update formula for the approximate Hessian matrix can be described by Eq. (16):

(16) $$B_{k+1}=B_k+{\displaystyle \frac{y_k{y}_k^T}{y_k^T{s}_k}-{\displaystyle \frac{B_k{s}_k{s}_k^T{B}_k}{s_k^T{B}_k{s}_k}}}$$where $s_k=x_{k+1}-x_k$, $y_k=\mathrm{\nabla }f\left(x_{k+1}\right)-\mathrm{\nabla }f\left(x_k\right)$, with B₀ being the identity matrix.

The BFGS Quasi-Newton method is employed to refine the optimal solution obtained, thereby enhancing the local search capability. It is worth noting that a random number between 0 and 1 is generated to determine whether the IDSMPSO will use the Quasi-Newton method or not. When the random number is less than or equal to a pre-set threshold, the algorithm updates each sub-swarm based on different learning mechanisms, otherwise, the Quasi-Newton method is executed. Extensive experiments have demonstrated that setting the threshold to 0.9 yields the best results. As described above, the pseudocode of IDSMPSO is outlined in Algorithm 1.

Test functions

To demonstrate the performance of IDSMPSO in unimodal, multimodal and complex scenarios, the CEC’17 and CEC’22 benchmark test suites are selected. CEC’17 consists of 29 functions (f₂ is excluded due to its instability in high-dimensional cases), including unimodal (f₁, f₃), multimodal (f₄–f₁₀), hybrid (f₁₁–f₂₀), and composition (f₂₁–f₃₀) functions. The search range for each function is shown in Table 1. Moreover, Table 2 presents the CEC’22 test suite. Specifically, within the test suite, f₁ belongs to the unimodal category, while f₂ to f₅ are classified as multimodal functions, f₆ to f₈ fall into the hybrid function category, and f₉ to f₁₂ are defined as composite functions, each variable is constrained within the range of [−100, 100].

Parameter settings

To evaluate the performance of our proposal, eight competitive algorithms are selected for comparison, including four state-of-the-art optimizers (Grey Wolf Optimizer (GWO) (Mirjalili, Mirjalili & Lewis, 2014), GOOSE Algorithm (GO) (Hamad & Rashid, 2024), Equilibrium Optimizer (EO) (Faramarzi et al., 2020), Kepler Optimization Algorithm (KOA) (Abdel-Basset et al., 2023)) and four classical PSO variants (MPSO (Liu, Zhang & Tu, 2020), EAPSO (Zhang, 2023), HPSOALS (Wang et al., 2024), DMPSORH (Tang et al., 2021)). To ensure a fair evaluation, the parameter settings of all the comparison algorithms are adopted from their original studies, with the main configurations summarized in Table 3. The parameters of IDSMPSO are as follows: w_max is set to 0.9 and w_min is set to 0.4 so as to strengthen the convergence rate of particles in the worst sub-swarm. As the number of iterations increases, w is adaptively adjusted based on the particle’s state. For the Levy flight strategy, β = 3/2 and α = 0.01 (step-size factor). The maximum number of population iterations (T_Max) is set to 1,000 and the swarm size N is set to 30. Each test is repeated independently 30 times to minimize statistical errors. Additionally, to prevent particles from moving beyond the boundary, IDSMPSO includes boundary handling mechanisms. If a particle’s position exceeds the predefined maximum value, its position is reassigned to the previous maximum. Likewise, if the position of a particle is less than the minimum value, it is reset to the previous minimum. The same boundary handling method is applied to the particle’s velocity.

Comparison on CEC’17 test suite

In this section, experiments are conducted on the CEC’17 with three different dimensions (D = 30, 50, 100). Without loss of generality, the metrics including the mean value (mv), standard deviation value (sdv), rank value (rv), average rank (AR) and final rank (FR) of the best results achieved are used to evaluate the peer algorithms. Tables 4–6 report the experimental results, and the mean value (mv), standard deviation value (sdv), rank value (rv), average rank (AR) and final rank (FR) of the best results are highlighted in bold.

Accuracy comparison

Based on the experimental settings described in Table 3, all algorithms have been reproduced as completely as possible in accordance with their original literature, and extensive simulation experiments have been conducted.

As can be clearly observed from Table 4, IDSMPSO outperforms other methods on functions f₁, f₃, f₄, f₁₀–f₂₀, f₂₃, f₂₅, f₂₆, f₂₈, and f₃₀ in terms of the mean value. On function f₁₆, its performance is nearly identical to that of MPSO. On functions f₅, f₇, f₈ and f₉, its results are comparable to those of DMPSORH. Similarly, although the mean values of our method on f₆, f₇, f₂₁, f₂₂, f₂₄ and f₂₇ are higher than those of EAPSO, and its mean values on f₅, f₈ and f₂₁ are higher than those of DMPSORH, IDSMPSO is still significantly superior to them in terms of mean value. The reason lies in that when dealing with multimodal functions, the multi-swarm structure can lead to slow information transmission within the population. Consequently, the information of the global optimal solution cannot be quickly propagated to all the other particles, which in turn results in an excessively long convergence time for the entire population to approach the optimal solution.

Table 5 presents the experimental results for 50-dimensional problems. Regarding the results of unimodal functions (f₁, f₃), it can be observed that IDSMPSO achieves the optimal performance. This is attributed to its excellent optimization ability on unimodal functions, enabling it to converge to the optimal values on f₁ and f₃. For the results of multimodal functions (f₅–f₇ and f₉), MPSO obtains better mean values and standard deviations than that of IDSMPSO. However, IDSMPSO still surpasses GWO, GO and EO significantly. In other words, our method is capable of handling high-dimensional and relatively complex multimodal problems. Compared with other peer algorithms, IDSMPSO yields superior results on functions f₁₁–f₁₅, f₁₈, and f₁₉, but its performance on f₁₆, f₁₇, and f₂₀ is slightly inferior to that of MPSO. With respect to the results of the 10 composite functions (f₂₁–f₃₀), it can be seen that our method also has achieved encouraging performance on most test functions.

As observed from Table 6, under the 100-dimensional case, IDSMPSO achieves the optimal performance on unimodal functions, which is consistent with its performance in 30-dimensional and 50-dimensional scenarios, and its performance becomes increasingly superior as the dimension increases. However, for multimodal functions, IDSMPSO only performs best on f₄, f₈, and f₁₀. In contrast, MPSO obtains the optimal results on f₅–f₇ and f₉. IDSMPSO continues to yield encouraging results on hybrid functions. In addition, even though the mean values of our method on f₂₄ is higher than that of several peer algorithms for composite functions, it has achieved smaller standard deviation. In sum, it can be clearly concluded from the extensive experiments that the adoption of the multi-swarm collaborative search strategy enables the adjustment of particle movement in the search space from multiple perspectives. Note that different inertia weights and distinct learning strategies for each sub-swarm can effectively maintain swarm diversity and suppress the premature convergence of particles. Finally, the BFGS quasi-Newton method is utilized to refine the obtained optimal solution, thereby enhancing the local convergence ability of the swarm. All these strategies play a good complementary role each other in the particles evolution and should be used together to obtain better optimization performance.

To further illustrate the superiority of IDSMPSO, Figs. 4–6 present histograms of the average value rankings for 30-, 50- and 100-dimensional problems, respectively. These histograms are primarily used to display the number of times each peer algorithm achieves rankings from 1st to 9th place. As shown in Fig. 4, under the 30-dimensional scenario, the number of times IDSMPSO secures the 1st place is nearly twice that of MPSO. Similarly, in the 50-dimensional scenario depicted in Fig. 5, our proposal also ranks as the top-performing algorithm, which significantly surpasses MPSO (ranks 2nd) and EAPSO (ranks 3rd). Furthermore, it is worth noting that in the 100-dimensional scenario shown in Fig. 6, IDSMPSO wins the 1st place slightly more times than MPSO. However, MPSO achieves the 2nd place far more frequently than IDSMPSO. This fully verifies the effectiveness of IDSMPSO in the task of numerical function optimization, at least on CEC’17 test suite.

Convergence analysis

To analyze the convergence speed and accuracy of the proposed IDSMPSO and the comparison algorithms, we recorded and stored the best fitness value at each iteration and plotted the corresponding convergence curves. Due to the limited space, only eight representative functions are selected for testing under D = 30, 50 and 100, including two unimodal functions (f₁, f₃), one multimodal function (f₄), two hybrid functions (f₁₂, f₁₃) and composition functions (f₂₅).

As shown in Figs. 7–9, compared with other algorithms, IDSMPSO exhibits significantly faster convergence on most functions and is able to converge to the optimal value without being trapped in local optima. Taking Fig. 7B as an example, the convergence curves of these algorithms display almost the same rate of decline during the initial 50 iterations. However, IDSMPSO maintains a faster convergence speed consistently and continues to decrease compared with its peer algorithms after this stage. It is worth noting that for f₁₂, MPSO converges toward the global optimal solution at the fastest speed, but eventually gets trapped in the local optimal value after a stable iteration phase. In contrast, although IDSMPSO takes 400 iterations to reach the optimal value, its curve basically stabilizes afterward, this indicates that IDSMPSO possesses stable local convergence capability at the end of iterations. Figure 8 presents the convergence curves of various algorithms under the 50-dimensional setting. Taking function f₂₅ in Fig. 8 as an example, all algorithms except KOA and GWO exhibit a fast evolutionary speed during the initial 150 iterations. After this stage, KOA shows fluctuations with poor stability. Notably, the convergence curves of GWO, HPSOALS and EO do not exhibit a significant decline afterward. In comparison, the iteration curves of other algorithms tend to stabilize after 400 iterations and converge to relatively small values. Taking functions f₄ and f₁₃ in Fig. 9 as examples, it is obvious that IDSMPSO converges to a smaller value. In contrast, although IDSMPSO converges slightly slower on composite functions, it outperforms the other eight algorithms by escaping from local optima and finding more promising accuracy. The results indicate that different inertia weights for particles in IDSMPSO indeed play an important role in accelerating the searching process. However, its effectiveness is only slightly demonstrated in multimodal functions.