A particle swarm optimization algorithm based on diversity-driven fusion of opposing phase selection strategies

The PSO algorithm is generated by simulating the foraging behavior of a flock of birds. Initially, the position of each bird is random, and each bird is flying in different directions. The flock of birds transmits information through interaction, gradually gathers into small groups, and flies in the same direction at the same speed, and finally, the whole flock gathers at the same location, which is the source of food. In the basic PSO algorithm, on the one hand, the independent individuals in the population will carry simple and limited own-specific information and continuously exchange local information. On the other hand, it will form a group, complete the information dissemination of the whole population in the way of group behavior, and show excellent cooperation ability and intelligence. Particles search continuously by learning their own historical cognition (pbest) and group cognition (gbest) to achieve the purpose of optimization. For a D-dimensional search space, let the number of swarm particles be n, and the position and velocity of the i-th particle are denoted as \(X_{i}=\left( x_{i,1},x_{i,2},\ldots ,x_{i,D} \right) \) and \(V_{i}=\left( v_{i,1},v_{i,2},\ldots ,v_{i,D} \right) \), \(i=1,2,\ldots ,n\), respectively. The formula for calculating the \(d\left( 1\le d\le D \right) \)-dimensional velocity and position of the i-th particle are as follows:Among them, \(v_{i,d}\left( t+1 \right) \) and \(x_{i,d}\left( t+1 \right) \) are the velocity and position of particle i in the \(t+1\) -th generation respectively, \(\omega \) is the inertia factor, the acceleration constants \(c_1\) and \(c_2\) are the weight parameters for adjusting the current optimal value and the global optimal value, \(r_1\) and \(r_2\) are [0, 1] uniformly distributed random numbers.

The combination of the PSO algorithm and OBL is usually divided into two ways, one is to use the opposite idea to improve the algorithm locally, and the other is to embed the OBL variant in the whole algorithm. In the BPSO algorithm, the idea of OBL is integrated into the individual optimal update equation of particles [59]. In [51], the ideas of opposition and rank are introduced to construct a new inertia weight, which is used to balance the degree of exploration and development of particle swarms, thereby speeding up the convergence speed.

The most basic OBL variant was first combined with the PSO algorithm called OPSO [52]. In OPSO, OBL is used in the population initialization phase and the local update strategy based on the jump rate. In the population initialization stage, the population \(X_{i}=( x_{i,1},x_{i,2},\ldots ,x_{i,D}) \) is randomly generated within the constraint range, and then n opposite solution\(\tilde{X}_{i}=(\tilde{x}_{i,1},\) \(\tilde{x}_{i,2},\ldots ,\tilde{x}_{i,D}) \) of the random population is obtained based on the OBL strategy. The specific calculation formula is as follows:where \(x_{i}\in \left[ a_{i},b_{i} \right] \), \(d=\,\,1,2,\ldots ,D\).

Finally, select N individuals with the best fitness from \(\left\{ X\left( N \right) \cup \tilde{X}\left( N \right) \right\} \) to form the initial population.

In the update process, after the population completes the speed and position update, if a random number rand(0,1) obeying a uniform distribution is less than the set jump rate Jr, the population shows opposition, that is, the local OBL strategy is executed, and the specific update steps Similar to the initialization phase. First, generate N opposing positions corresponding to the original N positions, select the best from the original population and the opposing population, and select the N solutions with the best fitness as the current new population:The boundary of the relative individual is different from the initialization stage, which changes with the iterative process, and \(a_{d}\) and \(b_{d}\) represent the interval boundary ( \(x_{i,d}\in \left[ a_{d},b_{d} \right] \) ) of the i-th variable d-dimension.

Tizhoosh et al. [49] proposed the idea of opposition learning in 2005, and then in 2008 provided the relevant definitions of the type-I opposite point and the type-II opposite point [60]. Type-I opposite points are aimed at the relationship between points in the search space, and Type-II opposite points consider the relationship between the target values of points in the space.

Example samples of Type-I and Type-II opposite points are shown in Fig. 1.

Because Type-II opposition needs to clarify the evaluation value of the target space in advance, it is very difficult to calculate Type-II opposition when solving black-box problems [50]. At present, most research are carried out for Type-I opposition. Considering the distance between the opposite solution and the current solution, two types of opposition are extended in Type-I opposition: Type-I Super-Opposition and Type-I Quasi-Opposition [60].

Seif et al. [61] propose the following extended opposition for Type-I Super-Opposition.

Based on the above two definitions, it can be found that reflected extended opposition randomly generates each initial solution between the current solution and the closest boundary. Whereas extended opposition randomly generates each solution between the solution’s opposing solution and the farther bound.

Similarly, Type-I Quasi-Opposition can also be divided into two categories: reflection and opposition.

Based on Definitions 5, 6, 7, and 8, it can be found that quasi-reflected opposition extends the range of opposite-based learning between the value center and the current solution, and quasi-opposition moves each solution between the solution’s opposite and the value center. Figure 2 shows the range of opposition points for Quasi opposition, Extended opposition, and Reflected extended opposition.

Currently, the vast majority of literature is on quasi opposition variants. In [53], Wang et al. proposed the definition of generalized opposite-based learning (GOBL):where k is a random number in the range[0,1].

Although GOBL can alleviate the problem of premature convergence of the algorithm to a certain extent, it will also increase the probability of falling into the local optimum. To this end, [64] employs two adaptive strategies as supplements to improve the performance of GOPSO.

The opposition based on the current optimal solution information was proposed by Xu et al. [65] The definition of COOBL is as follows:where \(x_{i}^{\mathrm{{best}}}\) is the optimal solution for the current population.

Rahnamayan et al. [66] proposed centroid opposition computing, which uses the center of gravity of the entire group as a reference point to calculate the reverse point, so that the reverse point includes the group search experience. Let the center of gravity of \(X=\left( x_1,x_2,\ldots ,x_{D} \right) \) be C, then the definition of the centroid opposition points \(\tilde{X}^{\mathrm{{co}}}=\left( \tilde{x}_{1}^{\mathrm{{co}}},\tilde{x}_{2}^{\mathrm{{co}}},\ldots ,\tilde{x}_{D}^{\mathrm{{co}}} \right) \) is as follows:

Tables 1 and 2 show the combination of the PSO algorithm and OBL, and give the most influential OBL-related variants in recent years. For the algorithm, the common purpose of adding the OBL strategy is to speed up the convergence speed of the population. The OBL variant is mainly used in the initialization phase of the population and the iterative update phase based on the hopping rate.

In the swarm intelligence algorithm, population diversity is used to describe the distribution of a single particle in the search area, which is an important factor affecting the global performance of the algorithm. When the PSO algorithm is searching, it is usually accompanied by the lack of population diversity, which causes the algorithm to fall into local optimum and premature [70]. Exploration and exploitation are two important stages of population evolution. When in the exploration stage, the population will explore a wider search area, and the positions of the particles will be relatively scattered; during the exploitation, the population will perform a fine search for the optimal solution area, and the particle positions will be relatively concentrated. To show how different OBL variants differ in population diversity, we introduce a diversity measure [71]. In a D-dimensional search space, the population diversity defined based on the mean is as follows:where N is the population size, \(x_{ij}\) is the value of the jth variable of the ith particle, and \(\bar{x}_j\) represents the average value of the jth dimension of the population, defined as follows:In order to show the variation trend of diversity of different OBL variants in the process of population iteration, the multimodal function Griewank is selected as the experimental function [72]:In the experiment, set the common parameters population size N = 100, dimension D = 30, and maximum function evaluation times to 100,000. After 25 independent runs of each algorithm, the curve of the mean population diversity with the number of iterations is shown in Fig. 3.

It can be clearly seen from the figure that as the number of iterations increases, EOBL can maintain a higher population diversity than other OBL variants, which means a wider range of search for optimal values, effectively avoiding premature trapping local optimum. However, the local search ability of particles is insufficient, and the convergence accuracy cannot be guaranteed. COBL relies on the advantage of centroid-based opposition to converge rapidly compared to other OBL variants. Premature convergence will lead to the inability to jump out of the local optimum, and the lack of searchability in the later stage is an obvious defect of COBL. To this end, how integrating the advantages of the above two basic variants and balancing the ability of population exploration and development is one of the research focuses of this paper.

In the original version of OBL, the selection of opposite points depended on the center point of each dimension boundary. But compared to the centroid point of the overall population, the center point is obviously not an appropriate choice. Therefore, Rahnamayan proposed in 2014 to replace the center point with the centroid point to calculate the corresponding opposite point [66]. In the D-dimensional search space, suppose there are N particles \(X=\left( x_1,x_2,\ldots ,x_{D} \right) \), and each particle in this space has a mass attribute. Then the mass center point of the particle population, that is, the Mean Center (MC) of the population can be expressed by the following formula:Among them, \(i=1,2,\ldots ,N\), N is the number of particles, \(d=1,2,\ldots ,D\), in the formula represents the d-th dimension value of the mean center, that is, the particle’s centroid value in the d-th dimension.

In physics, the center of mass is defined as the imaginary point of concentration of an object of uniform density and is used to characterize the average location of the mass of the object. However, the solution space where the particles are located is not always strictly uniform distribution, but there is a certain degree of offset [73], so it is not rigorous to describe the properties of the entire population with the centroid point. In the calculation of the center of mass, only the influence of the mass attribute is considered, but the density attribute of the particle is ignored. The opposite points obtained through the centroid points cannot reflect the distribution of data points in the neighborhood. Therefore, this paper takes the mean density of all particles as the benchmark and selects particles higher than the mean density to form a representative density subgroup. The particle density is defined by calculating the number of data points contained in the decentered neighborhood, the Euclidean distance between samples, and the corresponding fitness of samples. Assuming that \(X=\left( x_1,x_2,\ldots ,x_{D} \right) \) is N points in the D-dimensional search space, the average distance between particle points and particle points in the neighborhood is:Among them, \(N_{k}=\left\{ x_{j}\left| 0<\left| x_{i}-x_{j} \right| <\varepsilon \right. \right\} \) is the neighborhood of \(x_i\), \(\varepsilon \) is the radius of the neighborhood, and k is the number of particles in the neighborhood of particle \(x_i\).

Let \(x^k=\{ x_{i}|\)The number of neighbors of \( x_i\) is \(k \} \), \(D_{\max }^{k}={\max }_{x_{i}\in x^k}D\left( x_{i} \right) \), \(D_{\min }^{k}={\min }_{x_{i}\in x^k} D\left( x_{i} \right) \), introduce a linear normalization function to reduce the computational complexity, then the density of particle \(x_i\) is:Select particles that are better than the overall density mean to form a density subgroup, and the mean of the density subgroup is the Density Center (DC) of the population, which can be expressed by the following formula:Among them, n is the number of particles that are better than the mean density, and the formula represents the d-th dimension value of the density center, that is, the mean value of all the particles that are better than the density means in the d-th dimension.

In order to comprehensively consider the two factors of the mean center and the density center, this paper combines the advantages of the two to define a new center called the double mean center(DMC), which can be expressed by the following formula:where represents the d-th dimension value of DMC.

Judging the range of parameters according to the characteristics of data distribution can effectively determine the parameters in a relatively reasonable range. Kernel density estimation can characterize the data distribution [74] and is a nonparametric method. Assuming that the independent distribution F contains \(x_1\),\(x_2\),...,\(x_n\) sample points, the probability density function is f, and the kernel density is estimated as follows:Among them, h is the bandwidth, K(x) is the kernel function, and n is the number of samples.

In the kernel density estimation theory, the choice of the bandwidth value has a great influence on the estimator [75]. If h is too small, the density estimation is limited to the nearby area of the observed data, resulting in too few observed samples in the neighborhood. If h is too large, then the density estimation divides the probability densities too scattered, which filters out some important properties. According to the above analysis, if the size of h can be determined mathematically, the value of the parameter neighborhood radius \(\varepsilon \) can be determined. In statistics, the integral mean square error MISE(h) is usually used as the criterion for judging the quality of the density estimator [75].In,Minimizing MISE (h) is equivalent to minimizing AMISE (h). We must set the bandwidth h at some intermediate value to avoid excessively biased kernel estimates. First, take the partial derivative of AMISE (h) and set the derivative equal to 0:Then the optimal bandwidth value is:For the unknown quantity \(\int {\left[ f''\left( x \right) \right] ^2\mathrm{{d}}x}\) in the optimal bandwidth, Sliverman proposed a rule of thumb [76], replacing f with a normal density whose variance matches the estimated variance. This is equivalent to estimating \(\int {\left[ f''\left( x \right) \right] ^2\mathrm{{d}}x}\) using \(\int {\left[ \phi ''\left( x \right) \right] ^2\mathrm{{d}}x}/\hat{\sigma }\), where \(\phi \) is the standard normal density function. If the kernel function is the Gaussian density kernel function, \(\phi \) uses the sample variance \(\hat{\sigma }\), and uses the rule of thumb to obtain the adaptive neighborhood radius:Fig. 4 shows the particle distributions with reference to the convergence of different centers in the iterative process of the population. Four algorithms were selected and compared at 1, 10, and 30 iterations. Figure 4a–c are the particle distributions of the standard PSO algorithm, and Fig. 4d–f are the particle distributions of the standard PSO algorithm. The particle distribution of the OPSO algorithm, Fig. 4g–i are the particle distribution of the OPSO algorithm with the center of mass instead of the center, Fig. 4j–l are replaced by the double mean center. The particle distribution of the center’s OPSO algorithm. Overall, the four algorithms are randomly dispersed in the solution space in the early stage of evolution. With the increase in the number of iterations, OBL uses its own characteristics to guide the particles to gather the global optimal solution, which is obviously different from the ordinary PSO algorithm. In addition, the OPSO algorithm that replaces the center with MC and the OPSO algorithm that replaces the center with DMC can converge to the global optimal solution faster than the PSO algorithm.

In order to reflect the advantages of DMC, the test is carried out on 12 test functions of CEC2022. The population size is set to N = 10, the dimension D = 20, the maximum number of function evaluations is 5000, each test function is run independently 30 times, and the final result is taken as 30. The average value of the times, the experimental results obtained are shown in Table 3. Wilcoxon rank sum test and Friedman test were performed on the data in the table, and the rank means of DMC was the smallest, which verified that DMC has more advantages than MC. The specific DMC strategy is shown in Algorithm 1.

Refraction of light is a common physical phenomenon in nature. Snell gave the definition of light refraction, called Snell’s Law [77]. When light is refracted into another medium from a vacuum, the ratio n of the incident angle \(\alpha \) to the sine of the refraction angle \(\beta \) is called the absolute refractive index of the medium, which can be expressed by the following formula:The process of OBL strategy is improved by introducing the principle of refraction, as shown in Fig. 5, which is a schematic diagram of the Quasi opposition process based on the principle of refraction.

The discrete solution space is divided into two along the x-axis, the upper half of the space is in a vacuum state, and the lower half can be regarded as other medium space (such as water). Assuming that the value range of particle x on the x-axis is [a, b], point O is the midpoint of the value range of x. Make the normal of the x-axis along the point O, which is the y-axis. In order to apply the principle of refraction to the opposition learning process, the following assumption is made: there is a beam of light source \(x'\) (called the incident point) directly above the particle x, and a beam of incident ray \(Ox'\) is emitted to the midpoint O, and the distance of \(Ox'\) is h. The incident light is refracted at the midpoint O, the refracted ray is \(x^{*'}O\), \(x^{*'}\) is the refraction point, and the distance of \(x^{*'}O\) is \(h^*\). It is not difficult to obtain the following calculation formula from Fig. 5:Set \(k=h/h^*\), then Eq. (28) can be rewritten as:Extending Eq. (29) to multidimensional space, it can be rewritten as:where \(a_j\) and \(b_j\) are the maximum and minimum values of the jth dimension in the current population, \(x_{i,j}\) is the jth dimension value of the ith particle in the population, and \(x^*_{i,j}\) is the refraction solution of \(x_{i,j}\).

Replacing the midpoint O with the DMC mentioned in the previous section, Eq. (29) can be rewritten as:Extending Eq. (31) to multidimensional space, it can be rewritten as:where \(\mathrm{{DMC}}_{j}^{t}\) represents the double mean center of the j-th dimension of the population at the t-th iteration, \(x_{i,j}\) is the j-th dimension value of the i-th particle in the population, and \(x*_{i,j}\) is the refraction solution of \(x_{i,j}\).

The refractive index n is 4/3 in water, and the value of k is linearly decreasing:Among them, \(k_{\max }\) and \(k_{\min }\) are the maximum and minimum values of k, respectively, t is the current number of iterations, and \(t_{\max }\) is the maximum number of iterations.

Therefore, in this paper, the opposition learning strategy of refraction imaging is combined with the double mean center, and the Quasi opposition learning strategy of refraction imaging based on DMC is obtained.

Macroscopically, light reflected from an object contains innumerable photons, which are radiated in all directions due to diffuse reflection. The virtual image generated by the object through refraction is actually a collection of image points corresponding to each point on the object. What people see is like the intersection of the reverse extension lines of the light source emitted by the object and entering the human eye after being refracted, and the position of the virtual image can be determined by selecting any two of the many rays. As shown in Fig. 6, two rays are arbitrarily selected, the slopes of the straight lines \(l_0\) and \(l_1\) are set as \(k_0\) and \(k_1\) respectively, the coordinates of the intersection of the straight lines are (x, y), and the intercepts between the two straight lines and the coordinate axes are \(x_{0}^{'}\), \(x_{1}^{'}\), \(y_{0}^{'}\), \(y_{1}^{'}\), then the straight line equation of \(l_0\) and \(l_1\) can be expressed as:Substitute \(y_{0}^{'}=-k_0x_{0}^{'}\) and \(y_{1}^{'}=-k_1x_{1}^{'}\) into Eqs. (34) and (35) to obtain the coordinates of the intersection:If the intercept of a straight line changes continuously with the slope, then the numerator and denominator in Eqs. (36) and (37) are changed to differential equations, and the formula for the coordinates of the intersection of two approximately parallel straight lines can be obtained:It can be seen from Eq. (38) that the intersection coordinates of two approximately parallel straight lines can be obtained by derivation of the intercept of the straight line against the slope or the reciprocal of the slope.

When people look down at the chopsticks that are obliquely inserted in the bowl full of water, they will see that the chopsticks are deflected upwards. This is actually because the virtual image produced by the chopsticks is deflected upwards relative to the chopsticks themselves. As shown in Fig. 7, it is the learning process of Extended opposition based on the principle of refraction.

Insert the chopsticks \(AA'\) into the water diagonally in half, the angle \(\theta \) between the chopsticks and the water surface, point O is both the midpoint of the chopsticks and the origin of the coordinate axis. Assume that the abscissa corresponding to endpoint A is particle x, and the search space of x is [a, b]. The underwater object point \(A'\)is at a depth h from the water surface (x-axis), and an incident light beam is emitted from the object point \(A'\), and refraction occurs at the intersection point M of the water surface. The virtual image point \(A^*\) is obtained through the reverse extension line of the refracted light through point M, the incident angle is i, and the refraction angle is r. The projection of the virtual image point \(A^*\) on the x-axis is \(x^*\), and \(x^*\) is called the extended opposition solution of particle x based on the principle of refraction.

The midpoint O of Extended opposition learning based on refraction imaging, like OBL, is the midpoint (a+b)/2 of the particle x search space. It can be seen from Fig. 7 that \(x'=h\tan i+a+b-x\), \(y'\) = -(x’+x-a-b) \(\cot r=-h\tan i\cdot \cot r\), \(h=\tan \theta \cdot \left| x \right| \) are the slope \(k=\cot r\) of the refracted ray l. The position coordinates of the virtual image point in the direction of viewing angle r can be obtained by substituting the above parameters into Eq. (38). First, using \(n=\sin r/\sin i\) of Snell’s Law (when n is the refractive index of water, n = 4/3), \(\tan i\), \(x'\), and \(y'\) are expressed as:Substituting the above formula into the previously obtained intersection coordinate Eq. (38), we can get:It can be seen from Eqs. (42) and (43) that as i (or viewing angle r) increases, the coordinates x and y increase synchronously, which means that the virtual image point a moves upward to the right as the viewing angle r increases, until it moves to the water surface (y = 0), at this time \(\tan ^2i=1/\left( n^2-1 \right) , \) namely \(\sin i=1/n\), is obviously the critical position where total reflection occurs. At this time, the coordinate of the virtual image point \(A^*\) is \(x_{\max }=a+b-x+h\cdot \sqrt{1/\left( n^2-1 \right) }\), which is also the maximum value of the opposition solution \(x^*\). When the viewing angle is \(r\rightarrow 0\), the coordinate of the virtual image point is \(\left( a+b-x,-h/n \right) \), which is directly above the object point \(A'\), and the depth is about 75% of the depth of the object point \(A'\). At this time, the opposition solution corresponding to the virtual image point \(A'\) is actually the OBL opposition solution. In order to see the change trajectory of the virtual image point more clearly, with \(a+b-x\) as the origin, the trajectory equation can be obtained by eliminating \(\tan i\) from Eqs. (42) and (43) simultaneously:The specific trajectory is shown in Fig. 8.

Extending Eq. (42) to multidimensional space, it can be rewritten as:Among them, the angle \(\theta =30\) between the chopsticks and the water surface, the incident angle is i, the refractive index in the water is \(n=4/3\), \( x_{i,j} \) is the j-th dimension value of the i-th particle in the population, and \(x_{i,j}^{*}\) is the refraction solution of \( x_{i,j} \). The value of the incident angle i is linearly decreasing in the range [0\(^\circ \), 90\(^\circ \)):Among them, \(i_{\max }\) and \(i_{\min }\) are the maximum and minimum values of i, respectively, t is the current number of iterations, and \(i_{\max }\) is the maximum number of iterations.

According to the global search characteristics of the PSO algorithm, with the iterative operation of the algorithm, the particles gradually approach each other, and the diversity of the population decreases. At this time, the global exploration ability of the algorithm is weakened, and it is easy to fall into the local optimum. However, it is not enough to only enhance the global exploration ability. Neglecting the local development ability will cause the particles to be in the search stage all the time, unable to converge to the optimal value. Therefore, the degree of exploration and development needs to be balanced. Different evolutionary stages require different diversity and accurate control of population diversity helps population evolution. Given the above analysis, in order to prevent local convergence or premature maturity, the percentage of exploration and exploitation in each iteration of the algorithm needs to be determined according to the diversity metric in the Preliminaries section [78]:where Div is the diversity of the population in the current iteration and \(\mathrm{{Div}}_{\max }\) is the maximum diversity of the population in all iterations. \(Xpl\%\) and \(Xpt\%\) are iterative exploration and development percentages, respectively, and these two elements are conflicting and complementary.

In order to better balance the global exploration and local development capabilities of the population, a diversity-driven fusion-opposition strategy (SQOBL) is designed in this paper. Diversity is used to drive the mutual adaptive conversion of the two opposing strategies. When Xpl% is greater than the set threshold, the refraction principle based on the double mean center is used to quasi-opposition, which not only accelerates the algorithm convergence but also performs local fine search. However, when the population diversity gradually decreases, the particles are also prone to fall into the local optimum, that is, the case where Xpl% is lower than the threshold. At this time, the principle of refraction is used to expand the opposition, increase the diversity of the population, help the particles to jump out of the local optimum, and prevent premature maturity. Once Xpl% is above the threshold, the extended opposition learning strategy stops, switches to the quasi-opposition learning strategy again, and repeats this step until the maximum number of iterations is reached.

On the basis of the above research, a combination of the SQOBL strategy and PSO algorithm is proposed. The specific flow chart of the algorithm is shown in Fig. 9, and its pseudo-code is shown in Algorithm 2. The code is available at https://​github.​com/​ZooooZoooo/​SQOBL. The algorithm flow of the combination of SQOBL and PSO is shown in Algorithm 2. It can be seen from Algorithm 2 that the SQOPSO algorithm mainly includes four components: initialized population, initialized opposition population, DMC strategy, and fusion-opposition strategy based on diversity drive. Let the dimension of the objective function be D, the population size be N, and the maximum number of iterations be T. The initialization phase includes initializing the population and initializing the opposing population. The time complexity of the initializing population and the opposing population are both O(ND), so the time complexity of the entire initialization phase is O(2ND). In the iterative process, the time complexity of Algorithm 1 is O(NN), and the time complexity of the fusion-opposition strategy based on diversity drive is O(D), so the time complexity of the entire iterative process is \(O(T(NN + ND ))\), then the total time complexity of SQOPSO is \(O(2ND + T(NN + ND))\), which is only related to T, N, and D. Because the problem dimension D is often larger than or similar to the number of groups N, the low-order terms are ignored, and the time complexity of the SQOPSO algorithm is O(TND), which is consistent with the time complexity of the PSO algorithm.

In this subsection, the proposed SQOBL strategy is analyzed. First of all, the opposite solution with DMC as the reference point is closer to the optimal solution of the optimization problem than the current solution, and the proof derivation process refers to the literature [79].

Then, under the given assumptions, briefly analyze the convergence of the extended oppositional learning strategy based on the refraction imaging principle:

All experiments were performed in MATLAB R2022b and in a Windows 10 hardware environment with Intel Core i5-8500 with frequency 3.00 GHz and 8.0 GB of RAM. In the experiment, the population size N = 100, there are different problem dimension values on different benchmark function sets, the maximum number of function evaluations FE_max=100,000 and the iteration termination condition of each algorithm is that the current iteration number is equal to the maximum iteration number, and each algorithm is in 25 independent runs on each function.

To comprehensively and objectively evaluate the performance of SQOBL variants presented in the overall OBL variant framework, SQOBL was compared with 8 representative OBL variants. The OBL variants compared were: OBL [49], QOBL [63], QROBL [62], COOBL [65], COBL [66], GOBL [53], EOBL [55], and REOBL [61]. We embed all OBL variants into the PSO algorithm, such as QOPSO is to embed QOBL into PSO and QROPSO is to embed QROBL into PSO. In order to compare the differences between the algorithms fairly, the common parameters of each comparison algorithm are set uniformly, such as population size, maximum function evaluation times, etc.; other parameter settings are consistent with the original literature, and the specific parameter settings are shown in the appendix. The jump rate of OPSO, GOPSO, COOPSO, and COPSO is 0.3 [52, 53, 65, 66], and the jump rate of QOPSO, QRPSO, EOPSO, and REOPSO are 0.05 [55, 61‐63].

In order to further observe the performance of the SQOPSO algorithm, we select 8 newly proposed intelligent optimization algorithms and test them on the latest benchmark function set CEC2022. The smart optimization algorithms compared are: Sparrow Search Algorithm (SSA) [5], Dwarf Mongoose Optimization (DMO) [6], Northern Goshawk Optimization (NGO) [7], Tuna swarm optimization (TSO) [8], Golden jackal optimization (GJO) [9], Sand Cat Swarm Optimization (SCSO) [10], Generalized normal distribution optimization (GNDO) [11] and PSO. The parameter settings of each comparison algorithm are consistent with the original literature, and the specific parameter settings are shown in the appendix. The early warning value of the SSA algorithm is 0.6, the proportion of discoverers is 0.7, and the proportion of aware of dangerous sparrows is 0.2. The number of babysitters in the DMO algorithm is 3, and the Alpha female vocalization is 2. In the TSO algorithm, it is determined that the degree of tuna following the optimal in the initial stage is 0.7. The constants \(c_1\) and \(\beta \) of the GJO algorithm are both 1.5. The auditory characteristic of sand cats in the SCSO algorithm is 2.

In the SQOPSO algorithm, the refractive index n is 4/3, the angle of the chopsticks entering the water is 30\(^\circ \), the value of k is linearly decreasing, and the initial value is 0.75, which is near the middle value of the better interval [0, 1.4], the incident angle i is linearly decreasing in the interval [0\(^\circ \), 90\(^\circ \)], and the threshold T of the fusion strategy is equal to 0.1 [58].

We choose different benchmark function sets for comparative experiments, they are divided into three kinds with different characteristics. The first is an unconstrained optimization problem, which selects 15 optimization functions in CEC2015 [80]; the second is a constrained optimization problem, which selects 29 optimization functions in CEC2017 [61]; the third is the latest single-objective optimization Question, 12 optimization functions in CEC2022 are selected. Record the mean (Mean) and standard deviation (SD) of the difference between the actual results of each calculation and the optimal solution.

In recent years, in the field of intelligent computing, many more advantageous algorithms have been proposed and successfully applied to engineering practice. In order to further reflect the performance of SQOPSO algorithm, SQOPSO is compared with SSA, DMO, NGO, TSO, GJO, SCSO, GNDO and PSO. SQOPSO and 8 new intelligent optimization algorithms were tested in 10-D and 20-D on CEC2022. The specific experimental results are shown in Table 10, and the best results are highlighted in bold: It can be found that compared with the latest intelligent optimization algorithm, the performance of SQO-PSO algorithm is in the upper middle, but there is little difference with the optimal solution found by other algorithms. As a framework, SQOBL is applied to the PSO algorithm. After a number of tests, it can be clearly seen that it has been greatly improved compared with the PSO algorithm. If SQOBL is applied to the improvement of other intelligent optimization algorithms, it will have strong competitiveness.

This result further verifies that the proposed SQOBL strategy can greatly improve the optimization ability of the algorithm, and has a good competitiveness in the latest intelligent optimization algorithm.

In order to study the impact of variables of different dimensions on the performance of the algorithm, we conducted tests on 20-D. Table 11 lists the mean, standard deviation, Wilcoxon rank sum test and Friedman test results.