Research Article PSO with Mixed Strategy for Global Optimization

Particle swarm optimization (PSO) is an evolutionary algorithm for solving global optimization problems. PSO has a fast convergence speed and does not require the optimization function to be diferentiable and continuous. In recent two decades, a lot of researches have been working on improving the performance of PSO, and numerous PSO variants have been presented. According to a recent theory, no optimization algorithm can perform better than any other algorithm on all types of optimization problems. Tus, PSO with mixed strategies might be more efcient than pure strategy algorithms. A mixed strategy PSO algorithm (MSPSO) which integrates fve diferent PSO variants was proposed. In MSPSO, an adaptive selection strategy is used to adjust the probability of selecting diferent variants according to the rate of the ftness value change between ofspring generated by each variant and the personal best position of particles to guide the selection probabilities of variants. Te rate of the ftness value change is a more efective indicator of good strategies than the number of previous successes and failures of each variant. In order to improve the exploitation ability of MSPSO, a Nelder–Mead variant method is proposed. Te combination of these two methods further improves the performance of MSPSO. Te proposed algorithm is tested on CEC 2014 benchmark suites with 10 and 30 variables and CEC 2010 with 1000 variables and is also conducted to solve the hydrothermal scheduling problem. Experimental results demonstrate that the solution accuracy of the proposed algorithm is overall better than that of comparative algorithms.


Introduction
In recent years, optimization algorithms are applied more and more widely in various felds [1]. One of the most famous ones is PSO. In 1995, Kennedy and Eberhart developed particle swarm optimization (PSO) [2]. PSO as a global optimization method is an important tool to solve difcult optimization problems without a good problemspecifc approach efciently. Its original inspiration comes from birds focking behaviours. In PSO, each individual in the population is a particle. A particle represents a potential solution in solution space. Particles scan the search area and converge to the optimum by fying in the space and adjusting its fying velocity based on its personal best historical experience and the best solution in the population. PSO is a robust stochastic optimization algorithm that is easy to implement. Its parameter settings are negligible. On account of its simple realization and high efciency, PSO has been successfully applied to various real-world problems such as wireless sensor networks [3], feature selection [4], trafc control [5], road identifcation [6], task allocation [7], and crowd user selection [8].
Recently, various improvements of PSO are proposed to enhance these comprehensive performances. In this research, according to a recent theory [9], this study presented a mixed strategy PSO (MSPSO). In the theory, the hardest problem to one evolutionary algorithm might be the easiest for another algorithm and vice versa. Tus, the mixed strategy PSO algorithms might be more efcient than pure strategy PSO algorithms. Just as a company wants to run well, it needs talents who are good at management, good at marketing, and good at purchasing to work together. If a company employs talents who are good at management for all work, the company will not operate well because employees who are not good at what they do spend more time and get worse results. Inspired by this theory, MSPSO integrates fve diferent PSO variants and adopts a new probability update strategy according to the proportion of diferences in ftness values. According to our experimental verifcation, the probability of those variants is guided by the rate of change of ftness value between ofspring generated by each variant and the personal best positions of the particles. According to the rank of the rate of change, the variants are assigned the diferent probabilities. Using the rate of ftness change to guide the selection probabilities of variants could increase the probability of selecting excellent variants than using the number of previous successes and failures of each variant. In addition, in order to enhance the exploitation ability of MSPSO, a local search method inspired by the Nelder-Mead method is proposed.
In summary, we have made the following contributions: (1) We propose a cooperative strategy to integrate multiple PSO operators, and the integrated algorithm can achieve better generalization capability (2) We add a local search operator to the integrated algorithm, so that the algorithm can further obtain better performance (3) We also demonstrate the performance of MSPSO on benchmark suites and real-world problem instances Te rest of the study is organized as follows. First, related methods are reviewed in Section 2. Second, Section 3 introduces the MSPSO. Tird, Section 4 evaluates the proposed MSPSO and gives the results of the experiments. Finally, the conclusion of the study is shown in Section 5.

Canonical PSO.
In the optimization process, the velocity vector V i for the ith particle in the population is updated using (1) given in [2] iteratively through the guidance of pbest i and gbest.
Acceleration parameters c 1 and c 2 are usually set to 2.0. r 1id and r 2id are two random numbers within [0,1] for the dth dimension of the ith particle.
To avoid the premature convergence, the authors in [10] introduced an inertia weight ω to update the fying velocities of particles. Te particle velocity is adjusted through the following formula: v id � ω * v id + c 1 * r 1id * pbest id − x id + c 2 * r 2id * gbest d − x id . (2) In (2), ω commonly decreases linearly from 0.9 to 0.4 with generations to balance exploration capability and exploitation capability. A large value of ω enhances the exploration capability, whereas a small value of ω encourages the capability of convergence during the search process.

PSO Variants.
To enhance the performance of PSO on global optimization problems, a lot of researches have been working on improving PSO algorithms, and numerous PSO variants have been presented. Designing new strategies, new techniques and topological structures of PSO are an important research trend. Various topologies have been suggested. In PSO, the trajectory of particles is adjusted by their own personal best positions and the best position in the population. However, this may cause premature convergence when solving multimodal functions. Because the best particle in the population is the best solution for the whole population, it could be a local optimum for a multimodal function and is far away from the global optimum. Te authors in [11] proposed a social learning PSO (SL-PSO) which introduced social learning into PSO. Te advantage of social learning was that individuals could learn from others without paying for their own trials and mistakes. In SL-PSO, each particle was updated based on any better particles in the current population. Furthermore, to reduce parameter settings, SL-PSO proposed a dimension-dependent parameter control method. Compared with other optimization algorithms, SL-PSO could be implemented easily, be computed efciently, and require no complicated adjustment of the control parameters. In order to accelerate convergence speed and improve exploitation ability, the authors in [12] proposed prey-predator PSO (PP-PSO). PP-PSO achieved this goal by deleting or transforming "slothful particles" which were the particles with low velocities. It was hard for these slothful particles to fnd the global optimum, and this reduced the convergence speed. Furthermore, in order to enhance population diversity, PP-PSO designed a proportional-integral control parameter to control the population to fuctuate within a relatively stable range during the iterative process. Te above-mentioned PSO variant algorithm mainly improves the classical PSO algorithm from the perspective of designing a new information-sharing mode between particles and building a new particle search model.

Complexity
However, when solving some complex optimization problems, PSO and its variants are still prone to premature convergence in the search process. Furthermore, when trapped in the local optimum, it is difcult for particles to get rid of this region. Terefore, in the past decades, researchers have also tried to solve this problem by proposing various improvement strategies based on existing PSO algorithms. Another popular modifcation is to combine PSO with other mathematical methods or evolutionary computation techniques. Te authors in [13] integrated a PSO algorithm with the sine cosine algorithm (SCA) and the Lévy fight approach, to overcome the shortcoming that PSO tended to fall into a local optimum. Te solution in SCA was updated by sine and cosine functions to ensure the exploitation and exploration capabilities. In addition, SCA used Lévy distribution which was a more efective search to produce a random walk in the search space. Te combination of SCA, Lévy fight, and PSO enhanced the exploration capability of the original PSO and prevented being trapped in the local minimum. In addition, the hybridization of PSO with GAs has also been presented in [14,15]. A hybrid PSO with bat algorithm (BA) has been proposed in [16] for numerical optimization problems. A communicating strategy provided information fow between the population of PSO and the population of BA. In this work, several best individuals in BA replaced the worst individuals in PSO after fxed iterations, and on the contrary, the fnest particles of PSO replaced the poorer individuals of BA.
Multipopulation strategy and ensemble optimizer are also efective methods to optimize the performance of PSO. In order to avoid the phenomena of "oscillation" and "two steps forward, one step back" in PSO, the authors in [17] proposed a two-swarm learning PSO algorithm called TSLPSO. Te algorithm hybridized two diferent learning strategies which were dimensional learning strategy (DLS) and comprehensive learning strategy, respectively. One of the swarms used DLS to construct the learning exemplars. DLS used the information of the best particle in the population for the local search of the particles. However, in order to guide the global search, the other swarm used the comprehensive learning strategy to construct the learning exemplars. In [18], Xu et al. constructed a DMS-PSO-CLS algorithm that combined the dynamic multiswarm particle swarm optimizer (DMS-PSO) and a new cooperative learning strategy (CLS). In the CLS subpopulation, in order to learn more excellent examples, the two poor particles updated their dimensions with the better particle which was selected from two random subswarms using a tournament selection strategy. By using this method, particles could search the global optimum more easily. Te simulation results showed that the performance of the DMS-PSO-CLS algorithm was superior compared with other comparison PSO variants. Te above algorithms also have their limitations. For example, some hybrid algorithm frameworks require more computing resources to execute the iterative process of diferent algorithms, while most multipopulation strategies cannot perform fne local search at the later stage of the search process, so it is difcult to obtain the fnal search results with high accuracy. [19] proposed a complementary strategy theorem. According to this theorem, mixed strategy evolutionary algorithms might outperform pure strategy evolutionary algorithms. One advantage was that the overall performance of mixed strategy evolutionary algorithms might be the same as the best performance of pure strategy evolutionary algorithms. Theorem 1. If a pure strategy evolutionary algorithm PS 2 is better than another pure strategy evolutionary algorithm PS 1 , then for any initial population P, the expected hitting time of mixed strategy evolutionary algorithms MS derived from PS 1 and PS 2 satisfes that m MS (P) ≥ m PS2 (P) and m MS (P) > m PS2 (P) for some state P.

Complementary Strategy Teorem. Te authors in
Theorem 2. If a pure strategy evolutionary algorithm PS 2 is equivalent to another pure strategy evolutionary algorithm PS 1 , then for any initial population P, the expected hitting time of mixed strategy evolutionary algorithm MS derived from PS 1 and PS 2 satisfes that m MS (P) � m PS2 (P).

Theorem 3.
If a pure strategy evolutionary algorithm PS 2 complements with another pure strategy evolutionary algorithm PS 1 , then there exists a mixed strategy evolutionary algorithm MS derived from PS 1 and PS 2 , and its expected hitting time satisfes that m MS (P) ≤ m PS2 (P) for any initial population P and m MS (P) < m PS2 (P) for some initial population P.
Theorem 4 (complementary strategy theorem). Te condition that a pure strategy evolutionary algorithm PS 2 is complementary to another pure strategy evolutionary algorithm PS 1 is sufcient and necessary if there exists a mixed strategy evolutionary algorithm MS derived from them such that m MS (P) ≤ m PS2 (P) for any initial population P and m MS (P) < m PS2 (P) for some initial population P.
Te complementary strategy theorem can be interpreted intuitively as follows: (1) If one pure strategy evolutionary algorithm is better than another pure strategy evolutionary algorithm, then the design of a mixed strategy evolutionary algorithm with the same performance as the better pure strategy evolutionary algorithm is impossible. So mixed strategy evolutionary algorithms do not usually outperform pure strategy evolutionary algorithms that they derived from. (2) If one pure strategy evolutionary algorithm is complementary to another, then the design of a mixed strategy evolutionary algorithm better than both pure strategy evolutionary algorithms is possible. However, this does not mean all mixed strategy evolutionary algorithms will outperform pure strategy evolutionary algorithms that they derived from. (3) Te following principle should be followed when a better-mixed strategy evolutionary algorithm is designed: if a pure strategy evolutionary algorithm has a better performance than another at a state, then Complexity 3 the mixed strategy evolutionary algorithm should apply the pure strategy with a higher probability at that state.
3.1.1. MCLPSO. We proposed a modifed CLPSO (MCLPSO) in [20] a few years ago. Compared with CLPSO, MCLPSO could improve the convergence ability while maintaining the population diversity. Furthermore, MCLPSO has a better balance of exploration and exploitation than CLPSO. Te updating equation of MCLPSO for a particle velocity is given as follows: where g � [1, 2, . . . , Max Gen] is the current generation number, α is an adjustment coefcient between 0 and 1, Max Gen is the maximum number of generations, rand is a random number within range [0, 1], meanv d is the dth dimension of the average value of velocities in the whole population, and pbest fi(d)d represents the dth dimension of the best position of the particle located in a list of particles selected randomly from the whole population, and the rest of the parameters have the same meanings as those in (2). Using the above equation, the velocity of MCLPSO is fast with a high probability in the early stage of the search. Conversely, in the later stage, the velocity is slow with a high probability for better exploitation.

LIPS.
LIPS used the best experiences of adjacent particles rather than the global best experience of the population to guide the particles to the optimum [21]. Tis algorithm adopted the personal best position of neighbor particles measured by Euclidean distance to adjust the particle velocity. Te formula is given as follows: where nbest k is the best position of the kth neighbor particle of the ith particle, ϕ k is a random number that obeys uniform distribution within [0, 4.1/nsize], and nsize is the number of neighbor particles.

HPSO-TVAC.
In order to make the particles converge quickly to the global optimum, the authors in [22] designed a new formula for calculating velocity without using the previous velocity. Te formula is given as follows: where c 1 , r 1id , c 2 , and r 2id have the same meanings as those in (1).

FDR-PSO.
In order to avoid the premature convergence, the authors in [23] added the position of neighbor particle in the formula of particle velocity. Te formula is given as follows: where nbest id is the dth dimension of the best experience of the neighbor of the ith particle which minimizes the fneness-distance ratio (FDR), and the rest of the parameters have the same meanings as those in (2). Te formula of fneness-distance ratio for a minimization problem is given as follows: where P i denotes the best experience of other particles in the population except the ith particle.

Local Search.
A local search method inspired from the Nelder-Mead method is used in MSPSO, in order to improve its exploitation ability. Te Nelder-Mead method is a numerical algorithm that adapts to local landscapes [24]. It makes down-hill search using a simplex instead of derivatives. Introducing the Nelder-Mead method into MSPSO can further improve the performance of MSPSO. In our work, we make a modifcation of the Nelder-Mead method, in order to reduce time consumption. Te number of testing points is set to 3, rather than n + 1 (the dimension). Te following is the detail of the method. Given 3 test points x 1, x 2 , x 3 , a Nelder-Mead variant is given in Algorithm 1. In Line 1, three individuals are sorted in the order of the function value from low to high. In Line 2, various x o is the centre of triangle △x 1 x 2 x 3 . Lines 3-22 are used to implement the Nelder-Mead process.

Improved Adaptive Probability Adjustment Method.
In some ensemble evolutionary algorithms, the selection probability of diferent variants is adjusted based on the number of previous successes and failures of each variant at a fxed iteration interval. However, the number of successes 4 Complexity and failures is not a perfect indicator of a good strategy because it cannot measure the degree of improvement of successful ofspring generated by each variant. Tus, we use the rate of change of ftness value between ofspring generated by each variant and the personal best position to adjust the selection probabilities of variants. Te adaptive probability adjustment method is given as follows.
Step 5. Initialize the probability p k of the kth PSO variant to 1/K and the change rate Cr k of the kth variant as 0. In our work, K is equal to 5.
Step 6. Generate a random number rand pk . If 0 ≤ rand pk < p 1 , choose the frst variant to generate an ofspring. If k−1 i�1 p i ≤ rand pk < k i�1 p i , 1 < k < K, choose the kth variant to generate an ofspring. If k i�1 p i ≤ rand pk ≤ 1, choose the Kth variant to generate an ofspring.
Step 7. If the kth variant is selected to generate an ofspring for a particle, then the change rate is recorded as follows: where f o and f p are the ftness of the ofspring and the ftness of the personal best historical position of the particle, respectively. When the ftness of the ofspring is larger than the ftness of pbest, i.e., the ofspring is worse than the parent, the change rate is reduced. Otherwise, the change rate is increased.
Step 8. After lp generations, update the probability p k of each PSO variant, and set Cr k to 0. Te probability p k is updated by the following steps: (1) Sort the K strategies in descending order based on Cr k . Get a new sequence K′. In this study, we use the change rate rather than the diference between the ftness of the ofspring and the ftness of pbest to guide the adjustment of the probability. Because the diference cannot refect the merits and demerits of each strategy especially when the ftness values of a strategy and pbest are large, however, the ftness values of another strategy and pbest are small. In this situation, the strategy with a small diference in ftness value may be better than the strategy with a large diference. Furthermore, we use a fxed probability distribution to assign a signifcantly larger selection probability to a good strategy and a signifcantly smaller selection probability to a bad strategy.

Framework of MSPSO.
MSPSO integrates PSO, MCLPSO, LIPS, HPSO-TVAC, and FDR-PSO together. It adopts two subpopulations in the early stage and a whole population in the later stage of the search process. In the early stage, MSPSO adopts a subpopulation that implements MCLPSO and a subpopulation that implements ensemble Input: population P with three individuals.
(1) Sort the three points in the order: (2) Calculate x o as follows: Complexity 5 PSO. In the later stage, the whole population implements ensemble PSO. Furthermore, the Nelder-Mead method variant is used in this stage to improve the exploitation ability of MSPSO. In this way, the population diversity and convergence ability of the algorithm can be improved in the early stage. Its convergence ability can be improved in the later stage. Te pseudo-code of MSPSO is given in Algorithm 2. In lines 1-2, population and parameters are initialized. Lines 3-10 give the steps of the early stage of MSPSO. In this stage, the whole population is divided into two subpopulations, one of which is composed of μ 1 individuals and the other is composed of μ − μ 1 individuals. Lines 5-7 indicate that MCLPSO is implemented in the frst subpopulation, while lines 8-10 indicate that the ensemble PSO is implemented in the second subpopulation. Lines 11-16 give the steps of the late stage of MSPSO. In this stage, the whole population implements the ensemble PSO, and then, the Nelder-Mead method variant is implemented. In lines 17-22, the best ftness value of the current population is obtained, and the best ftness value of the algorithm is updated if necessary. Te framework fgure is given in Figure 1.  [10], CLPSO [26], LIPS [21], HPSO-TVAC [22], FDR-PSO [23], EPSO [27], OSC-PSO [28], and A-PSO [29]. All the selected peer algorithms are proposed in the last decade. EPSO is an ensemble PSO. OSC-PSO drives particles into oscillatory trajectories. A-PSO introduces the nonlinear dynamic acceleration coefcients, logistic map, and a modifed particle position update approach in PSO. In order to verify the efectiveness of all the improved strategies proposed by us, we compare MSPSO with the original CLPSO algorithm. Te parameter settings of these algorithms are listed in Table 1. Te parameter settings of MSPSO in diferent dimensions are given in Table 2. Te high dimension of the function is more complex compared to the low dimension of the function, so we use a larger population size for the high dimension of the function to maintain the diversity of the population. Parameters limit gp , α, and β can afect the population diversity and convergence of MSPSO. Te larger the limit gp , the greater the probability of updating particles in MCLPSO based on the global optimal position. Also, the larger the α, the greater the probability of updating particle velocities in MCLPSO based on the mean velocity of the population. Te smaller the β, the more times ensemble PSO is used in the whole population. Te benchmark problems with 30 variables in CEC 2014 are more complex than those with 10 variables. So, we use diferent parameter settings in MSPSO for diferent dimensions.

Benchmark Functions and Comparative
Experimental results in the CEC 2014 suite with 10 and 30 dimensions are reported in Tables 3 and 4, respectively. Te error is an absolute value of the diference between the best value for 30 runs and the actual optimal value of a specifc objective function.
Te nonparametric statistical test has become an important method to compare a group of evolutionary algorithms recently [30]. In this study, the Wilcoxon signed-rank test is employed to estimate MSPSO and other PSO variants with the signifcance level of 5%. For each algorithm, Tables 3 and 4 show the number of best/2nd best/worst ranking, the number of average ranking, and the number of +/�/− in the last three rows, respectively. Te algorithms are ranked according to the mean error of each algorithm. Symbol "+," "�," and "−" indicate that MSPSO is signifcantly better than, similar to, and worse than the compared PSO variant, respectively.
Te simulation results on 30 functions with 10 variables in CEC 2014 are shown in   Table 4. Te results show that MSPSO outperformed the other eight algorithms on functions F2, F3, F4, F5, F6, F7, F11, F15, F17, F20, and F27. For function F26, the mean error of MSPSO was equal to other optimal algorithms. Specifcally, for unimodal functions, MSPSO generally performs better than most of the other algorithms. Also, it is superior or equal to the other eight algorithms on all functions except F1. Furthermore, for simple multimodal functions, MSPSO exhibits a better performance. Also, it is superior to EPSO in eight functions.
Regarding hybrid functions, MSPSO shows a better performance compared with other comparative PSO variants. Also, it ranked within the top 3 on all functions. As for Input: ftness function f(x), dimension n, population size μ, subpopulation size μ 1 , maximum number of generation Max_Gen, MCLPSO limit value limit gp1 and limit gp2 , MCLPSO adjustment coefcient α 1 and α 2 , iterative parameter β, interval iteration lp.
(1) Generate an initial population P consisting of μ individuals at random.

end for
Output: the best ftness value f min .
ALGORITHM 2: MSPSO.             10 Complexity            Tables 3 and 4, we see that the performance of two ensemble PSO algorithms (EPSO and MSPSO) is better than other types of PSO algorithms. Tis result can be explained by the theory of easy and hard ftness functions [17]. According to that theory, the hardest problem to one evolutionary algorithm could be the easiest to another algorithm. Tus, given an ensemble of diferent PSO algorithms, a hard problem might be solved easily by one of them. Of course, if a problem is hard (or easy) to all of them, using an ensemble does not bring too much improvement.
Te convergent speed is evaluated in Figures 2-8. From Figure 2, we can see that MSPSO obtains a better performance than other PSO variants. Te convergent speed of MSPSO is not fast, but its optimization accuracy is higher than other competitors in many functions. Tis is because, in the early stage of the search, diferent particle generation strategies may interfere with the direction in which particles quickly fnd good positions. However, in the later stage of the search, diferent particle generation strategies increase the chances of particles fnding good positions.

Application to the Hydrothermal Scheduling Problem.
Te hydrothermal scheduling problem [31] is a complex optimization problem from the real world. Its main objective is to schedule the power generations of the thermal and hydro units in the system to meet the load demands, under the premise of satisfying the constraints of the hydraulic systems and the power system networks. In order to evaluate the performance of hydrothermal scheduling problem in dealing with real-world problems, we apply MSPSO to solving this problem. In the hydrothermal scheduling problem, decision variables are nonlinearly related to the major operation problem of hydrothermal systems. Te objective of the problem is to minimize the fuel cost of thermal units for 24 hours with four hydro units in the system, and the dimension of the problem is 96.
In order to meet load requirements during the scheduling period, the total fuel cost of the thermal system operation is expressed by F. Te objective function is given as follows: In the previous formula, P Ti is the power generation of an equivalent thermal unit at ith interval, and f i represents the cost function corresponding to P Ti . M is the total number of intervals considered for the short-term planning. Te cost function f i is expressed as follows: MSPSO is compared with fve other algorithms in three hydrothermal scheduling instances, which are CoBiDE [32], TLBO [33], ALC-PSO [34], DNS-PSO [35], and EPSO [27]. CoBiDE incorporates the covariance matrix learning and the bimodal distribution parameter setting into DE. TLBO designs an optimization mechanism inspired by the efect of the infuence of a teacher on learners. TLBO divides the optimization process into "Teacher Phase" and "Learner Phase." ALC-PSO transplants the aging mechanism to PSO to overcome the problem of premature convergence. DNS-PSO employs a diversity-enhancing mechanism and neighborhood search strategies in PSO to achieve a trade-of between exploration and exploitation abilities.
Te computational results of hydrothermal scheduling instances are shown in Table 5 [37], standard sine cosine algorithm (SCA) [38], and slap swarm algorithm (SSA) [39]. All experiments are tested 30 times in 1000 dimensions. Te mean and standard deviation of all algorithms are shown in Table 6. Te average rank and rank are also recorded in the last two rows of Table 6. From Table 6, it shows that MSPSO has outperformance than other comparative algorithms to solve the large-scale global optimization problems. For most CEC 2010 functions, the MSPSO improves the accuracy by some orders of magnitudes. Terefore, the experimental results demonstrate that MSPSO has a good performance in solving the large-scale optimization problems.

Conclusions
Te paper proposes a mixed-strategy PSO algorithm called MSPSO. MSPSO uses the rate of ftness change which measures the degree of improvement of successful ofspring generated by each variant to guide selection probabilities of variants. Compared with previous PSO algorithms which use the number of previous successes and failures of each variant to adjust selection probabilities, MSPSO can increase the probability of selecting excellent variants. Furthermore, the proposed Nelder-Mead variant method is introduced in MSPSO to improve the exploitation ability. Te proposed algorithm is tested on CEC 2014 benchmark suites with 10 and 30 variables. Experimental results demonstrate that MSPSO has a better overall performance than the other eight PSO algorithms on all problems in terms of the solution accuracy. MSPSO is also applied to three instances of the hydrothermal scheduling problem. Computational results show that the MSPSO algorithm also has a good performance in dealing with this real-world optimization problem. MSPSO is further tested on CEC 2010 with 1000 variables. Te experimental results show that MSPSO has a good performance in solving large-scale optimization problems. Our work shows a promising direction for designing effcient mixed strategy PSO algorithms; that is, the rate of ftness change guides the selection probabilities of variants. Tus, using the rate of ftness change to design other mixed strategy evolutionary algorithms will be left for testing as a future work.

Data Availability
Te data used to support the fndings of this study are included within the article.

Disclosure
Te short version of this study has been accepted by the 2021 IEEE International Conference on Space-Air-Ground Computing (SAGC 2021). We expanded that conference paper in this study.