Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization

: The particle swarm optimization algorithm (PSO) is not good at dealing with discrete optimization problems, and for the krill herd algorithm (KH), the ability of local search is relatively poor. In this paper, we optimized PSO by quantum behavior and optimized KH by simulated annealing, so a new hybrid algorithm, named the annealing krill quantum particle swarm optimization (AKQPSO) algorithm, is proposed, and is based on the annealing krill herd algorithm (AKH) and quantum particle swarm optimization algorithm (QPSO). QPSO has better performance in exploitation and AKH has better performance in exploration, so AKQPSO proposed on this basis increases the diversity of population individuals, and shows better performance in both exploitation and exploration. In addition, the quantum behavior increased the diversity of the population, and the simulated annealing strategy made the algorithm avoid falling into the local optimal value, which made the algorithm obtain better performance. The test set used in this paper is a classic 100-Digit Challenge problem, which was proposed at 2019 IEEE Congress on Evolutionary Computation (CEC 2019), and AKQPSO has achieved better performance on benchmark problems.

The swarm intelligence algorithms mainly come from the bionic idea, which is a set of methods summarized by referring to the laws of life and predatory behavior of animals in the biological world. A swarm intelligence algorithm is a kind of algorithm that works by studying the collective behavior of animals, and the most famous swarm intelligence algorithms are particle swarm optimization algorithm (PSO) [46] and ant colony optimization algorithm (ACO) [55]. An evolutionary algorithm is mainly a kind of method obtained by studying the process of biological genetic change, and these include genetic algorithm (GA) [44], genetic programming (GP) [62], evolutionary strategy (ES) [63,64], differential evolution (DE) [58] and other algorithms are based on genes. The krill herd (KH) [49] and quantum-behaved particle swarm optimization (QPSO) [65] algorithms mentioned in this paper belong to the swarm intelligence algorithm. The KH algorithm has the advantages of simplicity, flexibility, and high computational efficiency, and the QPSO algorithm has a relatively fast convergence speed, but when they are used to solve the 100-Digit Challenge problems, they do not do well. Owing to the poor local optimization ability of QPSO, it is easy to fall into the local optimal value, so it does not solve this problem well. In order to study an algorithm with very high accuracy, as well as to simultaneously optimize the exploitation and exploration and improve the accuracy of annealing krill quantum particle swarm optimization (AKQPSO), we studied the KH algorithm with strong exploitation and the QPSO algorithm with strong exploration. By combining their advantages, the new algorithm overcomes their original shortcomings and has a strong ability in exploration and exploitation. We combine the advantages of KH and QPSO, and optimize them, forming the annealing krill quantum particle swarm optimization (AKQPSO) algorithm. The new algorithm solved the 100-Digit Challenge better. Moreover, the study solved the problem of the poor accuracy of general algorithms, and exploitation and exploration cannot achieve the optimal at the same time.
In this paper, the 100-Digit Challenge problem is difficult to solve by traditional methods, and we can solve this problem better by using a swarm intelligence algorithm. In 2002, academician Nick Trefethen of Oxford University and the Society for Industrial and Applied Mathematics (SIAM) jointly developed the 100-Digit Challenge problem, which was proposed mainly to test high-precision computing [18]. The challenge consists of ten problems, each of which needs to be accurate to ten decimal places, with a total of 100 digits, so the problem is named the 100-Digit Challenge. However, traditional methods need a lot of computation to solve this challenge, and they cannot get good results, so we use swarm intelligence algorithm to solve these problems, and get satisfactory results.
The rest of paper is organized as follows. Most representative studies regarding the KH and QPSO algorithm are reviewed in Section 2, and these two algorithms are introduced in Section 3. The proposed algorithm is described in Section 4 and the experimental results of AKQPSO are presented in Section 5. Section 6 provides the main conclusions and highlights future work.

Related Work
In order to illustrate the hybrid method, we introduced KH and PSO algorithms. In the following, we will introduce the KH algorithm, PSO algorithm, and 100-Digit Challenge problem, respectively.

KH
At present, we have two kinds of optimizations for the KH algorithm [49]; one is to improve the KH algorithm itself, and the other is to improve the KH by other excellent operators or algorithms. Wang et al. [54] changed the parameters of the KH algorithm and improved the speed of global convergence through chaos theory. Then, in order to improve the performance, they [66] used harmony search to replace the physical diffusion, which greatly improved the performance and efficiency. These algorithms belong to the first category, which optimized the KH algorithm itself. The following are the second category, which optimized the algorithm with better strategies. Abualigah et al. [50] put forward a new algorithm, which combined harmony search algorithm with he KH algorithm to generate a new probability factor and improved exploration search ability, so as to get a better solution. Another algorithm is proposed by Abualigah et al. [67], which combined the objective function with the KH algorithm, and had better performance in solving the problem of text clustering. Niu et al. [68] sped up the exploration convergence and improved the efficiency by using the opposite learning strategy, using a sinusoidal graph to change the inertia weight, and modifying the search process according to the inertia weight and acceleration coefficient.

PSO
The PSO algorithm is one of the most famous swarm intelligence algorithms [47,69,70]. It was proposed in 1995, and it has the advantages of few parameters, a simple principle, and fast convergence. This algorithm has been widely used. For example, Sun et al. [71] used the agent model to assist the PSO algorithm to solve complex optimization problems. Through the combination of exploration and exploitation, they can better solve high-dimensional problems with limited resources. Similarly, for high-dimensional problems, Tran et al. [72] changed the fixed length of feature selection in the PSO algorithm, so that the particle swarm had a shorter length. This operation reduced the search space and gave shorter particles better performance. Thus, the overall efficiency of the PSO algorithm is improved, and it is also more suitable for high-dimensional problems. For the feature selection of the PSO algorithm, they [73] also optimized other methods. They improved the algorithm by discretizing the feature selection, and proved that the single variable discretization may reduce the performance of the feature selection stage, so they proposed a new discretization method, and got better performance. Zhang et al. [74] considered that there was no PSO algorithm that can work in noisy and noiseless environment at the same time, so they proposed the dual-environment PSO algorithm. This new algorithm is based on the top-k elite particles to search, which not only guaranteed the good performance in the noise environment, but also can be re-applied to the noise-free environment, so it filled a gap in the field of the PSO algorithm.

100-Digit Challenge
The 100-Digit Challenge problem was originally proposed by SIAM [75], which is a test method for accuracy calculation, and has been studied by many experts. Many algorithms have been applied to this challenge, and some of them are hybrid algorithms that are used to challenge this problem. Epstein et al. [76] combined the genetic algorithm (GA) with differential evolution (DE) algorithm to search the genetic space and find the latest solution. The new algorithm improved the ability to find the right answer through the fitness-based opposition and tide mutation. Brest et al. [77] used the self-adaptive differential evolution (jDE) algorithm in the DE algorithm to solve this problem, and also got better results. Different from the above algorithms, Zhang et al. [78] proposed a new DE algorithm named collective information powered DE. This new algorithm proposed a restart mechanism through collective population information to improve the robustness.

KH and PSO
KH and PSO are very efficient algorithms in the field of swarm intelligence, and we will respectively introduce them in the following.

KH
In the process of predation, the predator will change the distribution of krill population, which will make them move rapidly, and then cause their distribution density to decrease and the distance between the predator and the food to become more and more far, which is the initial stage of KH. In this process, the distribution of krill population is determined by the following three situations: the influence of other krill individuals, behavior of getting food, and random diffusion. The KH algorithm can be described as follows: where N i is the influence of other krill individuals, F i is the behavior of getting food, and D i is the behavior of random diffusion; i = 1, 2, . . . , N, and N is the population size.
For the influence of other krill individuals, the motion N i,new of krill i induced by other krill is defined as follows: where N max represents the maximum induced velocity, N i,old represents the previously induced movement, ω n represents the inertia weight and the value range is (0,1), and α i indicates that the individual i is affected by the induction direction of the surrounding neighbors. The next behavior F i is to get food, as follows: where V f is the maximum foraging speed, and its value is a constant, which is 0.02 (ms −1 ); ω f is the inertia weight of foraging movement, and its range is (0, 1); F i,old is the previous foraging movement; and β i is the foraging direction. The individual D i in the last behavior can be represented as follows: where D max represents the maximum random diffusion speed; δ represents the direction of random diffusion; and I and I max represent the current number and the maximum number of iterations, respectively. From above process, we can get the krill update process of the KH algorithm as follows: where ∆t is the time interval related to the specific application; NV is the dimension of the decision variable; step factor Ct is a constant between 0 and 2; and UB j and LB j are the upper and lower bounds of corresponding variable j (j = 1, 2, . . . , NV), respectively. The process of the KH algorithm (Algorithm 1) is as follows.
Step 1: Initialization. Initialize the generation counter G, the population P, V f , D max , and N max .

3.
Step 2: Fitness calculation. Calculate fitness for each krill according to its initial position.

4.
Step 3: While G < MaxGeneration do 5. Sort the population according to their fitness. 6. for i = 1:N (all krill) do 7. Perform the following motion calculation.

PSO
In the PSO algorithm, each individual is called a "particle", and each particle will find a potential optimal solution at each iteration. Assuming that the size of the population is N, each particle i (i = 1, 2, . . . , N) in the population has its own initial position , and they will search for the optimal solution in D-dimensional space according to their own individual extremum p best and global extremum g best . Individual extremum is the current best point found by each particle in the search space, and global extremum is the current best point found by the whole particle group in the search space. During the search process, the updating formula of particle's relevant state parameters is as follows: where η is the inertia weight that determines the specific gravity of the particle to the current velocity. If the η is large, the particle has a strong exploration ability and can span a longer distance to find the global optimal solution, but it may cause the particle to oscillate back and forth before and after the optimal solution. If the η is small, it means that particles have better ability to find the optimal solution locally, but they can easily to fall into the local optimization solution.
The flow of the standard PSO algorithm (Algorithm 2) is given below, where b up and b lo represent the upper and lower bounds of the problem domain, respectively, and D represents the dimension of the solution space.

Begin
Step 1: Initialization. 1. Initial position: the initial position of each particle obeys uniform distribution, that is X i ∼ U(b up , b lo ).
2. Initialize its own optimal solution and the overall optimal solution: the initial position is its own optimal solution p i = x i , and then calculating the corresponding value of each particle according to the defined utility function f, and find the global optimal solution g best .
3. Initial speed: the speed also obeys the uniform distribution.
Step 2: Update. According to Equations (7) and (8), the velocity and position of particles are updated, and the current fitness of particles is calculated according to the utility function f of the problem. If it is better than its own historical optimal solution, it will update its own historical optimal solution p best , otherwise it will not update. If the particle's own optimal solution p best is better than the global optimal solution g, then the global optimal solution g is updated, otherwise it is not updated.
Step 3: Determine whether to terminate: Determine whether the best solution meets the termination conditions, if yes, stop. Otherwise, return to Step 2. End. (7) and (8), during each iteration, the particles update the direction and speed of the next flight based on their own and group experience. The main characteristics of the PSO algorithm are as follows:

As shown in Equations
1.
Particles have memory. Each iteration of particles will transfer the optimal solution of the population to each other, and update the database of all particles. If the particles deviate, the direction and velocity can be corrected by self-cognition and group-cognition.

2.
PSO algorithm has fewer parameters, so it is easy to adjust, and the structure is simple, so it is easy to implement. 3.
The operation is simple, and it only searches for the optimal solution in the solution space according to the flight of particles.

AKQPSO
The AKQPSO algorithm is a mixed metaheuristic algorithm, which combines the advantages of annealing krill herd (AKH) and QPSO [79]. AKH solves the disadvantage that KH cannot escape from the local optimal solution. At the same time, the efficiency of KH algorithm is improved by the simulated annealing strategy. Quantum behaved PSO (QPSO) is a new algorithm proposed by Sun et al. [79] in 2004. By introducing the quantum behavior and combining it with the idea of simulated annealing, the search ability of AKQPSO is greatly improved, and the new algorithm has better performance. The principle of QPSO is shown below.
Quantum computing is a new computing mode, which follows the laws of quantum mechanics and regulates the quantum information unit. In quantum space, when the aggregation state property is satisfied, particles can search in the whole feasible solution space, thus greatly improving the exploration search ability of QPSO. According to the analysis theory of particle convergence trajectory, if every particle can converge to its local attraction point P i = (p i1 , p i2 . . . , p id ), then the algorithm has the possibility of convergence. The particle position update expression of the standard QPSO algorithm is as follows: where α is the only parameter that needs to be adjusted in the algorithm, called the compression-expansion factor, which is used to control the convergence rate of particles. During the iterative process, the calculation method of individual and global optimal position is the same as that of the PSO algorithm, and the biggest difference is that the QPSO algorithm removes the speed information.
First of all, we initialize the whole population, and all the individuals in the population are randomly generated. After initialization, we divide the population into two subpopulations according to the ratio of 3:7, which is discussed in Section 5.2. The population with the proportion of 3/10 is optimized by the improved KH algorithm, which is called subpopulation-AKH, and the population with the proportion of 7/10 is optimized by the QPSO algorithm, which is called subpopulation-QPSO. The two subpopulations will be re-integrated into a population after iterative optimization. If the optimization result of the new population meets the output conditions, then the result can be output. However, if it does not meet the output conditions, then it can be re-divided according to the proportion of 3:7, and repeat the above process until the results meet the termination conditions. In the process of decentralized and re-fusion of this population, the location information of individuals is shared, so it will greatly improve the efficiency of fusion search and get the best results faster. In the process of the AKQPSO algorithm, we also optimized the KH algorithm by the idea of simulated annealing, named annealing KH (AKH), so we used QPSO and simulated annealing strategy to guarantee the algorithm to escape from the local optimal value. The framework of AKQPSO is shown in Figure 1 and the process of the AKQPSO algorithm (Algorithm 3) is as follows.

Initialization:
N random individuals were generated. Set initialization parameters of AKH and QPSO.

Evaluation:
Evaluate all individuals based on location.

Partition:
The whole population was divided into subpopulation-AKH and subpopulation-QPSO. AKH process: Subpopulation-AKH individuals were optimized by AKH. Update through these three actions of the influence of other krill individuals, behavior of getting food and random diffusion.
The simulated annealing strategy is used to deal with the above behaviors. Update the individual position according to the above behavior.

QPSO process:
Subpopulation-QPSO individuals were optimized by QPSO. Update the particle's local best point P i and global best point P best . Update the position x t+1 i j by Equation (10).

Combination:
The population optimized by AKH and QPSO was reconstituted into a new population.

Finding the best solution:
The fitness of all individuals was calculated, and the best solution was found in the newly-combined population.

Determine whether to terminate:
Determine whether the best solution meets the termination conditions, if yes, stop. Otherwise, return to step Evaluation.
the idea of simulated annealing, named annealing KH (AKH), so we used QPSO and simulate nealing strategy to guarantee the algorithm to escape from the local optimal value. Th mework of AKQPSO is shown in Figure 1 and the process of the AKQPSO algorithm (Algorith is as follows. Update the position x(i) by Equation (11) Combine the two subpopulations into one population

Simulation Results
The 100-Digit Challenge problem comes from CEC 2019 and can be found through the website http://www.ntu.edu.sg/home/EPNSugan/index_files/CEC2019. This challenge includes 10 problems, which are 10 functions in our data. The goal is to calculate the accuracy of the function to 10 digits without a time limit. The proposed algorithm was written in MATLAB R2016a and was run in the following conditions: Intel(R) Core(TM) i5-8100 CPU 3.60 GHz, which possesses 8G of RAM on Windows 10. Table 1 shows the basic parameters of the 100-Digit Challenge. The 10 test problems are the definition of minimization and some other definitions are as follows: where D is the dimension, the global optimal value of the shift is randomly distributed in (−80, 80), and all testing problems are scalable.

The Comparison of AKQPSO and Other Algorithms
In this paper, we compared the AKQPSO algorithm with the following eight state-of-the-art algorithms.

2.
By adding a new hybrid differential evolution operator, the efficiency of the updating process is improved, and differential evolution KH (DEKH) [80] was proposed.

4.
By adding the stud selection and crossover operator, the efficiency was improved and stud krill herd (SKH) [81] was proposed.
Genetic algorithm (GA) [45] is basic algorithm of evolutionary computing. Table 2 is the common parameter setting of these algorithms. We set the population size (N) to 50; after 7500 iterations (max_gen) and 100 independent runs (max_run), all algorithm experiments are carried out under the condition that the basic parameters are consistent.  Table 3 shows the optimization results of AKQPSO and eight other algorithms for ten of the 100-Digits Challenge, with the best value in bold. From the experimental results, we can see that the optimal values of the ten problems are all calculated by the AKQPSO algorithm. AKQPSO is the best algorithm among these algorithms, and the result is the best in every problem, but QKH is not the second in every problem, so we can know that our algorithm improvement is very effective. Through Figure 2, we can clearly conclude that the performance of AKQPSO is the best in general, followed by QKH, and the performance of GA is the worst. In general, we rank the algorithms as follows: AKQPSO > QKH > CCS > BBKH > DEKH > SKH > VNBA > GA.    Figures 3-12 show the results of the nine algorithms on each problem, and the experimental results show that AKQPSO algorithm has obvious advantages over other algorithms. From these column charts, we can see that the fluctuation of problem 1 is the largest. In problem 1, the results of the GA algorithm with the worst performance are 735 times different from those of AKQPSO algorithm with the best performance, which can be said to be very different. Because the 100-Digits Challenge problem is used to test the computational accuracy, we can know that the computational accuracy of AKQPSO algorithm is the highest among the nine algorithms. In Figure 3, because the results of AKQPSO, QKH, and BBO are so different from those of GA, the columns of these algorithms are almost invisible, so they are not obvious in the same chart. However, in fact, the improvement of AKQPSO compared with QKH and BBO is significant. Figures 3-12 show the results of the nine algorithms on each problem, and the experimental results show that AKQPSO algorithm has obvious advantages over other algorithms. From these column charts, we can see that the fluctuation of problem 1 is the largest. In problem 1, the results of the GA algorithm with the worst performance are 735 times different from those of AKQPSO algorithm with the best performance, which can be said to be very different. Because the 100-Digits Challenge problem is used to test the computational accuracy, we can know that the computational accuracy of AKQPSO algorithm is the highest among the nine algorithms. In Figure 3, because the results of AKQPSO, QKH, and BBO are so different from those of GA, the columns of these algorithms are almost invisible, so they are not obvious in the same chart. However, in fact, the improvement of AKQPSO compared with QKH and BBO is significant.                              Among these ten problems, we divide the problems into four groups according to the ability of these problems to test the computational accuracy of AKQPSO. Some groups have a strong ability to test, so it is more obvious which algorithm is better. The first group: problem 1. According to the computational accuracy, the results displayed in this group are quite different. The second group: problems 2, 3, and 7. This group has higher requirements for the accuracy, so it is difficult to calculate the best value of the function. The third group: problems 4, 6, 8, and 10. This group has a slightly lower requirement for the computational accuracy, so the results of various algorithms are very close, but it is difficult to achieve the best results. The fourth group: problems 5 and 9. This group has the lowest requirements for the computational accuracy, so many algorithms can be very close to the optimal result 1.0000000000. Figures 13-16 show the advantage of AKQPSO over other algorithms in each group of the 100-Digits Challenge problem. Although the accuracy of GA is the worst when ranking from the whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and the computational accuracy of this algorithm is also the highest. 100-Digits Challenge problem. Although the accuracy of GA is the worst when ranking from the whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and the computational accuracy of this algorithm is also the highest.   whole, and in the first of the four groups, it is also the GA. In the other three groups, however, the algorithm with the worst accuracy is CCS algorithm, so in terms of grouping comparison, CCS algorithm is the worst. In addition, the accuracy of AKQPSO algorithm is still the best among the four groups, which has not changed. From the above analysis, AKQPSO is the best in all aspects, and the computational accuracy of this algorithm is also the highest.

Evaluation Parameter λ
In Section 4, after the initialization of the population, AKQPSO algorithm divided the population into two subpopulations: AKH and QPSO. We set the parameter λ, which represents the proportion of the two subpopulations. In the experiment, λ = 0.1, 0.2, …, 0.9. For example, if λ = 0.1, this means that the ratio of subpopulation-AKH/subpopulation-QPSO is 1:9, and other numbers mean the same thing. After a lot of experiments, the optimal parameter value is λ = 0.3. All the above experiments are based on the results obtained using λ = 0.3, and the following part will introduce the experiments on λ = 0.3. The parameters of this part of experiments are still the same as Table 2 to ensure that all experiments are tested under the same conditions. Table 4 shows the experimental results of 10 problems that are calculated by different λ in the AKQPSO algorithm in the 100-Digit Challenge. Bold font indicates the best value. From Table 4, we can see that, in the case of λ = 0.3, eight of the ten problems can get the best value. In the remaining two problems, even if λ = 0.3 does not reach the best value, its result is the second among the nine values, and it is very close to the best value. Therefore, when λ = 0.3, the performance of AKQPSO can reach the best.
As can be seen from Figure 17a, the fluctuation range of the result of problem 1 is the largest. As problems 1-10 are in the same chart, and the fluctuation of problem 1 is much larger than that of

Evaluation Parameter λ
In Section 4, after the initialization of the population, AKQPSO algorithm divided the population into two subpopulations: AKH and QPSO. We set the parameter λ, which represents the proportion of the two subpopulations. In the experiment, λ = 0.1, 0.2, . . . , 0.9. For example, if λ = 0.1, this means that the ratio of subpopulation-AKH/subpopulation-QPSO is 1:9, and other numbers mean the same thing. After a lot of experiments, the optimal parameter value is λ = 0.3. All the above experiments are based on the results obtained using λ = 0.3, and the following part will introduce the experiments on λ = 0.3. The parameters of this part of experiments are still the same as Table 2 to ensure that all experiments are tested under the same conditions. Table 4 shows the experimental results of 10 problems that are calculated by different λ in the AKQPSO algorithm in the 100-Digit Challenge. Bold font indicates the best value. From Table 4, we can see that, in the case of λ = 0.3, eight of the ten problems can get the best value. In the remaining two problems, even if λ = 0.3 does not reach the best value, its result is the second among the nine values, and it is very close to the best value. Therefore, when λ = 0.3, the performance of AKQPSO can reach the best. As can be seen from Figure 17a, the fluctuation range of the result of problem 1 is the largest. As problems 1-10 are in the same chart, and the fluctuation of problem 1 is much larger than that of other problems, the fluctuation range of problems 2-10 is not obvious in Figure 17a. In order to compare problems 2-10 more clearly, we used Figure 17b to show the fluctuation of other problems 2-10. Through Figure 17b, we can see that the fluctuation of problems 2 and 7 are also obvious. Therefore, among these ten problems, problems 1, 2, and 7 are the three problems with the largest fluctuation. Among these ten problems, the fluctuation of problem 7 is different from that of the other nine problems. With the increasing proportion of AKH and QPSO, the result of problem 7 tends to be better. For other problems, however, with the increasing proportion of AKH and QPSO, their results tend to be worse, which is the opposite of problem 7.

Complexity Analysis of AKQPSO
The main computing overhead of AKQPSO associated with Step "Partition" of the AKQPSO algorithm in Section 4. The following is a detailed analysis of the single computational complexity of Step "Partition" and other step in AKQPSO. N is the number of individuals in the population. From the results of Sections 5.1 and 5.2, problems 1, 2, and 7 belong to the first and the second of the four groups, respectively, and they are also the two groups with the highest requirements for the computational accuracy of AKQPSO. Therefore, the adjustment of subpopulation has a very direct effect on the accuracy of AKQPSO.

Complexity Analysis of AKQPSO
The main computing overhead of AKQPSO associated with Step "Partition" of the AKQPSO algorithm in Section 4. The following is a detailed analysis of the single computational complexity of Step "Partition" and other step in AKQPSO. N is the number of individuals in the population.

1.
Step "Partition": This step accounts for the main computational overhead, so we focus on this step.

•
Step "AKH process": The computational complexity of this step mainly includes "Subpopulation-AKH individuals were optimized by AKH", "Update through these three actions of the influence of other krill individuals, behavior of getting food and random diffusion", "The simulated annealing strategy is used to deal with the above behaviors", and "Update the individual position according to the above behavior", and their complexities are O(N), O(N 2 ), O(N), and O(N), respectively. • Step "QPSO process": The computational complexity of this step mainly includes "Subpopulation-QPSO individuals were optimized by QPSO", "Update the particle's local best point P i and global best point P best ", and "Update the position x t+1 ij by Equation (10)", and their complexities are all O(N).
• The computational complexity of Step "Initialization", Step "Evaluation", Step "Combination", and Step "Finding the best solution" are all O(N).
Therefore, in one generation AKQPSO, the worst-case complexity is O(N 2 ).

Conclusions
In this paper, aiming at the disadvantage of poor local search ability of KH algorithm, we optimized it by simulated annealing strategy and QPSO algorithm, and proposed a new algorithm: AKQPSO. This algorithm was compared with eight other excellent algorithms, and the computational accuracy of AKQPSO was tested by the 100-Digit Challenge problem. As predicted before the experiment, the computational accuracy of AKQPSO algorithm is the highest among these algorithms. Moreover, we also adjusted the parameters of AKQPSO through experiments to further improve the computational accuracy. By calculating the accuracy of the 100-Digit Challenge problem, our experiment provided a new method to study the accuracy of the algorithm, and the paper provided a very high accuracy algorithm. However, the research of this paper still has some limitations. The accuracy of the algorithm in 100-Digit Challenge problem is very high, but it is unclear whether AKQPSO still has high accuracy in general problems.
Thirdly, now we just use two kinds of algorithms to make them cooperate and optimize in a relatively simple way, and we can also study how to make them cooperate more efficiently through other methods. Finally, although the 100-Digit Challenge is a classical problem, it is still less used for testing computational accuracy of algorithms, and there is still a large development space in this area.

Conflicts of Interest:
The authors declare no conflict of interest.