# Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. KH

#### 2.2. PSO

#### 2.3. 100-Digit Challenge

## 3. KH and PSO

#### 3.1. KH

_{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.

_{i,new}of krill i induced by other krill is defined as follows:

_{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.

_{i}is to get food, as follows:

_{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 ${\beta}_{i}$ is the foraging direction.

_{i}in the last behavior can be represented as follows:

_{max}represents the maximum random diffusion speed; $\delta $ 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:

_{j}and LB

_{j}are the upper and lower bounds of corresponding variable j (j = 1, 2, …, NV), respectively.

Algorithm 1. KH [49] |

1. Begin |

2. 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: WhileG < MaxGeneration do |

5. Sort the population according to their fitness. |

6. for i = 1:N (all krill) do |

7. Perform the following motion calculation. |

8. Motion induced by other individuals |

9. Foraging motion |

10. Physical diffusion |

11. Implement the genetic operators. |

12. Update the krill position in the search space. |

13. Calculate fitness for each krill according to its new position |

14. end for i |

15. G = G + 1. |

16. Step 4: end while. |

17. End. |

#### 3.2. PSO

_{i}(X

_{i}= x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{id}) and initial velocity V

_{i}(V

_{i}= v

_{i}

_{1}, v

_{i}

_{2}, …, v

_{id}), 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:

_{up}and b

_{lo}represent the upper and lower bounds of the problem domain, respectively, and D represents the dimension of the solution space.

Algorithm 2. PSO [46] |

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. |

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

## 4. AKQPSO

_{i}= (p

_{i}

_{1}, p

_{i}

_{2}…, p

_{id}), then the algorithm has the possibility of convergence. The particle position update expression of the standard QPSO algorithm is as follows:

Algorithm 3 AKQPSO |

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 ${x}_{ij}^{t+1}$ 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. |

## 5. Simulation Results

#### 5.1. The Comparison of AKQPSO and Other Algorithms

- By improving krill migration operator, biogeography-based KH (BBKH) [51] was proposed.
- By adding a new hybrid differential evolution operator, the efficiency of the updating process is improved, and differential evolution KH (DEKH) [80] was proposed.
- By quantum behavior to optimize KH, quantum-behaved KH (QKH) [65] was proposed.
- By adding the stud selection and crossover operator, the efficiency was improved and stud krill herd (SKH) [81] was proposed.
- By adding chaos map to optimize cuckoo search (CS), the chaotic CS (CCS) [82] was proposed.
- By adding variable neighborhood (VN) search to bat algorithm (BA), VNBA [83] was proposed.
- By discussing physical biogeography and its mathematics, biogeography-based optimization (BBO) [84] was proposed.
- Genetic algorithm (GA) [45] is basic algorithm of evolutionary computing.

**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.

#### 5.2. Evaluation Parameter λ

**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.

#### 5.3. Complexity Analysis of AKQPSO

**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.

**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 ${x}_{ij}^{t+1}$ by Equation (10)”, and their complexities are all O(N).

- Other step.
- The computational complexity of
**Step “Initialization”**,**Step “Evaluation”**,**Step “Combination”**, and**Step “Finding the best solution”**are all O(N).

^{2}).

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Accuracy of AKQPSO and other algorithms on problem 1: Storn’s Chebyshev polynomial fitting problem. BBKH, biogeography-based krill herd; DEKH, differential evolution KH; QKH, quantum-behaved KH; SKH, stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat algorithm; BBO, biogeography-based optimization; GA, genetic algorithm.

**Figure 5.**Accuracy of AKQPSO and other algorithms on problem 3: Lennard–Jones minimum energy cluster.

**Figure 17.**The different subpopulations in different problems. (

**a**) The different subpopulations in problems 1–10 and (

**b**) the different subpopulations in problems 2–10.

No. | Functions | ${\mathbf{F}}_{\mathit{i}}^{*}={\mathit{F}}_{\mathit{i}}({\mathit{x}}^{*})$ | D | Search Range |
---|---|---|---|---|

1 | Storn’s Chebyshev Polynomial Fitting Problem | 1 | 9 | (−8192–8192) |

2 | Inverse Hilbert Matrix Problem | 1 | 16 | (−16,384–16,384) |

3 | Lennard–Jones Minimum Energy Cluster | 1 | 18 | (−4–4) |

4 | Rastrigin’s Function | 1 | 10 | (−100–100) |

5 | Griewangk’s Function | 1 | 10 | (−100–100) |

6 | Weierstrass Function | 1 | 10 | (−100–100) |

7 | Modified Schwefel’s Function | 1 | 10 | (−100–100) |

8 | Expanded Schaffer’s F6 Function | 1 | 10 | (−100–100) |

9 | Happy Cat Function | 1 | 10 | (−100–100) |

10 | Ackley Function | 1 | 10 | (−100–100) |

Parameters | N | max_gen | max_run |
---|---|---|---|

Value | 50 | 7500 | 100 |

**Table 3.**The minimum value of annealing krill quantum particle swarm optimization (AKQPSO) and other algorithms in the 100-Digit Challenge. BBKH, biogeography-based krill herd; DEKH, differential evolution KH; QKH, quantum-behaved KH; SKH, stud KH; CCS, chaotic cuckoo search; VNBA, variable neighborhood bat algorithm; BBO, biogeography-based optimization; GA, genetic algorithm. (The table results retain ten decimal places.).

Algorithm | Problem 1 | Problem 2 | Problem 3 |

AKQPSO(Algorithm 3) | 10,893.3861385383 | 215.6832766302 | 1.3068652046 |

BBKH | 291,803.2852933580 | 613.7135086878 | 5.9027057769 |

DEKH | 975,339.7308122220 | 708.0601338156 | 6.5872946609 |

QKH | 47,583.9014728552 | 404.4416682185 | 2.1853645745 |

SKH | 1,773,168.4809322700 | 402.7243166556 | 8.0692567398 |

CCS | 211,411.5538712060 | 358.9175375272 | 12.4818284785 |

VNBA | 2,550,944.5550316900 | 816.8071391350 | 4.6163144446 |

BBO | 72,287.9373622652 | 311.8124377894 | 2.9209583205 |

GA | 8,009,879.0206872900 | 2209.1904660292 | 9.7547945042 |

Algorithm | Problem 4 | Problem 5 | Problem 6 |

AKQPSO(Algorithm 3) | 4.0398322695 | 1.0434696313 | 1.1185826442 |

BBKH | 17.7900443880 | 1.8852367864 | 3.9551854227 |

DEKH | 29.4088567232 | 1.9507021960 | 3.1953451545 |

QKH | 7.1133424631 | 2.0695362381 | 3.1213610789 |

SKH | 50.9548872356 | 1.9350825919 | 10.2543233489 |

CCS | 61.4604530135 | 3.4583111771 | 6.6010211976 |

VNBA | 24.7278199397 | 3.0251295691 | 5.7134938818 |

BBO | 17.4206259198 | 1.0971964590 | 2.5154932374 |

GA | 53.9444360765 | 2.7126573212 | 6.2420012291 |

Algorithm | Problem 7 | Problem 8 | Problem 9 |

AKQPSO(Algorithm 3) | 121.8932375270 | 2.3131324015 | 1.0715438957 |

BBKH | 779.4066229661 | 4.3846591727 | 1.3554069744 |

DEKH | 1060.0387128932 | 4.4509969471 | 1.3614909209 |

QKH | 195.0770363640 | 2.5711387868 | 1.2989726797 |

SKH | 1348.8213407547 | 4.8795287164 | 1.5980935412 |

CCS | 2247.4406602539 | 5.4002626350 | 1.6453371395 |

VNBA | 644.6334455535 | 4.1477932707 | 1.3950038682 |

BBO | 950.8018632939 | 3.5941524656 | 1.2098037868 |

GA | 1276.4565372145 | 4.2393160129 | 1.3580083289 |

Algorithm | Problem 10 | - | - |

AKQPSO(Algorithm 3) | 21.0237707978 | - | - |

BBKH | 21.6420368754 | - | - |

DEKH | 21.6383024320 | - | - |

QKH | 21.1582649016 | - | - |

SKH | 21.6017549654 | - | - |

CCS | 21.9603961872 | - | - |

VNBA | 21.0799825759 | - | - |

BBO | 21.9988888954 | - | - |

GA | 21.4392031521 | - | - |

λ | Problem 1 | Problem 2 | Problem 3 | Problem 4 | Problem 5 |

0.1 | 49,649.1118923043 | 371.5625700159 | 1.4091799599 | 6.9697543426 | 1.1082189445 |

0.2 | 133,608.1106469210 | 370.0473039979 | 1.4091358439 | 3.9848771713 | 1.0983396055 |

0.3 | 10,893.3861385383 | 215.6832766302 | 1.3068652046 | 4.0398322695 | 1.0434696313 |

0.4 | 53,539.2035950009 | 346.1060025716 | 1.4091347288 | 4.9798362284 | 1.0588561989 |

0.5 | 86,346.3589566716 | 424.4795442904 | 1.4091497973 | 5.9747952855 | 1.0564120753 |

0.6 | 82,979.4098834243 | 422.2483723507 | 1.4094790359 | 5.0457481023 | 1.0861547616 |

0.7 | 99,948.7393233432 | 609.3844384539 | 7.7057897580 | 7.9647083618 | 1.0885926731 |

0.8 | 172,794.7224327620 | 462.1482898652 | 1.4091546130 | 13.9344627044 | 1.0689273036 |

0.9 | 29,013.1541012323 | 412.4776865757 | 3.9404220234 | 8.9597299262 | 1.0642761100 |

λ | Problem 6 | Problem 7 | Problem 8 | Problem 9 | Problem 10 |

0.1 | 1.4704093392 | 401.1855067018 | 3.0118025092 | 1.0958187632 | 21.1048204811 |

0.2 | 1.2289194470 | 336.4312675492 | 2.7987816724 | 1.0968167661 | 21.0708288643 |

0.3 | 1.1185826442 | 121.8932375270 | 2.3131324015 | 1.0715438957 | 21.0237707978 |

0.4 | 2.6943320338 | 119.5337169812 | 2.3299026264 | 1.1010758378 | 21.0389405345 |

0.5 | 2.9771884111 | 209.5578114314 | 2.5217968033 | 1.1257275250 | 21.0587530353 |

0.6 | 2.5706601162 | 253.1211550958 | 3.5227629532 | 1.1353127271 | 21.0976573470 |

0.7 | 1.4737157145 | 187.5895821815 | 3.0884949525 | 1.1344605652 | 21.1144506567 |

0.8 | 1.5487741806 | 144.3502633617 | 3.1612574448 | 1.1553497252 | 21.0835644844 |

0.9 | 4.0000000000 | 131.2707035329 | 4.0209018951 | 1.1615503990 | 21.0768903380 |

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**MDPI and ACS Style**

Wei, C.-L.; Wang, G.-G.
Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization. *Mathematics* **2020**, *8*, 1403.
https://doi.org/10.3390/math8091403

**AMA Style**

Wei C-L, Wang G-G.
Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization. *Mathematics*. 2020; 8(9):1403.
https://doi.org/10.3390/math8091403

**Chicago/Turabian Style**

Wei, Cheng-Long, and Gai-Ge Wang.
2020. "Hybrid Annealing Krill Herd and Quantum-Behaved Particle Swarm Optimization" *Mathematics* 8, no. 9: 1403.
https://doi.org/10.3390/math8091403