=Paper=
{{Paper
|id=Vol-3304/paper27
|storemode=property
|title=Improved Whale Optimization Algorithm Based on Chaos Strategy and Gaussian Mutation
|pdfUrl=https://ceur-ws.org/Vol-3304/paper27.pdf
|volume=Vol-3304
|authors=Haining Zhang,Tao Tao,Hongshen Liu
}}
==Improved Whale Optimization Algorithm Based on Chaos Strategy and Gaussian Mutation==
https://ceur-ws.org/Vol-3304/paper27.pdf
Improved Whale Optimization Algorithm Based on Chaos
Strategy and Gaussian Mutation
Haining Zhang, Tao Tao, Hongshen Liu*
School of Computer Science & Technology, Anhui University of Technology, Maβanshan 243032, China.
ABSTRACT
Aiming at the shortcomings of the basic whale optimization algorithm's slow convergence
speed and easy to fall into local optimum, an improved whale optimization algorithm IGWOA
is proposed, which uses the chaotic strategy to initialize the population, and the elite reverse
learning strategy to increase the diversity of the population. factor and nonlinear convergence
factor to speed up the global convergence; by adding an adaptive threshold, the global search
capability is improved. Finally, the Gaussian mutation strategy is used to improve the
convergence efficiency of the algorithm. Simulation experiments are carried out on 6 standard
test functions, and the results show the effectiveness of the improved whale optimization
algorithm.
Keywords
Whale Optimization Algorithm; Tent Chaos Map; Elite Reverse Learning; Adaptive Weights;
Gaussian Mutation
1. Introduction 1
The whale optimization algorithm (WOA) [1] is a new swarm intelligence optimization algorithm
proposed by Mirjalili of Griffith University in Australia in 2016. Its advantages are simple operation,
few parameters to adjust and jump out of the local optima. At the same time, there are many
shortcomings, easy to fall into the local optimum, slow convergence speed. Many scholars have made
various improvements to these shortcomings. For example, the literature [2] introduced an adaptive
strategy; Kaur et al. introduced the chaotic map into the whale algorithm [3] (CWOA) and introduced
10 different chaotic map formulas, among which Tent chaotic mapping has the best effect by
comparison, but there is still a lot of room for improvement in convergence accuracy and convergence
speed.Long Wen et al. proposed an improved whale algorithm (IWOA) based on nonlinear convergence
factors to solve large-scale problems [4]
In view of the lack of WOA, this paper proposes an initial population based on Tent chaotic mapping
combined with elite direction learning strategy, add adaptive weights, nonlinear convergence factors to
improve the global search performance, increase the adaptive threshold, improve the probability of early
search, and avoid falling into local extremum too early. At the same time, Gaussian mutation [5] was
carried out on the optimal value to improve the convergence efficiency of the algorithm, and the
effectiveness of the algorithm was demonstrated by six test functions.
2. Algorithm Improvement
2.1. Improved The Whale Optimization Algorithm
β’ Chaos Mapping and Elite Reverse Learning Strategies
ICBASE2022@3rd International Conference on Big Data & Artificial Intelligence & Software Engineering, October 21-
23, 2022, Guangzhou, China
* Corresponding author: e-mail: 695073265@qq.com (Hongshen Liu*)
Β© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
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The more evenly the initialized population is distributed in the search space, the more beneficial it
is to improve the optimization efficiency and solution accuracy of the algorithm. The chaotic sequence
generated by Tent map has good distribution and randomness, which can be used to significantly
improve the optimization performance of the algorithm. The Tent formula is:
2π , π < 0.5 (1)
π =
2(1 β π ) , π β₯ 0.5
π = ππ + (π’π β ππ)π¦ (2)
Among them: lb and ub are the lower and upper bounds of x, respectively.
β’ Elite Reverse Learning Strategies formula
π(π, π) = π +π βπ (3)
The basic steps are as follows:
1) Tent mapping initializes the population S, and selects the first N/2 individuals with better fitness
as the elite population E
2) Find the reverse population OE of the elite population E
3) Combine the population S and the reverse population OE to obtain a new population {SβͺOE},
calculate the objective function value of the new population, and select the N individuals with the best
fitness as the initial population.
β’ Nonlinear Convergence Factor and Adaptive Weights
In the original whale algorithm, the performance of global optimization and local exploration mainly
depends on the value of A. The change of A is affected by the change of the convergence factor a, and
the value of a decreases linearly from 2 to 0 as the iteration progresses. , the decreasing speed is the
same in the whole algorithm, which leads to the decline of the overall optimization ability of the
algorithm, and reduces the diversity and flexibility of the whale population. So to solve this problem, a
decreasing nonlinear convergence factor is introduced. The specific expression is as follows:
π = (π βπ ) β (π βπ ) β ((1 β (π β π‘)/π ) ) (4)
Among them: t is the current number of iterations, Tmax is the maximum number of iterations, afinal
and ainitial are the initial and final values of the control parameter a, respectively, and K is the nonlinear
adjustment coefficient.
By using a nonlinear convergence factor, the performance of the algorithm is improved to a certain
extent. However, when it comes to changing the strategy formula, the convergence factor alone cannot
effectively balance global optimization and local search. To solve this problem, the adaptive weight is
added at the time of change, and the formula is as follows:
π(π‘) = (π βπ ) + 0.5 β ((π β π‘)/π ) (5)
Among them: t is the current number of iterations, Tmax is the maximum number of iterations, Winital
and Wend are the initial and final values of the control parameter w, respectively, and Ο is a constant
coefficient. In summary, the position update formula defined in this paper is defined as:
π(π‘ + 1) = π (π‘) β π β (π‘) β π΄π· (6)
π(π‘ + 1) = π (π‘) β π β (π‘) + π· π πππ (2ππ) (7)
π(π‘ + 1) = π (π‘) β π β π΄π· (8)
β’ Adaptive Threshold
The probability threshold p of the original algorithm is set as 0.5, and this equal probability predation
mode will fall into the local optimum.Aiming at this problem, an adaptive threshold p' is proposed to
balance the global optimization and local search capabilities:
πβ = 1 β (π‘/π ) (9)
Among them, ΞΌ is a constant variable. At the beginning of the iteration, P>P', the algorithm selects
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the spiral method to update the leader's position with high probability. The latter period decreases to 0
at a slow rate as the probability threshold decreases. By iterative adaptive threshold, the convergence
accuracy of the algorithm is improved.
β’ Gaussian Mutation
In the basic whale algorithm, the group update strategy is to generate a new individual near the
current optimal individual Xbest(t), which will cause other individuals in the group to move like the
optimal individual, thus causing the group to lose diversity and fall into the local optimal value. The
solution is to introduce mutation operations. In this paper, Gaussian mutation is used, and a certain
probability p* is used to perform Gaussian mutation operation on the current optimal solution Xbest(t):
π (π‘ + 1) = π (π‘)(1 + πΊππ’π π πππ(π)) (10)
Among them, Xbest(t+1) represents the position of the individual after mutation, and Gaussion(Ο) is
a random variable that satisfies the Gaussian distribution. The global optimal position is updated as
follows:
π (π‘ + 1) , ππ‘βπππ€ππ π
π (π‘ + 1) =
π (π‘), π(π (π‘ + 1) > π π (π‘) πππ ππππ < πβ (11)
Among them, rand represents a random variable between [0,1], p* is the selection probability of
survival of the fittest, and f(x) is the fitness value of the individual. It can be seen from formula (11)
that by performing mutation operation on the current global optimal solution Xbest(t), the population can
evolve towards the optimal solution, and at the same time, the search efficiency of the algorithm can be
effectively improved.
2.2. IGWOA Algorithm Process
In summary, the IGWOA algorithm process provided in this article:
The pseudo-code:
Begin
Initialization parameters, initialize the whale population based on Tent chaos and elite reverse
learning strategy, and generate an initial population with good fitness.
While(t=1) do
According to formula (8) whales move towards random individuals
Else if (p >= p') do
Repel prey with bubble net according to formula (7)
End if
End for
Record the optimal whale position, Gaussian mutation, generate rand, probability p*, compare the
individual fitness values, and determine whether to receive the position of the new whale after the
mutation
t=t+1
End while
End
2.3. Simulation Experiments and Analysis
β’ Benchmark Function
In order to verify the performance of IGWOA, the whale optimization algorithm (CWOA), standard
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Whale algorithm (WOA) and Gray Wolf algorithm (GWO) based on chaotic search strategy are
simultaneously conducted 20 times of comparative experiments under the 6 benchmark test functions
in Table 1. We selected F(1)-F(6), including F(1), F(2) a total of 2 unimodal functions and F(3, F(4),
F(5), F(6) a total of 4 multimodal functions are tested, as shown in Table 1:.
Table 1. Benchmark Function
Function Search Area Theoretical
F(1)οΌSphere [-100,100] n
0
F(2)οΌSchwefel 2.22 [-10,10] n 0
F(3)οΌGeneralized Rastrgin [-5.12,5.12] n 0
F(4)οΌAckley [-32,32] n 0
F(5)οΌGeneralized Griewank [-600,600] n
0
F(6)οΌKowalik [-5,5] n 0.0003075
β’ Parameter Setting and Analysis
The main parameters of each algorithm are shown in Table 2:
Table 2. Algorithm Parameter Settings
Algorithms Parameter
IGWOA ainitial = 2;afinal = 0;winitial = 0.8;wend= 0.4; ΞΌ=2;Ο=2;k=2
CWOA ainitial = 2;afinal = 0;winitial = 0.9;wfinal = 0.2;
WOA amax=2;amin=0
GWO amax=2;amin=0
Results are shown in Table 3:
Table 3. Algorithm experiment results
Function Algorithm Optimal value Worst value Average Standard
value deviation
IGWOA 0 0 0 0
CWOA 0 0 0 0
F(1)οΌSphere
WOA 7.8E-190 1.4E-206 5.4E-191 0
GWO 9.83E-84 6.97E-88 9.38E-85 2.38E-84
IGWOA 0 0 0 0
F(2)οΌSchwefel CWOA 0 0 0 0
2.22 WOA 1.2E-112 1.2E-122 9.9E-114 3.1E-113
GWO 1.16E-48 7.41E-50 3.2E-49 2.86E-49
IGWOA 0 0 0 0
F(3) Generalized CWOA 0 0 0 0
Rastrgin WOA 0 0 0 0
GWO 0 0 0 0
IGWOA 8.88E-16 8.88E-16 8.88E-16 2.02E-31
CWOA 8.88E-16 8.88E-16 8.88E-16 2.02E-31
F(4) Ackley
WOA 7.99E-15 8.88E-16 4.8E-15 2.55E-15
GWO 1.15E-14 7.99E-15 9.77E-15 1.82E-15
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IGWOA 0 0 0 0
F(5) Generalized CWOA 0 0 0 0
Griewank WOA 0.066589 0 0.003329 0.01489
GWO 0 0 0 0
IGWOA 0.000393 0.000308 0.000329 1.98E-05
CWOA 0.000431 0.000308 0.000331 3.09E-05
F(6) Kowalik
WOA 0.001239 0.000308 0.000601 0.000351
GWO 0.020363 0.000307 0.002313 0.006173
The unimodal function is usually used to evaluate the development ability of the function. In F(1),
F(2), IGWOA found the optimal value of 0.The function curve in Figure 1 shows that IGWOA has the
fastest convergence rate.
Multimodal functions are often used to evaluate the search ability of a function. IGWOA found the
optimal value 0 on the F(3) and F(4) functions, and showed the optimal mean and standard value in the
test of the F(4) F(6) function. Fig. 1 It is proved that the initial population adjusted by the chaotic
strategy converges faster than the traditional algorithm at the beginning of all the convergence curves.
By comparing the numerical value and the curve as a whole, the IGWOA algorithm has better
optimization performance than the CWOA, WOA and GWO functions.
F1(Sphere) F2(Schwefel 2.22) F3(Generalized Rastrgin))
F4(Ackley) F5(Generalized Griewank) F6(Kowalik)
Figure 1: Test Function Average Convergence Curve
3. Conclusion
In order to improve the performance of WOA, this paper firstly uses chaos Tent mapping to initialize
the population to make the population more uniform, then solves the fitness value, and uses the elite
reverse learning strategy to obtain the initialization population with higher fitness value, which
effectively improves the diversification of the initialization population. In the search and predation stage
of the algorithm, nonlinear convergence factor and adaptive weight are added to improve the global
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search ability of the population, so as to avoid the algorithm falling into the local optimal value too
early. Moreover, by adding a custom threshold, the algorithm has a higher probability of searching prey
in the early stage, so as to avoid falling into local extremum too early. Finally, Gaussian mutation is
carried out on the optimal individual to accelerate the convergence performance of the algorithm and
improve the local search ability to balance the global search and local search. Simulation results show
that the improved algorithm in this paper has faster convergence speed and higher search accuracy.
4. References
[1] Seyedali Mirjalili,Andrew Lewis. The Whale Optimization Algorithm[J]. Advances in Engineering
Software,2016,95.
[2] Guo Zhenzhou, Wang Ping, Ma Yunfeng, Wang Qi, Gong Changqing. Whale optimization
algorithm based on adaptive weight and Cauchy variation [J]. Microelectronics and Computer,
2017, 34(09): 20-25 .DOI: 10.19304/j.cnki.issn1000-7180.2017.09.005.
[3] Gaganpreet Kaur,Sankalap Arora. Chaotic whale optimization algorithm[J]. Journal of
Computational Design and Engineering,2018,5(3).
[4] Zhong Minghui, Long Wen. A Whale Optimization Algorithm for Randomly Adjusting Control
Parameters [J]. Science Technology and Engineering, 2017, 17(12): 68-73.
[5] Long Wen, Cai Shaohong, Jiao Jianjun, Tang Mingzhu, Wu Tiebin. An improved whale
optimization algorithm for solving large-scale optimization problems [J]. System Engineering
Theory and Practice, 2017, 37(11): 2983-2994.
[6] Qu Liangdong, He Dengxu. Artificial Fish Swarm Algorithm Based on Adaptive Gaussian
Mutation [J]. Computer Engineering, 2009, 35(15): 182-184+189.
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