JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (3): 51-60.doi: 10.6040/j.issn.1671-9352.7.2023.4633

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Grey wolf optimization algorithm based on multi-strategy combination and its application

Hongwu QIN1,2(),Lizheng WANG1,*(),Yu FU1,Muxuan SUI1,Binggao HE1,2   

  1. 1. College of Electronic Information Engineering, Changchun University, Changchun 130000, Jilin, China
    2. Jilin Provincial Key Laboratory of Human Health Status Identification and Function Enhancement (Changchun University), Changchun 130022, Jilin, China
  • Received:2023-04-29 Online:2024-03-20 Published:2024-03-06
  • Contact: Lizheng WANG E-mail:qinhongwu@ccu.edu.cn;wangliz0117@163.com

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Abstract:

The standard grey wolf optimizer (GWO) algorithm has issues such as difficulty balancing local exploration and global development. A multi-strategy grey wolf optimization algorithm (MSGWO), based on the fusion of various strategies, is presented to address such problems. First, the grey wolf algorithm introduces the Tent map and a nonlinear convergence factor. Then, to coordinate attempts in the GWO optimization process, the paper applies three learning strategies: extensive learning, elite learning, and coordinated learning. Finally, the paper uses roulette wheel for strategy selection to obtain more diverse wolf positions and globally representative individuals and utilizes benchmark function testing to compare algorithm variations. The outcomes demonstrate that the MSGWO algorithm has a faster convergence speed and a good balance between local development and global search. Based on this, the echo state networks (ESN) hyperparameter for regression prediction is optimized using the MSGWO method. The experiment demonstrates that the MSGWO algorithm performs optimally with an average absolute percentage error of 0.38 percent and a fitting degree of 0.98.

Key words: grey wolf optimizer, multiple strategies, roulette, convergence factor, echo state network

CLC Number: 

  • TP301.6

Fig.1

MSGWO algorithm flowchart"

Table 1

Unimodal standard reference function"

表达式 维度 搜索区间 最优值
$f_1(x)=\sum\limits_{i=1}^n x_i^2$ 30 [-100, 100] 0
$f_2(x)=\sum\limits_{i=1}^n\left|x_i\right|+\prod\limits_{i=1}^n\left|x_i\right|$ 30 [-10, 10] 0
$f_3(x)=\sum\limits_{i=1}^n\left(\sum\limits_{j-1}^i x_j\right)^2$ 30 [-100, 100] 0
$f_4(x)=max_i\left\{\left|x_i\right|, 1 \leqslant i \leqslant n\right\}$ 30 [-100, 100] 0
$f_5(x)=\sum\limits_{i=1}^{n-1}\left[100\left(x_{i+1}-x_i^2\right)^2+\left(x_i-1\right)^2\right]$ 30 [-30, 30] 0
$f_6(x)=\sum\limits_{i=1}^n\left(\left[x_i+0.5\right]\right)^2$ 30 [-100, 100] 0
$f_7(x)=\sum\limits_{i=1}^n i x_i^4+\operatorname{random}[0, 1)$ 30 [-1.28, 1.28] 0

Table 2

Multimodal standard reference function"

表达式 维度 搜索区间 最优值
$f_8(x)=\sum\limits_{i=1}^n-x_i \sin \left(\sqrt{\left|x_i\right|}\right)$ 30 [-500, 500] -2 094.9
$f_9(x)=\sum\limits_{i=1}^n\left[x_i^2-10 \cos \left(2 \pi x_i\right)+10\right]$ 30 [-5.12, 5.12] 0
$f_{10}(x)=-20 \exp \left(-0.2 \sqrt{1 / n \sum\limits_{i=1}^n x_i^2}-\exp \left(1 / n \sum\limits_{i=1}^n \cos \left(2 \pi x_i\right)\right)\right)+20+\mathrm{e}$ 30 [-32, 32] 0
$f_{11}(x)=1 / 4000 \sum\limits_{i=1}^n x_i^2-\prod\limits_{i=1}^n \cos \left(x_i / \sqrt{i}\right)+1$ 30 [-600, 600] 0
$\begin{aligned}f_{12}(x)= & \pi / n\left\{10 \sin \left(\pi y_1\right)+\sum\limits_{i=1}^{n-1}\left(y_i-1\right)^2\left[1+\sin \left(\pi y_{i+1}\right)\right]+\left(y_n-1\right)^2\right\} \\& +\sum\limits_{i=1}^n u\left(x_i, 10, 100, 4\right), y_i=1+\frac{x_i+1}{4}\end{aligned}$ 30 [-50, 50] 0

Table 3

Comparison of experimental data for unimodal functions"

GWO MIGWO MAGWO PSO MSGWO
f1 平均值 7.50E-28 2.78E-36 2.22E+03 3.85E-04 0.00E+00
标准差 1.46E-27 4.03E-36 9.25E+02 1.40E-03 0.00E+00
f2 平均值 1.18E-16 8.47E-22 1.71E+01 3.93E-02 5.25E-219
标准差 1.23E-16 1.15E-21 4.42E+00 4.76E-02 0.00E+00
f3 平均值 2.63E-05 1.56E-06 4.42E+04 8.44E+01 0.00E+00
标准差 9.73E-05 4.85E-06 9.28E+03 2.34E+01 0.00E+00
f4 平均值 5.02E-07 2.32E-09 8.57E+01 1.10E+00 7.80E-291
标准差 5.07E-07 3.34E-09 4.88E+00 2.05E-01 0.00E+00
f5 平均值 2.71E+01 2.68E+01 2.47E+06 9.96E+01 2.79E+01
标准差 6.72E-01 6.40E-01 1.58E+06 6.51E+01 7.37E-01
f6 平均值 8.51E-01 6.86E-01 2.73E+03 1.48E-04 1.81E+00
标准差 4.45E-01 3.55E-01 8.68E+02 1.38E-04 7.56E-01
f7 平均值 2.00E-03 1.50E-03 1.33E+00 1.84E-01 5.30E-05
标准差 1.10E-03 6.94E-04 6.06E-01 6.87E-02 6.84E-05

Table 4

Comparison of experimental data on multimodal functions"

GWO MIGWO MAGWO PSO MSGWO
f8 平均值 -6.03E+03 -5.55E+03 -4.49E+03 -4.89E+03 -5.82E+03
标准差 9.65E+02 1.23E+03 2.87E+02 1.28E+03 1.47E+03
f9 平均值 2.74E+00 1.82E-01 2.44E+02 5.94E+01 0.00E+00
标准差 3.49E+00 9.98E-01 3.42E+01 1.56E+01 0.00E+00
f10 平均值 9.62E-14 2.29E-14 2.00E+01 2.76E-01 4.20E-15
标准差 1.37E-14 4.70E-15 5.70E-03 5.55E-01 9.01E-16
f11 平均值 7.65E-04 8.53E-04 2.73E+01 7.60E-03 0.00E+00
标准差 2.90E-03 3.30E-03 1.42E+01 8.70E-03 0.00E+00
f12 平均值 5.27E-02 3.59E-02 2.75E+06 1.04E-02 1.46E-01
标准差 2.97E-02 1.52E-02 2.58E+06 4.17E-02 7.06E-02

Fig.2

Convergence diagram of function f1—f12 curve"

Table 5

Comparison of experimental results and data"

MAE MAPE RMSEP R2
LSTM 3.700 4 0.008 1 4.708 6 0.945 05
ELM 3.262 7 0.007 2 4.181 9 0.938 56
ESN 3.532 7 0.007 8 4.452 1 0.930 25
GWO-ESN 1.967 1 0.004 3 2.506 8 0.977 88
MAGWO-ESN 1.986 2 0.004 4 2.524 6 0.977 57
MSGWO-ESN 1.725 1 0.003 8 2.198 8 0.982 99

Fig.3

Comparison of prediction results"

Fig.4

Prediction result error chart"

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