Improved grey wolf optimizer algorithm based on dual convergence factor strategy

doi:10.11772/j.issn.1001-9081.2022091389

Journal of Computer Applications ›› 2023, Vol. 43 ›› Issue (9): 2679-2685.DOI: 10.11772/j.issn.1001-9081.2022091389

• 2022 10th CCF Conference on Big Data • Previous Articles Next Articles

Improved grey wolf optimizer algorithm based on dual convergence factor strategy

Yun OU¹(), Kaiqing ZHOU¹, Pengfei YIN², Xuewei LIU³

^1.School of Communication and Electronic Engineering，Jishou University，Jishou Hunan 416000，China
^2.Academic Affairs Office，Jishou University，Jishou Hunan 416000，China
^3.College of Computer Science and Engineering，Jishou University，Jishou Hunan 416000，China

Received:2022-09-06 Revised:2022-09-30 Accepted:2022-10-08 Online:2022-10-17 Published:2023-09-10
Contact: Yun OU
About author:ZHOU Kaiqing， born in 1984， Ph. D.， associate professor. His research interests include clinical assistant decision-making system， fuzzy Petri net， swarm intelligence algorithm.
YIN Pengfei， born in 1978， Ph. D.， lecturer. His research interests include expert recommendation system， swarm intelligence algorithm.
LIU Xuewei， born in 2001. Her research interests include swarm intelligence algorithm.
Supported by:
National Natural Science Foundation of China(62066016);Research Foundation of Department of Education of Hunan Province(21C0383);Natural Science Foundation of Hunan Province(2020JJ5458)

双收敛因子策略下的改进灰狼优化算法

欧云¹(), 周恺卿¹, 尹鹏飞², 刘雪薇³

^1.吉首大学通信与电子工程学院, 湖南吉首 416000
^2.吉首大学教务处, 湖南吉首 416000
^3.吉首大学计算机科学与工程学院, 湖南吉首 416000

通讯作者: 欧云
作者简介:周恺卿（1984—），男，湖南长沙人，副教授，博士，CCF会员，主要研究方向：临床辅助决策系统、模糊Petri网、群智能算法
尹鹏飞（1978—），男，湖南益阳人，讲师，博士，主要研究方向：专家推荐系统、群智能算法
刘雪薇（2001—），女，黑龙江齐齐哈尔人，主要研究方向：群智能算法。
基金资助:
国家自然科学基金资助项目(62066016);湖南省教育厅科学研究项目(21C0383);湖南省自然科学基金资助项目(2020JJ5458)

Abstract

Abstract:

Aiming at the drawbacks of standard Grey Wolf Optimizer （GWO） algorithm， such as slow convergence and being easy to fall into local optimum， an improved Grey Wolf Optimizer with Two Headed Wolves guide （GWO-THW） algorithm was proposed by utilizing a dual nonlinear convergence factor strategy. Firstly， the chaotic Cubic mapping was used to initialize the population for improving the uniformity and diversity of the population distribution. And the wolves were divided into hunter wolves and scout wolves through the average fitness values. The different convergence factors were used to two types of wolves to seek after and round up their prey under the leadership of their respective leader wolf. Secondly， an adaptive weight factor of position updating was designed to improve the search speed and accuracy. Meanwhile， a Levy flight strategy was employed to randomly update the positions of wolves for jumping out of local optimum， when no prey was found in a certain period of time. Ten benchmark functions were selected to test the performance and effectiveness of GWO-THW. Experimental results show that compared with standard GWO and related variants， GWO-THW achieves higher optimization accuracy and faster convergence on eight benchmark functions， especially on the multi-peak functions， the algorithm can converge to the ideal optimal value within 200 iterations， indicating that GWO-THW has better optimization performance.

Key words: Grey Wolf Optimizer (GWO) algorithm, dual convergence factor strategy, Levy flight, adaptive weight factor, two headed wolves guide

摘要：

针对标准灰狼优化算法（GWO）的收敛速度慢、易陷入局部最优等缺点，提出一种在非线性双收敛因子策略下基于双头狼引领的改进灰狼优化（GWO-THW）算法。首先，利用混沌Cubic映射初始化种群，提升种群分布的均匀性和多样性，并通过平均适应度值将狼群分为捕猎狼和侦察狼，两类狼群采用不同的收敛因子，在各自的头狼带领下寻找和围捕猎物；其次，为提升搜索速度和精度，设计了一种位置更新的自适应权重因子；同时，为跳出局部最优，当一定时间内未发现猎物时，狼群采用莱维（Levy）飞行策略随机更新位置。在10个常用的基准测试函数上验证GWO-THW的有效性。实验结果表明，与标准GWO及相关变体相比，GWO-THW在8个基准测试函数上都取得了较高的寻优精度和收敛速度，尤其在多峰函数上，200次迭代内就能收敛到理想最优值，从而验证了GWO-THW具有更好的寻优性能。

关键词: 灰狼优化算法, 双收敛因子策略, 莱维飞行, 自适应权重因子, 双头狼引领

CLC Number:

TP183

Yun OU, Kaiqing ZHOU, Pengfei YIN, Xuewei LIU. Improved grey wolf optimizer algorithm based on dual convergence factor strategy[J]. Journal of Computer Applications, 2023, 43(9): 2679-2685.

欧云, 周恺卿, 尹鹏飞, 刘雪薇. 双收敛因子策略下的改进灰狼优化算法[J]. 《计算机应用》唯一官方网站, 2023, 43(9): 2679-2685.

Figures/Tables 6

Fig.1 Curves of dual convergence factors and original convergence factor

Tab. 1 Ten benchmark functions

函数名	函数表达式	维度	范围
Sphere	$F 1 = ∑ i = 1 D x i 2$	30	［-100，100］
Schwefel2.22	$F 2 = ∑ i = 1 D x i + ∏ i = 1 D x i$	30	［-10，10］
Schwefel1.2	$F 3 = ∑ i = 1 D ∑ j = 1 i x j 2$	30	［-100，100］
Schwefel2.21	$F 4 (x) = m a x i x i, 1 ≤ i ≤ n$	30	［-100，100］
Rosenbrock	$F 5 = ∑ i = 1 D - 1 100 x i + 1 - x i 2 2 + (x i - 1) 2$	30	［-30，30］
Step	$F 6 = ∑ i = 1 D x i + 0.5 2$	30	［-100，100］
Quartic	$F 7 = ∑ i = 1 D i x i 4 + r a n d o m [0,1)$	30	［-128，128］
Rastrigin	$F 8 = ∑ i = 1 D x i 2 - 10 c o s 2 π x i + 10$	30	［-5.12，5.12］
Ackley	$F 9 = - 20 e x p - 0.2 1 D ∑ i = 1 D x i 2 - e x p 1 D ∑ i = 1 D c o s 2 π x i + 20 + e$	30	［-32，32］
Griewank	$F 10 = ∑ i = 1 D x i 2 4 000 - ∏ i = 1 D c o s x i i + 1$	30	［-600，600］

Tab. 1 Ten benchmark functions

函数名	函数表达式	维度	范围
Sphere	$F 1 = ∑ i = 1 D x i 2$	30	［-100，100］
Schwefel2.22	$F 2 = ∑ i = 1 D x i + ∏ i = 1 D x i$	30	［-10，10］
Schwefel1.2	$F 3 = ∑ i = 1 D ∑ j = 1 i x j 2$	30	［-100，100］
Schwefel2.21	$F 4 (x) = m a x i x i, 1 ≤ i ≤ n$	30	［-100，100］
Rosenbrock	$F 5 = ∑ i = 1 D - 1 100 x i + 1 - x i 2 2 + (x i - 1) 2$	30	［-30，30］
Step	$F 6 = ∑ i = 1 D x i + 0.5 2$	30	［-100，100］
Quartic	$F 7 = ∑ i = 1 D i x i 4 + r a n d o m [0,1)$	30	［-128，128］
Rastrigin	$F 8 = ∑ i = 1 D x i 2 - 10 c o s 2 π x i + 10$	30	［-5.12，5.12］
Ackley	$F 9 = - 20 e x p - 0.2 1 D ∑ i = 1 D x i 2 - e x p 1 D ∑ i = 1 D c o s 2 π x i + 20 + e$	30	［-32，32］
Griewank	$F 10 = ∑ i = 1 D x i 2 4 000 - ∏ i = 1 D c o s x i i + 1$	30	［-600，600］

Tab. 2 Ablation experiment results

测试函数	指标	GWO	GWO1	GWO2	GWO3	GWO4	GWO-THW
F₁	均值	2.12E-27	5.91E-292	1.23E-76	1.83E-37	1.90E-292	3.01E-293
F₁	标准差	4.06E-27	0	6.73E-76	4.61E-37	0	0
F₂	均值	7.44E-17	1.59E-178	3.00E-49	1.28E-22	2.61E-178	1.42E-177
F₂	标准差	7.71E-17	0	8.09E-49	1.09E-22	0	0
F₃	均值	4.27E-06	1.59E-243	5.52E-48	4.53E-02	6.86E-244	1.03E-246
F₃	标准差	6.63E-06	0	2.03E-47	1.30E-01	0	0
F₄	均值	7.29E-07	3.68E-148	7.17E-30	2.07E-07	4.63E-149	2.12E-149
F₄	标准差	5.78E-07	1.99E-147	1.71E-29	4.10E-07	1.67E-148	6.78E-149
F₅	均值	2.69E+01	2.65E+01	2.63E+01	2.66E+01	2.72E+01	2.67E+01
F₅	标准差	6.49E-01	4.15E-01	3.43E-01	4.00E-01	7.42E-01	4.23E-01
F₆	均值	8.21E-01	8.92E-03	9.86E-05	6.95E-06	6.73E-01	1.20E-05
F₆	标准差	4.16E-01	3.41E-02	2.82E-05	2.97E-06	3.94E-01	5.42E-06
F₇	均值	2.60E-03	1.55E-04	8.64E-04	3.09E-03	1.21E-04	1.48E-04
F₇	标准差	1.02E-03	1.20E-04	6.44E-04	2.03E-03	1.24E-04	1.29E-04
F₈	均值	3.22E+00	0	0	8.51E+00	0	0
F₈	标准差	4.10E+00	0	0	1.55E+01	0	0
F₉	均值	1.00E-13	0	4.74E-16	1.60E-14	0	0
F₉	标准差	1.87E-14	0	1.23E-15	3.20E-15	0	0
F₁₀	均值	4.63E-03	0	0	1.96E-03	0	0
F₁₀	标准差	9.20E-03	0	0	6.03E-03	0	0

Tab. 3 Comparison results of the proposed algorithm and other swarm intelligence algorithms

测试函数	算法	最差值	最佳值	平均值	标准差	测试函数	算法	最差值	最佳值	平均值	标准差
F₁	DE	6.75E-01	3.73E-02	2.57E-01	1.63E-01	F₆	DE	6.96E-01	7.13E-02	2.34E-01	1.42E-01
	FA	5.96E-01	3.77E-01	4.88E-01	5.44E-02		FA	6.27E-01	3.92E-01	4.95E-01	5.18E-02
	PSO	2.09E-06	3.92E-16	7.81E-08	3.75E-07		PSO	5.36E+00	4.23E+00	4.78E+00	3.09E-01
	GWO-THW	8.94E-292	0	3.01E-293	0		GWO-THW	2.57E-05	4.67E-06	1.20E-05	5.42E-06
F₂	DE	8.31E-01	1.75E-01	4.17E-01	1.81E-01	F₇	DE	3.96E+02	1.45E+01	7.34E+01	7.86E+01
	FA	3.84E+00	2.92E+00	3.24E+00	2.31E-01		FA	8.71E-01	3.79E-01	6.00E-01	1.21E-01
	PSO	3.79E-06	6.93E-11	6.27E-07	1.07E-06		PSO	1.54E-02	1.97E-04	3.82E-03	4.01E-03
	GWO-THW	3.38E-176	8.96E-186	1.42E-177	0		GWO-THW	5.99E-04	4.32E-06	1.48E-04	1.29E-04
F₃	DE	7.61E+03	2.44E+04	1.49E+04	4.40E+03	F₈	DE	2.46E+02	1.79E+02	2.10E+02	1.54E+01
	FA	1.96E+00	9.67E-01	1.37E+00	2.25E-01		FA	2.86E+02	1.69E+02	2.27E+02	2.93E+01
	PSO	7.88E+01	3.39E-04	5.80E+00	1.54E+01		PSO	3.70E-05	0	1.55E-06	6.70E-06
	GWO-THW	1.60E-262	2.95E-245	1.03E-246	0		GWO-THW	0	0	0	0
F₄	DE	3.35E+01	1.32E+01	2.15E+01	5.91E+00	F₉	DE	2.00E+01	1.14E-01	4.52E+00	7.79E+00
	FA	3.49E-01	2.54E-01	3.10E-01	2.36E-02		FA	2.00E+01	1.91E+01	1.97E+01	2.51E-01
	PSO	3.30E-02	6.45E-07	4.70E-03	7.87E-03		PSO	8.92E-05	5.38E-09	7.97E-06	1.77E-05
	GWO-THW	2.90E-148	8.81E-158	2.12E-149	6.78E-149		GWO-THW	0	0	0	0
F₅	DE	1.74E+03	4.66E+01	2.69E+02	3.85E+02	F₁₀	DE	8.70E-01	9.10E-02	5.34E-01	2.39E-01
	FA	6.68E+02	6.37E+01	1.48E+02	1.63E+02		FA	6.12E-02	2.10E-02	3.43E-02	1.00E-02
	PSO	2.89E+01	2.72E+01	2.81E+01	4.54E-01		PSO	8.89E-06	9.66E-15	3.82E-07	1.59E-06
	GWO-THW	2.72E+01	2.60E+01	2.67E+01	4.23E-01		GWO-THW	0	0	0	0

Tab. 4 Experimental results of GWO and its variants

测试函数	指标	GWO	EGWO	LGWO	NGWO	TGWO	OGWO	DGWO2	GWO-THW
F₁	均值	2.12E-27	1.17E-36	3.17E-30	1.16E-47	3.13E-76	5.09E-39	8.56E-63	3.01E-293
F₁	标准差	4.06E-27	1.05E-36	4.07E-20	2.72E-47	3.62E-76	1.43E-38	2.29E-62	0
F₂	均值	7.44E-17	2.84E-23	5.39E-19	2.92E-28	2.67E-41	1.40E-23	1.20E-35	1.42E-177
F₂	标准差	7.71E-17	1.47E-23	1.07E-02	4.16E-28	3.16E-41	2.43E-23	1.49E-35	0
F₃	均值	4.27E-06	6.19E-05	8.12E-08	9.98E-12	2.08E-39	4.83E-37	1.56E-60	1.03E-246
F₃	标准差	6.63E-06	1.21E-04	2.05E+00	1.99E-11	3.30E-39	1.88E-36	3.66E-60	0
F₄	均值	7.29E-07	2.43E-08	1.17E-08	7.15E-13	2.56E-29	2.34E-12	2.23E-14	2.12E-149
F₄	标准差	5.78E-07	4.08E-08	1.32E+00	7.60E-13	4.34E-29	5.21E-12	4.54E-14	6.78E-149
F₅	均值	2.69E+01	4.86E+01	8.35E+00	2.61E+01	—	2.67E+01	2.70E+01	2.67E+01
F₅	标准差	6.49E-01	9.44E+00	5.34E+00	3.96E-01	—	4.99E-01	5.91E-01	4.23E-01
F₆	均值	8.21E-01	0	2.69E-04	5.62E-01	—	5.59E-01	7.47E-01	1.20E-05
F₆	标准差	4.16E-01	0	2.30E-05	2.32E-01	—	3.34E-01	3.34E-01	5.42E-06
F₇	均值	2.60E-03	4.27E-03	3.02E-03	1.05E-03	—	1.69E-04	2.26E-03	1.48E-04
F₇	标准差	1.02E-03	1.50E-03	1.10E-03	2.32E-04	—	1.32E-04	1.14E-03	1.29E-04
F₈	均值	3.22E+00	1.35E+02	9.46E-02	0	0	1.89E-15	6.49E-01	0
F₈	标准差	4.10E+00	1.27E+01	2.16E+01	0	0	1.04E-14	1.34E+00	0
F₉	均值	1.00E-13	1.72E+00	2.12E-15	1.05E-14	0	1.03E-14	2.42E-14	0
F₉	标准差	1.87E-14	0	4.30E-02	2.39E-15	0	6.07E-15	5.39E-15	0
F₁₀	均值	4.63E-03	1.31E-08	2.42E-05	0	4.44E-15	1.31E-03	6.79E-03	0
F₁₀	标准差	9.20E-03	3.12E-08	8.39E-05	0	0	7.17E-03	1.27E-02	0

Fig. 2 Convergence curves of different algorithms on benchmark functions

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Improved grey wolf optimizer algorithm based on dual convergence factor strategy

双收敛因子策略下的改进灰狼优化算法

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