Golden Eagle Optimizer Algorithm Combining Levy Flight and Brownian Motion

doi:10.16182/j.issn1004731x.joss.23-0232

Abstract

Abstract:

Aiming at the slow attenuation and low convergence precision of golden eagle optimization algorithm, a new algorithm combining Levy fight and Brownian motion is proposed. In order to increase the diversity, Fuch chaotic map is introduced to initialize the golden eagle individuals. Levy flight mechanism and Brownian motion mechanism are introduced into the position update formula of golden eagle individual to improve the search accuracy and help to the jump out of local optimum. The reduction factor is introduced into the overall position update formula of the golden eagle individual to improve the convergence speed. Compared with 9 original algorithms and 5 improved algorithms under 14 benchmark test functions, the experimental results show that the improved golden eagle optimization algorithm has better performance, which are verified by three engineering applications.

Key words: Levy flight, Brownian motion, reduction factor, golden eagle optimization

CLC Number:

TP391.9

Jiaxin Deng, Damin Zhang, Qing He, Jianping Zhao. Golden Eagle Optimizer Algorithm Combining Levy Flight and Brownian Motion[J]. Journal of System Simulation, 2023, 35(6): 1290-1307.

Figures/Tables 20

Fig. 1

Table 1

Table 2

Table 3

Comparison of LRBGEO and classical basic algorithms test results

函数	算法	最优值	平均值	标准差	SR/%	函数	算法	最优值	平均值	标准差	SR/%
F1	BOA	1.44E-14	1.69E-14	9.40E-16	100	F8	BOA	0	3.39E+00	2.40E+01	94
	FA	1.10E-16	1.49E-16	1.83E-17	100		FA	9.95E-01	5.63E+00	2.52E+00	0
	GWO	2.00E-73	3.97E-70	6.76E-70	100		GWO	0	8.32E-02	4.66E-01	96
	PSO	3.59E-15	4.11E-11	1.68E-10	100		PSO	2.62E+01	3.99E+01	9.80E+00	0
	SCA	3.37E-08	8.80E-04	2.38E-03	26		SCA	2.68E-06	1.55E+01	2.54E+01	4
	SSA	5.86E-09	9.40E-09	1.90E-09	100		SSA	2.29E+01	4.74E+01	1.59E+01	0
	SOA	6.45E-22	1.72E-16	4.94E-16	100		SOA	0	5.23E-14	1.91E-13	100
	WOA	1.05E-189	9.88E-173	0	100		WOA	0	0	0	100
	GEO	7.70E-13	6.91E-12	7.57E-12	100		GEO	9.95E-01	1.24E+01	4.37E+00	0
	LRBGEO	0	0	0	100		LRBGEO	0	0	0	100
F2	BOA	1.86E-12	9.17E-12	2.87E-12	100	F9	BOA	6.99E-12	1.12E-11	1.59E-12	100
	FA	3.70E-10	6.40E-10	9.05E-11	100		FA	7.10E-10	1.08E-09	1.32E-10	100
	GWO	1.86E-42	4.83E-41	5.26E-41	100		GWO	7.99E-15	1.28E-14	2.76E-15	100
	PSO	5.70E-08	1.39E-05	2.44E-05	68		PSO	2.11E-07	1.88E-06	2.74E-06	98
	SCA	7.75E-09	6.19E-06	1.15E-05	80		SCA	1.35E-04	1.27E+01	9.67E+00	0
	SSA	1.75E-04	6.16E-01	8.07E-01	0		SSA	1.71E-05	1.53E+00	9.92E-01	0
	SOA	1.08E-16	2.29E-14	5.05E-14	100		SOA	1.34E-11	1.55E-09	4.12E-09	100
	WOA	7.31E-119	3.96E-109	1.93E-108	100		WOA	8.88E-16	4.01E-15	2.23E-15	100
	GEO	4.58E-02	5.92E-01	3.91E-01	0		GEO	3.25E-07	5.09E-01	6.77E-01	56
	LRBGEO	0	0	0	100		LRBGEO	8.88E-16	8.88E-16	0	100
F3	BOA	1.47E-14	1.70E-14	1.03E-15	100	F10	BOA	0	9.93E-16	1.16E-15	100
	FA	5.25E-18	9.03E-18	2.07E-18	100		FA	7.40E-03	4.64E-02	2.68E-02	0
	GWO	8.31E-24	2.52E-19	1.53E-18	100		GWO	0	1.37E-03	4.41E-03	90
	PSO	2.01E+00	7.16E+00	3.23E+00	0		PSO	5.22E-15	7.88E-03	8.82E-03	40
	SCA	1.33E+02	2.65E+03	2.95E+03	0		SCA	2.86E-06	1.60E-01	2.18E-01	2
	SSA	5.28E+00	5.77E+01	4.93E+01	0		SSA	1.57E-08	8.17E-03	8.52E-03	38
	SOA	1.97E-02	1.55E+01	5.95E+01	0		SOA	0	1.60E-03	1.13E-02	98
	WOA	1.14E+03	1.12E+04	6.32E+03	0		WOA	0	2.36E-03	9.51E-03	94
	GEO	2.36E+04	5.44E+04	1.41E+04	0		GEO	1.29E-07	6.40E-03	7.70E-03	50
	LRBGEO	2.92E-21	1.57E-20	8.24E-21	100		LRBGEO	0	0	0	100
F4	BOA	1.05E-11	1.15E-11	5.77E-13	100	F11	BOA	2.46E-04	2.98E-03	2.37E-03	0
	FA	7.11E-10	1.43E-09	2.10E-10	100		FA	9.28E-26	2.77E-23	2.82E-23	100
	GWO	7.43E-19	1.73E-17	2.17E-17	100		GWO	4.79E-09	4.03E-05	2.72E-04	96
	PSO	1.86E-01	3.91E-01	1.16E-01	0		PSO	0	3.31E-31	3.93E-31	100
	SCA	2.54E+00	1.49E+01	8.86E+00	0		SCA	3.53E-05	5.09E-03	5.01E-03	0
	SSA	3.50E-01	4.23E+00	3.24E+00	0		SSA	1.13E-16	1.89E-14	2.31E-14	100
	SOA	2.12E-04	1.09E-02	1.73E-02	0		SOA	1.45E-07	1.46E-05	2.69E-05	60
	WOA	8.11E-06	3.32E+01	3.13E+01	2		WOA	7.88E-13	3.55E-07	8.35E-07	100
	GEO	5.36E+01	6.41E+01	3.82E+00	0		GEO	1.60E-02	3.05E+00	2.55E+00	0
	LRBGEO	7.74E-12	1.40E-11	3.54E-12	100		LRBGEO	0	0	0	100

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Fig. 2

Fig. 3

Table 9

Comparison of holm follow-up verification results between LRBGEO and classical basic algorithms

i	算法	$p ¯$ 值	α/i	是否拒绝
1	WOA	0.118 649	0.050 000	否
2	GWO	0.080 512	0.025 000	否
3	FA	0.056 94	0.016 667	否
4	SOA	0.012 534	0.012 500	否
5	BOA	0.003 028	0.010 000	是
6	PSO	0.002 467	0.008 333	是
7	SSA	4.300E-05	0.007 143	是
8	GEO	2.000E-06	0.006 250	是
9	SCA	8.178E-07	0.005 556	是

Table 9

Table 10

Comparison of holm follow-up verification results between LRBGEO and Improved algorithms

i	算法	$p ¯$ 值	α/i	是否拒绝
1	CASSA	0.919 538	0.050 000	否
2	RDSSA	0.419 020	0.050 000	否
3	ISOA	0.043 352	0.025 000	否
4	GEO_DLS	0.009 998	0.016 667	是
5	EIW_PSO	0.000 336	0.012 500	是

Table 10

Fig. 4

Fig. 5

Table 11

Fig. 6

Table 12

Fig. 7

Table 13

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函数	名称	f^*	定义域	维度	特征
F1	Sphere	0	[-100, 100]	30	单峰
F2	Schwefel 2.22	0	[-10, 10]	30	单峰
F3	Schwefel 1.2	0	[-100, 100]	30	单峰
F4	Schwefel 2.21	0	[-100, 100]	30	单峰
F5	Beale	0	[-4.5, 4.5]	2	单峰
F6	Matyas	0	[-10,10]	2	单峰
F7	Three-hump camel	0	[-5, 5]	2	单峰
F8	Rastrigin	0	[-5.12, 5.12]	30	多峰
F9	Ackley1	0	[-32, 32]	30	多峰
F10	Griewank	0	[-600, 600]	30	多峰
F11	Himmelblau	0	[-5, 5]	2	多峰
F12	Periodic	0.9	[-50, 50]	30	多峰
F13	Salomon	0	[-100, 100]	30	多峰
F14	Yang 4	-1	[-10, 10]	30	多峰

函数	名称	最优值	平均值	标准差	SR/%	函数	名称	最优值	平均值	标准差	SR/%
F1	ISOA	2.10E-72	2.62E-67	6.86E-67	100	F8	ISOA	0	0	0	100
	RDSSA	2.44E-53	8.31E-52	1.62E-51	100		RDSSA	0	0	0	100
	EIW_PSO	9.54E-16	5.80E-10	3.46E-09	100		EIW_PSO	2.09E+01	4.83E+01	1.21E+01	0
	CASSA	9.54E-107	1.53E-106	2.79E-107	100		CASSA	0	0	0	100
	GEO_DLS	4.12E-32	2.63E-19	1.40E-18	100		GEO_DLS	0	2.13E-16	1.51E-15	100
	LRBGEO	0	0	0	100		LRBGEO	0	0	0	100
F2	ISOA	5.90E-45	1.82E-43	2.65E-43	100	F9	ISOA	8.88E-16	1.46E-15	1.32E-15	100
	RDSSA	3.74E-28	8.04E-27	8.40E-27	100		RDSSA	8.88E-16	8.88E-16	0	100
	EIW_PSO	1.81E-07	5.42E-04	1.16E-03	30		EIW_PSO	1.93E-08	7.55E-02	3.08E-01	72
	CASSA	5.04E-53	5.72E-53	2.44E-54	100		CASSA	8.88E-16	8.88E-16	0	100
	GEO_DLS	7.48E-14	9.46E-10	3.39E-09	100		GEO_DLS	7.99E-15	3.65E-11	8.14E-11	100
	LRBGEO	0	0	0	100		LRBGEO	8.88E-16	8.88E-16	0	100
F3	ISOA	3.17E-11	1.34E-06	6.64E-06	96	F10	ISOA	0	7.73E-03	1.17E-02	52
	RDSSA	9.52E-53	6.11E-50	3.05E-49	100		RDSSA	0	0	0	100
	EIW_PSO	3.92E-02	5.07E-01	3.83E-01	0		EIW_PSO	6.66E-16	9.75E-03	1.10E-02	40
	CASSA	1.68E-104	3.46E-104	7.55E-105	100		CASSA	0	0	0	100
	GEO_DLS	1.64E+02	3.40E+04	3.58E+04	0		GEO_DLS	0	8.88E-18	6.28E-17	100
	LRBGEO	2.06E-21	1.50E-20	9.11E-21	100		LRBGEO	0	0	0	100
F4	ISOA	1.32E-13	7.43E-11	3.70E-10	100	F11	ISOA	5.37E-10	5.10E-08	6.51E-08	100
	RDSSA	8.93E-29	2.04E-27	1.72E-27	100		RDSSA	3.91E-03	2.33E-01	2.54E-01	0
	EIW_PSO	4.03E-02	1.26E-01	6.23E-02	0		EIW_PSO	0	2.52E-31	3.72E-31	100
	CASSA	3.01E-54	4.30E-54	6.13E-55	100		CASSA	7.84E-15	3.26E-13	3.15E-13	100
	GEO_DLS	2.67E-13	7.77E-09	1.66E-08	100		GEO_DLS	6.61E-09	4.55E-06	1.54E-05	96
	LRBGEO	8.58E-12	1.48E-11	3.48E-12	100		LRBGEO	0	0	0	100
F5	ISOA	7.49E-11	1.40E-09	1.68E-09	100	F12	ISOA	1.07E+00	1.34E+00	1.94E-01	0
	RDSSA	1.31E-03	4.23E-02	1.16E-01	0		RDSSA	9.00E-01	9.00E-01	2.24E-16	100
	EIW_PSO	0	1.82E-02	9.00E-02	96		EIW_PSO	1.00E+00	1.00E+00	4.91E-09	0
	CASSA	9.81E-17	1.22E-14	1.02E-14	100		CASSA	9.00E-01	9.00E-01	2.24E-16	100
	GEO_DLS	2.59E-10	4.28E-07	1.67E-06	98		GEO_DLS	1.44E+00	2.09E+00	3.24E-01	0
	LRBGEO	0	1.37E-31	9.61E-31	100		LRBGEO	9.00E-01	9.00E-01	2.24E-16	100

函数	BOA	FA	GWO	PSO	SCA	SSA	SOA	WOA	GEO
F1	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F2	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F3	7.07E-18+	7.07E-18+	1.02E-02+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
F4	9.41E-05+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
F5	6.63E-20+	6.63E-20+	6.63E-20+	0.159 390-	6.63E-20+	6.63E-20+	6.63E-20+	6.63E-20+	6.63E-20+
F6	3.31E-20+	3.31E-20+	2.06E-17+	3.31E-20+	3.31E-20+	3.31E-20+	NAN=	NAN=	3.31E-20+
F7	3.31E-20+	3.31E-20+	NAN=	3.31E-20+	3.31E-20+	3.31E-20+	NAN=	3.31E-20+	3.31E-20+
F8	1.82E-03+	3.12E-20+	0.082 227-	3.31E-20+	3.31E-20+	3.31E-20+	1.49E-08+	NAN=	3.31E-20+
F9	3.31E-20+	3.31E-20+	1.03E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	6.52E-14+	3.31E-20+
F10	5.21E-13+	3.31E-20+	2.31E-02+	3.31E-20+	3.31E-20+	3.31E-20+	2.70E-09+	0.082 227-	3.31E-20+
F11	3.31E-20+	3.31E-20+	3.31E-20+	2.97E-07+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F12	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.98E-12+	3.31E-20+
F13	3.31E-20+	3.24E-20+	3.31E-20+	2.80E-20+	3.31E-20+	3.27E-20+	3.31E-20+	3.04E-20+	3.31E-20+
F14	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	2.32E-15+	3.31E-20+
+/-/=	14/0/0	14/0/0	12/1/1	13/1/0	14/0/0	14/0/0	12/0/2	11/1/2	14/0/0

函数	ISOA	RDSSA	GEO_DLS	EIW_PSO	CASSA
F1	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F2	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F3	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+	7.07E-18+
F4	5.89E-03+	7.07E-18+	1.34E-12+	7.07E-18+	7.07E-18+
F5	6.63E-20+	6.63E-20+	6.63E-20+	0.975 702-	6.63E-20+
F6	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F7	NAN=	3.31E-20+	3.31E-20+	3.31E-20+	3.31E-20+
F8	NAN=	NAN=	0.327 086-	3.31E-20+	NAN=
F9	3.43E-03+	NAN=	3.31E-20+	3.31E-20+	NAN=
F10	3.44E-08+	NAN=	0.327 086-	3.31E-20+	NAN=
F11	3.31E-20+	3.31E-20+	3.31E-20+	1.44E-05+	3.31E-20+
F12	3.31E-20+	NAN=	3.31E-20+	3.31E-20+	NAN=
F13	3.31E-20+	3.31E-20+	3.31E-20+	1.74E-20+	3.31E-20+
F14	3.31E-20+	NAN=	3.31E-20+	3.31E-20+	NAN=
+/-/=	12/0/2	9/0/5	12/2/0	13/1/0	9/0/5

算法	排名平均值	排名	算法	排名平均值	排名
LRBGEO	2.43	1	CASSA	2.50	2
RDSSA	3.00	3	ISOA	3.86	4
GEO_DLS	4.25	5	EIW_PSO	4.96	6