混合方法优化的自适应引力搜索算法

doi:10.3969/j.issn.1001-506X.2020.01.20

Abstract

Abstract:

In order to overcome the shortcomings of premature convergence, trapping in local optimum easily and lower search accuracy of gravitational search algorithm (GSA), an adaptive GSA improved by hybrid methods is proposed. Firstly, sobol sequence is used to initialize the population and enhance the global search ability. Secondly, hamming nearness degree is introduced to calculate the population maturity and judge whether the population is premature. Thirdly, logistic chaos is introduced to search the population chaotically and update the particle which has fallen into the local optimum. Finally, based on the precocious convergence judgment factor, the gravitational coefficient is improved, and the shrinkage factor is added to the particle position formula to accelerate the population departure from the local optimum. The simulation results of nine different types of benchmark functions show that the new algorithm can effectively improve the premature convergence problem and has better optimization performance.

Key words: gravitational search algorithm (GSA), low-discrepancy sequence, nearness degree, chaos

CLC Number:

TP273

Ao LOU, Minli YAO, Weimin JIA, Ding YUAN. Adaptive gravitational search algorithm improved by hybrid methods[J]. Systems Engineering and Electronics, 2020, 42(1): 148-156.

Figures/Tables 12

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Table 1

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References 20

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算法	参数	取值
GSA	G₀, N, T, α	G₀=100, N=50 T=1 000, α=20
I-GSA/PSO	c₁, c₂	c₁=c₂=-0.5e^-(4t/T)²+1
GGSA	c₁, c₂	c₁=-2(t/T)³+2 c₂=2(t/T)³
GPS	c₁, c₂ c₃, c₄	c₁=c₂=1 c₃=c₄=0.5
HMAGSA	γ s_max, s_min	γ=0.89 s_max=0.9, s_min=0.1

类型	函数名	函数表达式	维数	区间
	Sphere (F₁)	${F_1} = \sum\limits_{i = 1}^n {x_i^2} $	30	[-100, 100]
高维单峰	Schwefel (F₂)	${F_2} = \sum\limits_{i = 1}^n {\left\| {{x_i}} \right\|} + \prod\limits_{i = 1}^n {\left\| {{x_i}} \right\|} $	30	[-10, 10]
	Quartic (F₃)	${F_3} = \sum\limits_{i = 1}^n i x_i^4 + {\mathop{\rm random}\nolimits} [0, 1)$	30	[-1.28, 1.28]
	Rastrigrin (F₄)	${F_4} = \sum\limits_{i = 1}^n {\left[ {x_i^2 - 10\cos \left( {2{\rm{ \mathsf{ π} }}{x_i}} \right) + 10} \right]} $	30	[-5.12, 5.12]
高维多峰	Criewank (F₅)	${F_5} = \frac{1}{{4000}}\sum\limits_{i = 1}^n {x_i^2} - \prod\limits_{i = 1}^n {\cos } \left( {\frac{{{x_i}}}{{\sqrt i }}} \right) + 1$	30	[-600, 600]
	Ackley (F₆)	${F_6} = - 20\exp \left( { - 0.2\sqrt {\frac{1}{2}\sum\limits_{i = 1}^n {x_i^2} } - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^n {\cos } \left( {2{\rm{ \mathsf{ π} }}{x_i}} \right)} \right) + 20 + e} \right.$	30	[-32, 32]
	Kowalik (F₇)	${F_7} = \sum\limits_{i = 1}^{11} {{{\left[ {{a_i} - \frac{{{x_1}\left( {b_i^2 + {b_i}{x_2}} \right)}}{{b_i^2 + {b_i}{x_3} + {x_4}}}} \right]}^2}} $	4	[-5, 5]
低维多峰	Camel (F₈)	${F_8} = 4x_1^2 - 2.1x_1^4 + \frac{1}{3}x_1^6 + {x_1}{x_2} - 4x_2^2 + 4x_2^4$	2	[-5, 5]
	Hartman (F₉)	${F_9} = - \sum\limits_{i = 1}^4 {{c_i}} \exp \left( { - \sum\limits_{j = 1}^3 {{a_{ij}}} {{\left( {{x_j} - {p_{ij}}} \right)}^2}} \right)$	3	[1, 3]

函数	理论最优	项目	GSA	I-GSA/PSO	GGSA	GPS	HMAGSA
F₁	0	Best	9.01e-14	4.41e-16	9.59e-17	4.56e-17	1.01e-58
		Mean	1.46e-13	6.69e-16	2.04e-16	8.27e-17	8.30e-57
		Std	3.99e-14	1.94e-16	9.08e-17	2.85e-17	2.28e-56
F₂	0	Best	1.59e-06	1.06e-07	8.79e-08	2.79e-08	7.38e-29
		Mean	2.23e-06	1.26e-07	1.43e-07	4.52e-08	5.84e-27
		Std	9.35e-07	1.29e-08	6.68e-08	1.17e-08	1.79e-26
F₃	0	Best	6.88e+01	0.53e-02	0.12e-01	0.29e-02	1.38e-06
		Mean	1.06e+02	0.84e-02	0.19e-01	0.61e-02	4.03e-05
		Std	2.57e+01	0.29e-02	0.46e-02	0.21e-02	2.66e-05
F₄	0	Best	1.80e+01	3.38e+01	1.29e+01	2.29e+01	0.00e+00
		Mean	3.10e+01	5.17e+01	2.28e+01	3.22e+01	0.00e+00
		Std	8.02e+00	1.55e+01	5.96e+00	8.56e+00	0.00e+00
F₅	0	Best	2.33e-14	0.00e+00	0.00e+00	0.00e+00	0.00e+00
		Mean	1.72e-13	0.12e-01	4.81e-16	0.37e-02	0.00e+00
		Std	1.49e-13	0.11e-01	2.29e-16	0.74e-02	0.00e+00
F₆	0	Best	1.40e+01	1.99e+01	1.39e+01	4.03e-09	8.88e-16
		Mean	1.94e+01	2.00e+01	1.85e+01	6.48e-09	3.73e-15
		Std	1.87+00	0.23e-01	2.13e+00	1.41e-09	1.49e-15
F₇	3.00e-04	Best	0.19e-02	6.18e-04	6.02e-04	7.26e-04	3.74e-04
		Mean	0.41e-02	0.10e-02	0.12e-02	9.72e-04	5.72e-04
		Std	0.25e-02	2.60e-04	4.02e-04	2.06e-04	1.38e-04
F₈	-1.03e+00	Best	-1.03e+00	-1.03e+00	-1.03e+00	-1.03e+00	-1.03e+00
		Mean	-1.03e+00	-1.03e+00	-0.67e+00	-1.03e+00	-1.03e+00
		Std	0.00e+00	0.00e+00	0.47e+00	0.00e+00	0.00e+00
F₉	-3.86e+00	Best	-3.84e+00	-3.86e+00	-3.86e+00	-3.86e+00	-3.86e+00
		Mean	-3.50e+00	-3.86e+00	-3.54e+00	-3.86e+00	-3.86e+00
		Std	0.20e+00	0.00e+00	0.34e+00	0.00e+00	0.00e+00

函数	目标值
F₁	1.00e-15
F₂	1.00e-07
F₃	1.00e-02
F₄	3.00e+01
F₅	2.00e-16
F₆	6.00e-09
F₇	8.00e-04
F₈	-1.03e+00
F₉	-3.86e+00

函数	GSA		I-GSA/PSO		GGSA		GPS		HMAGSA
函数	SR/%	IT	SR/%	IT	SR/%	IT	SR/%	IT	SR/%	IT
F₁	0	-	90	994	100	985	100	942	100	424
F₂	0	-	0	-	43	912	100	955	100	432
F₃	0	-	87	411	0	-	100	305	100	135
F₄	47	326	0	-	87	323	53	244	100	126
F₅	0	-	60	989	0	-	57	916	100	406
F₆	0	-	0	-	0	-	47	996	100	449
F₇	0	-	43	189	23	254	40	171	100	123
F₈	100	125	100	160	40	111	100	136	100	98
F₉	0	-	100	330	30	267	100	279	100	144

Adaptive gravitational search algorithm improved by hybrid methods

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