Particle swarm optimization algorithm incorporating premature detection mechanism and opposite random walk strategy

doi:10.11772/j.issn.1001-9081.2023121868

Abstract

Abstract:

Considering the issues of premature and slow convergence of the existing Particle Swarm Optimization （PSO） algorithms， a PSO algorithm incorporating Premature Detection mechanism and Opposite Random Walk strategy （PDORW-PSO） was proposed. Firstly， a modified Sigmoid function was designed by introducing a translation parameter to ensure that the output value of the function remained relatively small with small independent variable. Secondly， the times of the global extremum remained unchanged continuously was utilized as the independent variable for the modified Sigmoid function to calculate the probability of population premature. Finally， the particle positions were updated based on two randomly selected candidate solutions and the opposite solution of historical optimal particle solution， so as to enhance the population's ability to escape from local optima. Experimental results show that compared with the classical PSO algorithm and five improved PSO algorithms on eight classical test functions， PDORW-PSO achieves the best convergence accuracy and convergence speed among six comparison algorithms on five test functions. It can be seen that the convergence accuracy and convergence speed of PDORW-PSO are greatly improved compared with the comparison algorithms.

Key words: Particle Swarm Optimization (PSO) algorithm, modified Sigmoid function, premature detection, Opposition-Based Learning (OBL), random walk

摘要：

针对现存粒子群优化（PSO）算法易早熟和收敛速度慢的问题，提出一种融合早熟检测机制和对立随机游走策略的粒子群优化算法（PDORW-PSO）。首先，通过引入平移参数的方法改进Sigmoid函数，以确保在自变量较小时，函数输出值也较小；其次，将全局极值连续未变的次数作为改进后Sigmoid函数的自变量，以计算种群早熟的概率；最后，基于2个随机候选解和粒子历史最优解的反向解更新粒子位置，从而增强种群逃离局部最优的能力。所提算法与经典PSO算法以及5种改进后的PSO算法在8种经典测试函数上的对比实验的结果表明，所提算法的收敛精度和收敛速度和6种对比算法相比，在5种测试函数上排名第一。可见，PDORW-PSO的收敛精度和收敛速度较对比算法有较大提升。

关键词: 粒子群优化算法, 改进Sigmoid函数, 早熟检测, 对立学习, 随机游走

CLC Number:

TP301

Jianhua CHEN, Zhangqian WU, Wei SONG. Particle swarm optimization algorithm incorporating premature detection mechanism and opposite random walk strategy[J]. Journal of Computer Applications, 0, (): 123-128.

陈健华, 吴张倩, 宋威. 融合早熟检测机制和对立随机游走策略的粒子群优化算法[J]. 《计算机应用》唯一官方网站, 0, (): 123-128.

Figures/Tables 9

参数	值	参数	值
N	30	w	0.9
D	30	$c 1, c 2$	2
T	10 000	$c l$ ， $c u$	0.5，2.5

参数	值	参数	值
N	30	w	0.9
D	30	$c 1, c 2$	2
T	10 000	$c l$ ， $c u$	0.5，2.5

函数	函数	维数	函数表达式	变量范围
Sphere	$f 1 (x)$	30	$∑ i = 1 D x i 2$	［-100，100］
Rosenbrock	$f 2 (x)$	30	$∑ i = 1 D - 1 100 x i + 1 - x i 2) 2 + (x i - 1) 2]$	［-30，30］
Rastrinin	$f 3 (x)$	30	$∑ i = 1 D (x i 2 - 10 c o s 2 π x i + 10)$	［-5.12，5.12］
Schwefel 1.2	$f 4 (x)$	30	$∑ i = 1 D ∑ j = 1 i x j 2$	［-100，100］
Griewank	$f 5 (x)$	30	$1 + 1 4 000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i / i$	［-600，600］
Penalized 1	$f 6 (x)$	30	$π D 10 s i n 2 (π y 1) + ∑ i = 1 i = 1 (y i - 1) 2 1 + 10 s i n 2 (π y (i + 1)) + (y D - 1) 2 + ∑ i = 1 D u (x i)$ ； $y i = 1 + 1 / 4 (x i + 1),$ $u (x i) = 100 (- x i - 10) 4, x i < - 10 0, x i ≤ 10 100 (x i - 10) 4, x i > 10$	［-50，50］
Step	$f 7 (x)$	30	$∑ i = 1 D x i + 0.5 2$	［-100，100］
Penalized 2	$f 8 (x)$	30	$0.1 s i n 2 (3 π x 1) + ∑ i = 1 D - 1 (x i - 1) 2 (1 + s i n 2 (3 π x i +))) + (x D - 1) 2 (1 + s i n 2 (2 π x D)) + ∑ i = 1 D u (x i);$ $u (x i) = 100 (- x i - 5) 4, x i < - 5 0, x i ≤ 5 100 (x i - 5) 4, x i > 5$	［-50，50］

函数	函数	维数	函数表达式	变量范围
Sphere	$f 1 (x)$	30	$∑ i = 1 D x i 2$	［-100，100］
Rosenbrock	$f 2 (x)$	30	$∑ i = 1 D - 1 100 x i + 1 - x i 2) 2 + (x i - 1) 2]$	［-30，30］
Rastrinin	$f 3 (x)$	30	$∑ i = 1 D (x i 2 - 10 c o s 2 π x i + 10)$	［-5.12，5.12］
Schwefel 1.2	$f 4 (x)$	30	$∑ i = 1 D ∑ j = 1 i x j 2$	［-100，100］
Griewank	$f 5 (x)$	30	$1 + 1 4 000 ∑ i = 1 D x i 2 - ∏ i = 1 D c o s x i / i$	［-600，600］
Penalized 1	$f 6 (x)$	30	$π D 10 s i n 2 (π y 1) + ∑ i = 1 i = 1 (y i - 1) 2 1 + 10 s i n 2 (π y (i + 1)) + (y D - 1) 2 + ∑ i = 1 D u (x i)$ ； $y i = 1 + 1 / 4 (x i + 1),$ $u (x i) = 100 (- x i - 10) 4, x i < - 10 0, x i ≤ 10 100 (x i - 10) 4, x i > 10$	［-50，50］
Step	$f 7 (x)$	30	$∑ i = 1 D x i + 0.5 2$	［-100，100］
Penalized 2	$f 8 (x)$	30	$0.1 s i n 2 (3 π x 1) + ∑ i = 1 D - 1 (x i - 1) 2 (1 + s i n 2 (3 π x i +))) + (x D - 1) 2 (1 + s i n 2 (2 π x D)) + ∑ i = 1 D u (x i);$ $u (x i) = 100 (- x i - 5) 4, x i < - 5 0, x i ≤ 5 100 (x i - 5) 4, x i > 5$	［-50，50］

$s h i f t$	指标	$f 1 (x)$	$f 2 (x)$	$f 3 (x)$	$f 4 (x)$	$f 5 (x)$	$f 6 (x)$	$f 7 (x)$	$f 8 (x)$
0	Mean	0.00E+00	2.89E-02	0.00E+00	4.60E-13	0.00E+00	7.89E-04	6.29E-04	9.96E-04
0	Std	1.31E-14	1.63E+01	2.01E+03	1.18E-15	7.01E+02	8.46E+04	4.91E-01	1.18E+02
3	Mean	0.00E+00	5.00E-03	0.00E+00	4.16E-17	0.00E+00	4.98E-05	5.09E-04	1.35E-04
3	Std	8.31E-23	1.56E+01	1.95E+02	4.18E-18	6.21E+02	2.03E+04	3.45E-01	8.12E+01
5	Mean	1.30E-251	4.52E-02	0.00E+00	2.80E-66	0.00E+00	4.89E-05	1.16E-03	4.28E-04
5	Std	2.31E-44	1.41E+00	1.15E+02	8.18E-38	5.09E+01	3.03E+03	1.15E-02	7.34E-01
7	Mean	4.71E-204	9.68E-03	0.00E+00	1.36E-79	0.00E+00	2.80E-04	1.39E-04	8.77E-04
7	Std	8.11E-67	1.21E-01	1.45E+01	2.13E-42	3.32E-01	7.02E-03	7.55E-03	7.41E-03
9	Mean	8.60E-230	1.85E-03	0.00E+00	1.15E-87	0.00E+00	4.07E-05	1.73E-07	9.37E-05
9	Std	3.61E-74	8.63E-02	1.01E+01	1.98E-65	1.01E-02	1.15E-04	1.28E-03	1.99E-03
10	Mean	1.01E-148	1.23E-01	0.00E+00	2.98E-80	0.00E+00	2.71E-05	4.97E-03	1.80E-04
10	Std	4.22E-46	9.27E-02	1.12E+01	1.28E-55	3.71E-01	2.71E-04	1.19E-02	5.11E-03

$s h i f t$	指标	$f 1 (x)$	$f 2 (x)$	$f 3 (x)$	$f 4 (x)$	$f 5 (x)$	$f 6 (x)$	$f 7 (x)$	$f 8 (x)$
0	Mean	0.00E+00	2.89E-02	0.00E+00	4.60E-13	0.00E+00	7.89E-04	6.29E-04	9.96E-04
0	Std	1.31E-14	1.63E+01	2.01E+03	1.18E-15	7.01E+02	8.46E+04	4.91E-01	1.18E+02
3	Mean	0.00E+00	5.00E-03	0.00E+00	4.16E-17	0.00E+00	4.98E-05	5.09E-04	1.35E-04
3	Std	8.31E-23	1.56E+01	1.95E+02	4.18E-18	6.21E+02	2.03E+04	3.45E-01	8.12E+01
5	Mean	1.30E-251	4.52E-02	0.00E+00	2.80E-66	0.00E+00	4.89E-05	1.16E-03	4.28E-04
5	Std	2.31E-44	1.41E+00	1.15E+02	8.18E-38	5.09E+01	3.03E+03	1.15E-02	7.34E-01
7	Mean	4.71E-204	9.68E-03	0.00E+00	1.36E-79	0.00E+00	2.80E-04	1.39E-04	8.77E-04
7	Std	8.11E-67	1.21E-01	1.45E+01	2.13E-42	3.32E-01	7.02E-03	7.55E-03	7.41E-03
9	Mean	8.60E-230	1.85E-03	0.00E+00	1.15E-87	0.00E+00	4.07E-05	1.73E-07	9.37E-05
9	Std	3.61E-74	8.63E-02	1.01E+01	1.98E-65	1.01E-02	1.15E-04	1.28E-03	1.99E-03
10	Mean	1.01E-148	1.23E-01	0.00E+00	2.98E-80	0.00E+00	2.71E-05	4.97E-03	1.80E-04
10	Std	4.22E-46	9.27E-02	1.12E+01	1.28E-55	3.71E-01	2.71E-04	1.19E-02	5.11E-03

函数	指标	AMBPSO	MSPSO	RPSO	EAMSPSO	MFPSO	PSO	PDORW-PSO
$f 1 (x)$	Mean	8.31E-184	1.73E-03	1.30E-34	1.79E-60	1.81E-320	2.55E+03	8.60E-230
$f 1 (x)$	Std	3.81E-14	2.88E-02	5.61E-34	8.14E-59	8.12E-20	1.45E+03	3.61E-74
$f 2 (x)$	Mean	4.63E+03	3.13E+02	1.10E+02	2.25E+04	1.25E+03	3.95E+05	1.85E-03
$f 2 (x)$	Std	2.06E+04	6.95E+02	4.32E+02	2.18E+04	2.18E+04	1.12E+05	8.63E-02
$f 3 (x)$	Mean	9.46E+01	6.21E+01	2.69E+01	4.01E+01	0.00E+00	1.93E+02	0.00E+00
$f 3 (x)$	Std	3.28E+01	2.15E+01	1.00E+01	2.23E+01	3.11E+01	2.22E+01	1.01E+01
$f 4 (x)$	Mean	8.01E+03	1.01E+03	2.00E+02	7.98E+03	4.15E-99	1.14E+04	1.15E-87
$f 4 (x)$	Std	4.17E+03	1.81E+03	9.90E+02	8.01E+03	7.23E+03	5.67E+03	1.98E-65
$f 5 (x)$	Mean	1.87E+00	4.92E-02	2.34E-02	1.76E+05	1.12E-55	2.42E+01	0.00E+00
$f 5 (x)$	Std	7.28E+01	7.86E-02	2.25E-02	1.18E+01	1.28E+01	1.32E+01	1.01E-02
$f 6 (x)$	Mean	6.93E-02	6.01E-02	4.15E-03	1.01E-02	1.04E-02	3.11E+01	4.07E-05
$f 6 (x)$	Std	1.49E-01	1.88E-01	2.05E-02	2.94E-02	3.14E-02	1.12E+01	1.15E-04
$f 7 (x)$	Mean	1.03E+09	2.38E-02	2.96E-03	5.11E-04	4.00E+02	1.06E+05	1.73E-07
$f 7 (x)$	Std	4.01E+08	3.78E-02	3.33E-03	3.21E-03	3.21E-03	5.98E+04	1.28E-03
$f 8 (x)$	Mean	4.11E+02	1.60E-01	0.00E+00	3.89E+02	1.27E-04	2.47E+03	9.37E-05
$f 8 (x)$	Std	1.98E+02	5.48E-01	5.18E-01	2.01E+03	1.98E+03	1.41E+03	1.99E-03

函数	指标	AMBPSO	MSPSO	RPSO	EAMSPSO	MFPSO	PSO	PDORW-PSO
$f 1 (x)$	Mean	8.31E-184	1.73E-03	1.30E-34	1.79E-60	1.81E-320	2.55E+03	8.60E-230
$f 1 (x)$	Std	3.81E-14	2.88E-02	5.61E-34	8.14E-59	8.12E-20	1.45E+03	3.61E-74
$f 2 (x)$	Mean	4.63E+03	3.13E+02	1.10E+02	2.25E+04	1.25E+03	3.95E+05	1.85E-03
$f 2 (x)$	Std	2.06E+04	6.95E+02	4.32E+02	2.18E+04	2.18E+04	1.12E+05	8.63E-02
$f 3 (x)$	Mean	9.46E+01	6.21E+01	2.69E+01	4.01E+01	0.00E+00	1.93E+02	0.00E+00
$f 3 (x)$	Std	3.28E+01	2.15E+01	1.00E+01	2.23E+01	3.11E+01	2.22E+01	1.01E+01
$f 4 (x)$	Mean	8.01E+03	1.01E+03	2.00E+02	7.98E+03	4.15E-99	1.14E+04	1.15E-87
$f 4 (x)$	Std	4.17E+03	1.81E+03	9.90E+02	8.01E+03	7.23E+03	5.67E+03	1.98E-65
$f 5 (x)$	Mean	1.87E+00	4.92E-02	2.34E-02	1.76E+05	1.12E-55	2.42E+01	0.00E+00
$f 5 (x)$	Std	7.28E+01	7.86E-02	2.25E-02	1.18E+01	1.28E+01	1.32E+01	1.01E-02
$f 6 (x)$	Mean	6.93E-02	6.01E-02	4.15E-03	1.01E-02	1.04E-02	3.11E+01	4.07E-05
$f 6 (x)$	Std	1.49E-01	1.88E-01	2.05E-02	2.94E-02	3.14E-02	1.12E+01	1.15E-04
$f 7 (x)$	Mean	1.03E+09	2.38E-02	2.96E-03	5.11E-04	4.00E+02	1.06E+05	1.73E-07
$f 7 (x)$	Std	4.01E+08	3.78E-02	3.33E-03	3.21E-03	3.21E-03	5.98E+04	1.28E-03
$f 8 (x)$	Mean	4.11E+02	1.60E-01	0.00E+00	3.89E+02	1.27E-04	2.47E+03	9.37E-05
$f 8 (x)$	Std	1.98E+02	5.48E-01	5.18E-01	2.01E+03	1.98E+03	1.41E+03	1.99E-03

函数	指标	PSO	PD-PSO	ORW-PSO	PDORW-PSO
$f 1 (x)$	Mean	2.55E+03	9.31E-124	3.19E-127	8.60E-230
$f 1 (x)$	Std	1.45E+03	3.18E-14	2.28E-12	3.61E-74
$f 2 (x)$	Mean	3.95E+05	5.13E+00	3.13E+01	1.85E-03
$f 2 (x)$	Std	1.12E+05	2.10E+03	6.05E+02	8.63E-02
$f 3 (x)$	Mean	1.93E+02	1.21E-114	3.21E-109	0.00E+00
$f 3 (x)$	Std	2.22E+01	3.28E+01	2.15E+01	1.01E+01
$f 4 (x)$	Mean	1.14E+04	1.11E-44	3.21E-54	1.15E-87
$f 4 (x)$	Std	5.67E+03	3.17E+03	1.21E+03	1.98E-65
$f 5 (x)$	Mean	2.42E+01	1.21E-41	3.21E-51	0.00E+00
$f 5 (x)$	Std	1.32E+01	7.09E+01	7.26E-03	1.01E-02
$f 6 (x)$	Mean	3.11E+01	6.93E-02	6.01E-02	4.07E-05
$f 6 (x)$	Std	1.12E+01	1.29E-01	1.28E-01	1.15E-04
$f 7 (x)$	Mean	2.47E+03	1.19E-01	1.56E-02	1.73E-07
$f 7 (x)$	Std	1.41E+03	1.98E+02	5.48E-01	1.28E-03
$f 8 (x)$	Mean	1.06E+05	2.14E-02	4.28E-02	9.37E-05
$f 8 (x)$	Std	5.98E+04	4.11E+03	3.08E-02	1.99E-03

函数	指标	PSO	PD-PSO	ORW-PSO	PDORW-PSO
$f 1 (x)$	Mean	2.55E+03	9.31E-124	3.19E-127	8.60E-230
$f 1 (x)$	Std	1.45E+03	3.18E-14	2.28E-12	3.61E-74
$f 2 (x)$	Mean	3.95E+05	5.13E+00	3.13E+01	1.85E-03
$f 2 (x)$	Std	1.12E+05	2.10E+03	6.05E+02	8.63E-02
$f 3 (x)$	Mean	1.93E+02	1.21E-114	3.21E-109	0.00E+00
$f 3 (x)$	Std	2.22E+01	3.28E+01	2.15E+01	1.01E+01
$f 4 (x)$	Mean	1.14E+04	1.11E-44	3.21E-54	1.15E-87
$f 4 (x)$	Std	5.67E+03	3.17E+03	1.21E+03	1.98E-65
$f 5 (x)$	Mean	2.42E+01	1.21E-41	3.21E-51	0.00E+00
$f 5 (x)$	Std	1.32E+01	7.09E+01	7.26E-03	1.01E-02
$f 6 (x)$	Mean	3.11E+01	6.93E-02	6.01E-02	4.07E-05
$f 6 (x)$	Std	1.12E+01	1.29E-01	1.28E-01	1.15E-04
$f 7 (x)$	Mean	2.47E+03	1.19E-01	1.56E-02	1.73E-07
$f 7 (x)$	Std	1.41E+03	1.98E+02	5.48E-01	1.28E-03
$f 8 (x)$	Mean	1.06E+05	2.14E-02	4.28E-02	9.37E-05
$f 8 (x)$	Std	5.98E+04	4.11E+03	3.08E-02	1.99E-03

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