粒子群优化算法的性能分析和参数选择

王东风; 孟丽

doi:10.16383/j.aas.2016.c150774

粒子群优化算法的性能分析和参数选择

doi: 10.16383/j.aas.2016.c150774

王东风,
孟丽^,

华北电力大学自动化系保定 071003

基金项目:

中央高校基本科研基金 20140139

国家自然科学基金 61203041

教育部高等学校博士学科点专项科研基金 20120036120013

详细信息

作者简介:
王东风博士, 华北电力大学控制与计算机工程学院教授.主要研究方向为群智能优化算法和智能控制.E-mail:wangdongfeng@ncepu.edu.cn

通讯作者:
孟丽华北电力大学控制与计算机工程学院博士研究生.主要研究方向为智能优化算法及其应用.本文通信作者.E-mail:mengli2014@163.com

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- 被引次数: 0
出版历程
- 收稿日期: 2015-11-18
- 录用日期: 2016-03-26
- 刊出日期: 2016-10-20

Performance Analysis and Parameter Selection of PSO Algorithms

WANG Dong-Feng,
MENG Li^,

Department of Automation, North China Electric Power University, Baoding 071003

Funds:

the Fundamental Research Funds for the Central Universities 20140139

National Natural Science Foundation of China 61203041

Specialized Research Fund for the Doctoral Program of Higher Education 20120036120013

More Information

Author Bio:
Ph. D., professor at the School of Control and Computer Engineering, North China Electric Power University. His research interest covers swarm intelligence-based optimization and intelligent control.E-mail:

Corresponding author: MENG Li Ph. D. candidate at the School of Control and Computer Engineering, North China Electric Power University. Her research interest covers swarm intelligence-based optimization and its application. Corresponding author of this paper.E-mail:mengli2014@163.com

摘要

摘要: 惯性权重和加速因子是影响粒子群算法优化性能的重要参数.基于常用的12个测试函数，本文通过实验研究了不同参数组合下粒子的探索能力和算法的优化性能，在此基础上推荐了一组固定的参数组合.通过惯性权重和加速因子的不同变化策略组合对算法性能影响的实验分析，推荐了一种变化的参数设置方法.基于CEC2015发布的15个基准函数进一步验证了本文推荐的参数选取方法的有效性.最后讨论了粒子群优化（Particle swarm optimization，PSO）算法在连续优化和离散优化方面的应用问题.
- 粒子群优化 /
- 探索能力 /
- 算法性能 /
- 参数选取
Abstract: Inertia weight and acceleration factors have significant impact on the performance of particle swarm optimization (PSO) algorithm. Through simulation experiments on twelve classical benchmark functions, this paper studies the algorithm's exploitation ability and optimization performance with different parameters. Based on the experimental results, we recommend a setting for fixed parameters. Furthermore, we study the situation where inertia weight remains unchanged and acceleration factors change with iterations. Then a setting for varying parameters is recommended. The recommended parameters setting methods are verified through 15 benchmark functions that were published in CEC2015. At the end of the paper, a discussion of the PSO application issue on continuous optimization problems and discrete optimization problems is given.
- Particle swarm optimization(PSO) /
- exploitation ability /
- optimization performance /
- parameter selection

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图 1 具有不同参数的种群优化函数Sphere时粒子的Pnum值

Fig. 1 The Pnum value of function Sphere optimized by PSO with different population scale

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图 2 粒子探索能力与参数的关系

Fig. 2 Relationship between particle exploration ability and its parameters

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图 3 算法成功率与参数的关系

Fig. 3 Relationship between the algorithm success rate and particle parameters

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图 4 算法在12个测试函数上的性能总和

Fig. 4 Total optimization performance on the 12 test functions

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图 5 c₁所占比率对算法成功率的影响

Fig. 5 The influence of ratio c₁ on algorithm0s success rate

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图 6 变化的c₁和c₂取值策略对算法成功率的影响

Fig. 6 The influence of varying c₁ and c₂ on algorithm0s success rate

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表 1 5组参数的运行结果对比

Table 1 Comparison of running results on 5 parameters

函数	PARA	min	max	mean±std	SR
Fun 1	1	1.47E-25	8.64E-22	6.32E-23±1.34E-22	1.00
	2	9.05E-36	5.64E-31	1.73E-32±7.16E-32	1.00
	3	1.98E-40	2.94E-34	1.31E-35±4.50E-35	1.00
	4	1.81E-57	2.84E-33	2.86E-35±2.84E-34	1.00
	5	1.63E-27	1.45E-23	2.71E-25±1.46E-24	1.00
Fun 2	1	2.72E-02	7.29E+00	2.40E+00±1.43E+00	0.84
	2	3.18E-03	1.20E+02	3.75E+00±1.19E+01	0.76
	3	1.20E-03	8.27E+00	3.45E+00±1.96E+00	0.30
	4	6.03E-02	8.96E+00	4.30E+00±1.57E+00	0.07
	5	2.21E-02	8.45E+00	3.45E+00±1.40E+00	0.31
Fun 3	1	6.55E-10	8.21E-08	1.38E-08±1.58E-08	1.00
	2	5.68E-14	1.07E-10	9.80E-12±1.70E-11	1.00
	3	2.98E-11	7.36E-09	8.82E-10±1.32E-09	1.00
	4	7.06E-12	4.01E-04	6.10E-06±4.09E-05	0.94
	5	1.23E-10	1.02E-08	1.63E-09±1.77E-09	1.00
Fun 4	1	1.06E-14	1.00E+01	5.00E-01±2.19E+00	0.95
	2	7.73E-20	1.00E+01	1.00E-01±1.00E+00	0.99
	3	7.38E-23	7.89E-20	9.10E-21±1.60E-20	1.00
	4	8.37E-20	1.00E+01	1.00E-01±1.00E+00	0.98
	5	1.64E-15	1.12E-13	1.82E-14±1.87E-14	1.00
Fun 5	1	1.14E-25	1.00E+04	2.00E+02±1.40E+03	0.00
	2	6.54E-35	1.82E-29	2.24E-31±1.82E-30	1.00
	3	1.74E-39	2.36E-33	1.04E-34±3.60E-34	1.00
	4	2.45E-54	6.74E-34	6.74E-36±6.74E-35	1.00
	5	1.44E-26	7.31E-24	5.72E-25±9.94E-25	0.32
Fun 6	1	9.91E-13	1.39E+01	5.73E+00±3.06E+00	0.55
	2	9.94E-01	1.59E+01	6.61E+00±3.23E+00	0.43
	3	0.00E+00	1.39E+01	4.87E+00±2.94E+00	0.68
	4	9.94E-01	4.47E+01	9.87E+00±7.05E+00	0.23
	5	1.66E-13	7.95E+00	3.29E+00±1.84E+00	0.85
Fun 7	1	1.47E-02	1.64E-01	7.49E-02±3.35E-02	0.83
	2	7.39E-03	1.84E-01	7.77E-02±3.52E-02	0.77
	3	1.47E-02	2.21E-01	7.94E-02±4.10E-02	0.74
	4	1.72E-02	3.05E-01	1.12E-01±5.99E-02	0.53
	5	1.11E-15	1.69E-01	6.15E-02±3.39E-02	0.86
Fun 8	1	2.38E-13	1.99E+01	1.99E-01±1.99E+00	0.99
	2	3.55E-15	7.10E-15	3.94E-15±1.11E-15	1.00
	3	3.55E-15	7.10E-15	3.97E-15±1.16E-15	1.00
	4	3.55E-15	1.15E+00	1.15E-02±1.15E-01	0.97
	5	1.42E-14	6.18E-13	1.51E-13±1.19E-13	1.00
Fun 9	1	0.00E+00	1.30E+03	6.41E+02±2.28E+02	0.35
	2	0.00E+00	1.42E+03	6.83E+02±2.86E+02	0.30
	3	1.18E+02	1.54E+03	6.41E+02±3.04E+02	0.44
	4	1.18E+00	1.19E+03	6.40E+02±2.51E+02	0.36
	5	0.00E+00	9.49E+02	4.48E+02±1.86E+02	0.67
Fun 10	1	3.55E-10	1.57E+00	1.46E-01±4.41E-01	0.86
	2	0.00E+00	3.07E+00	1.69E-01±5.28E-01	0.86
	3	0.00E+00	1.50E+00	6.00E-02±2.96E-01	0.96
	4	1.81E-13	3.00E+00	1.40E-01±4.50E-01	0.46
	5	0.00E+00	3.46E-02	3.46E-04±3.46E-03	0.99
Fun 11	1	5.76E-27	5.30E-21	5.90E-23±5.30E-22	1.00
	2	4.71E-32	1.90E-31	4.94E-32±1.53E-32	1.00
	3	4.71E-32	3.11E-01	1.55E-02±6.81E-02	0.95
	4	4.71E-32	3.11E-01	1.24E-02±6.12E-02	0.95
	5	5.16E-29	3.55E-24	5.37E-26±2.57E-25	1.00
Fun 12	1	6.86E-27	6.65E-22	2.13E-23±7.32E-23	1.00
	2	1.34E-32	4.65E-31	2.13E-32±4.68E-32	1.00
	3	1.34E-32	2.72E-26	5.45E-28±3.85E-27	1.00
	4	1.34E-32	4.23E-13	4.23E-15±4.26E-14	0.99
	5	2.78E-28	1.93E-23	2.82E-25±1.93E-24	1.00

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表 2 本文与文献[17]参数设置的优化结果比较

Table 2 Comparison of optimization results between the parameters set in this paper and [17]

函数M	Method	min	max	mean±std	SR
Fun 1	Ours [17]	3.91E-33	1.42E-29	1.20E-30±2.51E-30	1.00
Fun 1	Ours [17]	5.01E-33	6.95E-29	3.09E-30±9.83E-30	1.00
Fun 2	Ours [17]	9.79E-02	2.26E+00	1.30E+00±5.31E-01	1.00
Fun 2	Ours [17]	1.00E+00	5.72E+00	2.40E+00±7.33E-01	0.84
Fun 3	Ours [17]	5.90E-13	2.27E-10	2.13E-11±3.88E-11	1.00
Fun 3	Ours [17]	6.03E-13	2.48E-10	2.99E-11±4.90E-11	1.00
Fun 4	Ours [17]	5.58E-18	7.20E-16	1.60E-16±1.52E-16	1.00
Fun 4	Ours [17]	8.12E-18	2.87E-13	6.86E-15±4.09E-14	1.00
Fun 5	Ours [17]	7.24E-32	3.86E-28	1.60E-29±5.49E-29	1.00
Fun 5	Ours [17]	2.12E-32	3.44E-28	1.65E-29±5.70E-29	1.00
Fun 6	Ours [17]	9.94E-01	7.95E+00	3.34E+00±1.55E+00	0.92
Fun 6	Ours [17]	1.98E+00	1.09E+01	5.49E+00±2.36E+00	0.54
Fun 7	Ours [17]	0.00E+00	1.42E-01	5.20E-02±2.94E-02	0.96
Fun 7	Ours [17]	0.00E+00	1.84E-01	7.81E-02±3.58E-02	0.76
Fun 8	Ours [17]	3.55E-15	7.10E-15	4.68E-15±1.67E-15	1.00
Fun 8	Ours [17]	3.55E-15	2.13E-14	7.81E-15±3.58E-15	1.00
Fun 9	Ours [17]	1.18E+02	1.04E+03	4.71E+02±1.99E+02	0.64
Fun 9	Ours [17]	1.18E+02	1.19E+03	5.48E+02±2.16E+02	0.42
Fun 10	Ours [17]	0.00E+00	1.44E-02	6.04E-04±2.51E-03	0.94
Fun 10	Ours [17]	0.00E+00	1.15E-01	3.03E-03±1.64E-02	0.84
Fun 11	Ours [17]	4.71E-32	2.90E-27	9.03E-29±4.35E-28	1.00
Fun 11	Ours [17]	4.71E-32	1.34E-28	3.09E-30±1.90E-29	1.00
Fun 12	Ours [17]	1.34E-32	6.16E-29	3.88E-30±9.45E-30	1.00
Fun 12	Ours [17]	1.34E-32	1.09E-02	2.19E-04±1.55E-03	0.98

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表 3 本文给出的固定参数推荐值在CEC2015基准函数F1～F15上的运行结果

Table 3 Running results on CEC2015 Benchmark functions F1～F15 with fixed parameters

PARA	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12	F13	F14	F15
[6]	0.02	0.02	0.30	0.24	0.45	0.24	0.95	0.08	0.90	0.56	0.90	0.92	0.04	0.12	0.40
[7]	0.08	0.12	0.40	0.14	0.40	0.10	1.00	0.30	0.85	0.68	0.80	0.91	0.02	0.16	0.48
Ours	0.02	0.10	0.10	0.18	0.50	0.14	1.00	0.15	0.90	0.60	0.95	0.95	0.04	0.30	0.50

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表 4 本文给出的时变参数设置方式在CEC2015基准函数F1～F15上的运行结果

Table 4 Running results on CEC2015 Benchmark functions F1～F15 with time-varying parameters

Method	F1	F2	F3	F4	F5	F6	F7	F8	F9	F10	F11	F12	F13	F14	F15
Ours	0.70	0.26	0.30	0.94	1.00	0.64	1.00	0.50	1.00	0.95	1.00	1.00	0.25	0.40	0.96
[17]	0.16	0.14	0.01	0.82	0.85	0.25	1.00	0.22	1.00	0.85	1.00	1.00	0.20	0.40	0.90

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粒子群优化算法的性能分析和参数选择

doi: 10.16383/j.aas.2016.c150774

作者简介:
王东风博士, 华北电力大学控制与计算机工程学院教授.主要研究方向为群智能优化算法和智能控制.E-mail:wangdongfeng@ncepu.edu.cn

通讯作者:
孟丽华北电力大学控制与计算机工程学院博士研究生.主要研究方向为智能优化算法及其应用.本文通信作者.E-mail:mengli2014@163.com

计量

Performance Analysis and Parameter Selection of PSO Algorithms

Author Bio:
Ph. D., professor at the School of Control and Computer Engineering, North China Electric Power University. His research interest covers swarm intelligence-based optimization and intelligent control.E-mail:

Corresponding author: MENG Li Ph. D. candidate at the School of Control and Computer Engineering, North China Electric Power University. Her research interest covers swarm intelligence-based optimization and its application. Corresponding author of this paper.E-mail:mengli2014@163.com

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粒子群优化算法的性能分析和参数选择

doi: 10.16383/j.aas.2016.c150774

作者简介: 王东风 博士, 华北电力大学控制与计算机工程学院教授.主要研究方向为群智能优化算法和智能控制.E-mail:wangdongfeng@ncepu.edu.cn

通讯作者: 孟丽 华北电力大学控制与计算机工程学院博士研究生.主要研究方向为智能优化算法及其应用.本文通信作者.E-mail:mengli2014@163.com

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出版历程

Performance Analysis and Parameter Selection of PSO Algorithms

Author Bio: Ph. D., professor at the School of Control and Computer Engineering, North China Electric Power University. His research interest covers swarm intelligence-based optimization and intelligent control.E-mail:

Corresponding author: MENG Li Ph. D. candidate at the School of Control and Computer Engineering, North China Electric Power University. Her research interest covers swarm intelligence-based optimization and its application. Corresponding author of this paper.E-mail:mengli2014@163.com

计量

出版历程

目录

作者简介:
王东风博士, 华北电力大学控制与计算机工程学院教授.主要研究方向为群智能优化算法和智能控制.E-mail:wangdongfeng@ncepu.edu.cn

通讯作者:
孟丽华北电力大学控制与计算机工程学院博士研究生.主要研究方向为智能优化算法及其应用.本文通信作者.E-mail:mengli2014@163.com

Author Bio:
Ph. D., professor at the School of Control and Computer Engineering, North China Electric Power University. His research interest covers swarm intelligence-based optimization and intelligent control.E-mail: