一种自适应模拟退火粒子群优化算法

doi:10.19665/j.issn1001-2400.2021.04.016

摘要/Abstract

摘要：

为了提高粒子群算法的寻优速度和精度,避免陷入局部优解,提出一种自适应模拟退火粒子群优化算法。采用双曲正切函数来控制惯性权重系数,进行非线性自适应变化;利用线性变化策略控制社会学习因子和自我学习因子,达到改变不同阶段寻优重点的目的;引入模拟退火操作,根据种群的初始状态设置一个温度,根据米特罗波利斯准则和温度指导种群以一定的概率接受差解,保证了算法跳出局部最优解的能力。为验证这种算法的效果,选择7种典型测试函数与已有文献中提出的5种粒子优化算法进行对比实验,根据寻优结果的平均值、标准差以及迭代次数等数据,证明文中所提算法在迭代精度、收敛速度以及稳定性上都有很大的提升,有效地弥补了经典粒子群算法的缺陷。

关键词: 粒子群优化, 模拟退火, 惯性权重系数, 自适应调整策略

Abstract:

Particle swarm optimization is widely used in various fields because of the few parameters to be set and the simple calculation structure.In order to improve the optimization speed and accuracy of the PSO,and to avoid falling into the local optimal solution,an adaptive simulated annealing PSO is proposed,which uses the hyperbolic tangent function to control the inertia weight factor for nonlinear adaptive changes,uses linear change strategies to control 2 learning factors,introduces the simulation annealing operation,set a temperature according to the initial state of the population,guide the population to accept the difference solution with a certain probability according to the Metropolis criterion,and ensure the ability to jump out of the local optimal solution.To verify the effect of the algorithm proposed in this paper,7 typical test functions and 5 algorithms proposed in the literature are selected for comparison and testing.According to the average value,standard deviation and number of iterations of the optimization results,the algorithm proposed in this paper has greatly improved the iteration accuracy,convergence speed and stability so as to overcome the shortcomings of particle swarm optimization.

Key words: particle swarm optimization, simulated annealing, inertia weight factor, self-adaptive adjust tactics

中图分类号:

TP301.6

闫群民,马瑞卿,马永翔,王俊杰. 一种自适应模拟退火粒子群优化算法[J]. 西安电子科技大学学报, 2021, 48(4): 120-127.

YAN Qunmin,MA Ruiqing,MA Yongxiang,WANG Junjie. Adaptive simulated annealing particle swarm optimization algorithm[J]. Journal of Xidian University, 2021, 48(4): 120-127.

图/表 8

图1

表1

测试函数信息表"

函数	函数表达式	搜索空间
Sphere	f₁= $∑ i = 1 D x i 2$	[-100,100]
Levy	f₂=sin²(πω_i)+ $∑ i = 1 D - 1$ (ω_i-1)²[1+10sin²(πω_i+1)]+ (ω_D-1)²[1+sin²(2πω_D)], ω_i=1+(x_i-1)/4	[-10,10]
Ackley	f₃=-20exp $- 0.2 1 / D ∑ i = 1 D x i 2 1 / 2$ -exp((1/D) $∑ i = 1 D$ cos(2π $x i 2$ ))+20+e	[-30,30]
Griewank	f₄= $∑ i = 1 D x i 2 4000$ - $∏ i = 1 D$ cos $x 1 i$ +1	[-600,600]
Rastrigin	f₅= $∑ i = 1 D$ [ $x i 2$ -10cos(2πx_i)+10]	[-5.12,5.12]
Rosenbrock	f₆= $∑ i = 1 D - 1$ [100( $x i 2$ -x_i+1)²+(x_i-1)²]	[-2.408,2.408]
Schwefel	f₇=418.982 9D- $∑ i = 1 D$ x_isin\|x_i\|^1/2	[-500,500]

表1

表2

表3

图2

图3

图4

图5

参考文献 16

[1]	KENNEDY J, EBERHART R. Particle Swarm Optimization[C]//Proceedings of the IEEE International Conference on NeuralNetworks.Piscataway:IEEE, 1995:1942-1948.
[2]	SHI Y, EBERHART R. A Modified Particle Swarm Optimizer[C]//Proceedings of the 1998 IEEE International Conference on Evolutionary Computation Proceedings.Piscataway:IEEE, 1998:69-73.
[3]	TANG Y, WANG Z, FANG J A. Feedback Learning Particle Swarm Optimization[J]. Applied Soft Computing Journal, 2011, 11(8):4713-4725. doi: 10.1016/j.asoc.2011.07.012
[4]	YAN C, LU G, LIU Y, et al. A Modified PSO Algorithm with Exponential Decay Weight[C]//Proceedings of the 2017 13th International Conferenceon on Natural Computation,Fuzzy Systems and Knowledge Discovery.Piscataway:IEEE, 2017:239-242.
[5]	仝秋娟, 李萌, 赵岂. 基于分类思想的改进粒子群优化算法[J]. 现代电子技术, 2019, 42(19):11-14.
	TONG Qiujuan, LI Meng, ZHAO Qi. An Improved Particle Swarm Optimization Algorithm Based on Classification[J]. Modern Electronics Technique, 2019, 42(19):11-14.
[6]	周敏, 李太勇. 粒子群优化算法中的惯性权值非线性调整策略[J]. 计算机工程, 2011, 37(5):204-206.
	ZHOU Min, LI Taiyong. Nonlinear Adjustment Strategy of Inertia Weight in Particle Swarm Optimization Algorithm[J]. Computer Engineering, 2011, 37(5):204-206.
[7]	CHEN W, ZHANG N. Particle Swarm Optimization with An Aging Leader and Challengers[J]. IEEE Transactions on Evolutionary Computation, 2013, 17(2):241-258. doi: 10.1109/TEVC.2011.2173577
[8]	李江杰, 常安定, 陈童, 等. 融合牛顿-最速下降算子的自适应粒子群算法[J]. 微电子学与计算机, 2020, 37(2):1-7.
	LI Jiangjie, CHANG Anding, CHEN Tong, et al. An Adaptive Particle Swarm Optimization Algorithm with Newton-steepest Descent Operator[J]. Microelectronics and Computer, 2020, 37(2):1-7.
[9]	XIA X W, XIE C W, WEI B, et al. Particle Swarm Optimization Using Multi-level Adaptation and Purposeful Detection Operators[J]. Information Sciences, 2017, 385:174-195.
[10]	齐欣, 靳雁霞, 张晋瑞, 等. 大数据环境中简化粒子群算法的改进研究[J]. 微电子学与计算机, 2020, 37(2):25-29.
	QI Xin, JIN Yanxia, ZHANG Jinrui, et al. Improvement of Simplified Particle Swarm Optimization in Big Data Environment[J]. Microelectronics and Computer, 2020, 37(2):25-29.
[11]	刘道华, 原思聪, 兰洋, 等. 混沌映射的粒子群优化方法[J]. 西安电子科技大学学报, 2010, 37(4):764-769.
	LIU Daohua, YUAN Sicong, LAN Yang, et al. Method of Particle Swarm Optimization Based on the Chaos Map[J]. Journal of Xidian University, 2010, 37(4):764-769.
[12]	邓先礼, 魏波, 曾辉, 等. 基于多种群的自适应迁移PSO算法[J]. 电子学报, 2018, 46(8):1858-1865.
	DENG Xianli, WEI Bo, ZENG Hui, et al. A Multi-Population Based Self-Adaptive Migration PSO[J]. Acta Electronica Sinica, 2018, 46(8):1858-1865.
[13]	黄少荣. 基于随机参数的粒子群优化算法[J]. 重庆师范大学学报:自然科学版, 2013, 30(6):123-127.
	HUANG Shaorong. A New Particle Swarm Optimization with Random Parameters[J]. Journal of Chongqing Normal University:Natural Science, 2013, 30(6):123-127.
[14]	滕志军, 吕金玲, 郭力文, 等. 基于动态加速因子的粒子群优化算法研究[J]. 微电子学与计算机, 2017, 34(12):125-129.
	TENG Zhijun, LV Jinling, GUO Liwen, et al. Research on Particle Swarm Optimization Based on Dynamic Acceleaation Coefficients[J]. Microelectronics and Computer, 2017, 34(12):125-129.
[15]	徐浩天, 季伟东, 孙小晴, 等. 基于正态分布衰减惯性权重的粒子群优化算法[J]. 深圳大学学报:理工版, 2020, 37(2):208-213.
	XU Haotian, JI Weidong, SUN Xiaoqing, et al. A PSO Algorithm with Inertia Weight Decay by Normal Distribution[J]. Journal of Shenzhen University:Science and Engineering, 2020, 37(2):208-213.
[16]	姜建国, 田旻, 王向前, 等. 采用扰动加速因子的自适应粒子群优化算法[J]. 西安电子科技大学学报, 2012, 39(4):74-80.
	JIANG Jianguo, TIAN Min, WANG Xiangqian, et al. Adaptive Particle Swarm Optimization Via Disturbing Acceleration Coefficents[J]. Journal of Xidian University, 2012, 39(4):74-80.

函数	准则	MSMPSO	RPPSO	PSO-DAC	NDPSO	APSO-DA	文中算法
f₁	avg	2.48×10^-2	8.24×10	5.72×10²	4.24×10^-2	2.14×10^-4	0.00
	std	4.16×10^-2	8.99×10	3.65×10²	8.67×10^-2	1.14×10^-3	0.00
f₂	avg	3.91×10^-1	1.36	2.47	3.87×10^-2	8.56×10^-2	3.70×10^-2
	std	6.88×10^-1	1.23	1.80	9.86×10^-2	1.01×10^-1	4.34×10^-2
f₃	avg	1.78	4.87	7.90	3.89×10^-1	8.88×10^-16	8.88×10^-16
	std	9.44×10^-1	2.02	1.99	5.12×10^-1	9.86×10^-32	9.86×10^-32
f₄	avg	3.10×10^-1	2.17	7.05	1.67×10^-1	3.36×10^-3	0.00
	std	1.70×10^-1	2.25	4.65	2.05×10^-1	1.78×10^-2	0.00
f₅	avg	9.76	2.14×10	2.19×10	2.47	1.33×10^-1	0.00
	std	5.38	9.54	8.32	5.11	7.03×10^-1	0.00
f₆	avg	6.92	1.56×10	5.58×10	7.14	7.41	6.81
	std	1.76	1.30×10	3.41×10	1.65	2.60	3.17
f₇	avg	9.38×10²	1.58×10³	1.72×10³	8.81×10²	8.30×10²	8.22×10²
	std	2.87×10²	4.25×10²	3.07×10²	4.05×10²	2.86×10²	3.50×10²

函数	准则	MSMPSO	RPPSO	PSO-DAC	NDPSO	APSO-DA	文中算法
f₁	avg	1.93×10³	6.46×10³	9.74×10³	7.86×10²	2.04	0.00
	std	9.43×10²	2.48×10³	2.79×10³	2.53×10³	7.13	0.00
f₂	avg	8.68	2.05×10	2.55×10	1.10×10	2.16	2.04
	std	3.48	6.89	9.43	7.70	2.48	2.31
f₃	avg	8.74	1.30×10	1.44×10	6.32	1.83×10^-1	8.88×10^-16
	std	1.88	1.05	1.07	5.63	5.16×10^-1	5.07×10^-22
f₄	avg	1.85×10	5.82×10	9.99×10	1.92×10	6.47×10^-2	4.83×10^-2
	std	8.19	1.79×10	3.71×10	4.97×10	2.47×10^-1	2.64×10^-1
f₅	avg	1.05×10²	1.49×10²	1.72×10²	7.98×10	1.16×10	3.63
	std	2.06×10	2.49×10	2.77×10	3.86×10	1.70×10	8.68
f₆	avg	1.55×10²	4.54×10²	6.58×10²	4.05×10	2.87×10	2.87×10
	std	7.72×10	1.95×10²	1.96×10²	2.41×10	2.13×10^-1	4.03×10^-1
f₇	avg	4.74×10³	6.78×10³	7.65×10³	4.40×10³	4.13×10³	4.05×10³
	std	9.14×10²	5.80×10²	6.10×10²	1.02×10³	7.36×10²	6.62×10²