噪声环境下基于蒲丰距离的依概率多峰优化算法

王霞; 王耀民; 施心陵; 高莲; 李鹏

doi:10.16383/j.aas.c190474

噪声环境下基于蒲丰距离的依概率多峰优化算法

doi: 10.16383/j.aas.c190474

王霞^{1, 2,},
王耀民^1,,
施心陵^1,,
高莲^1,,
李鹏^1,

1.
云南大学信息学院昆明 650500
2.
云南民族大学电气信息工程学院昆明 650500

基金项目: 国家自然科学基金(61763046, 61762093, 61662089, 61761048), 云南省第十七批中青年学术和技术带头人资助项目(2014HB019), 云南省高校科技创新团队支持计划资助, 云南省教育厅科学研究基金(2017ZZX229), 云南省应用基础研究重点项目(2018FA036)资助

详细信息

作者简介:
王霞：云南大学博士研究生. 主要研究方向为智能优化算法. E-mail: wangxiacsu@163.com

王耀民：云南大学博士研究生. 主要研究方向为SDN, 数据中心和智能优化算法. E-mail: 18988081898@189.cn

施心陵：云南大学信息学院教授. 主要研究方向为智能优化算法, 自适应信号处理与信息系统, 医学电子学. 本文通信作者. E-mail: xlshi@ynu.edu.cn

高莲：云南大学信息学院讲师. 主要研究方向为生物医学信号处理. E-mail: ylbgl123@sina.com

李鹏：云南大学信息学院副教授. 主要研究方向为输变电系统安全诊断、预警与维护决策研究. E-mail: lipeng@ynu.edu.cn

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出版历程
- 收稿日期: 2019-07-01
- 录用日期: 2019-11-16
- 网络出版日期: 2021-09-16
- 刊出日期: 2021-11-18

Probabilistic Multimodal Optimization Algorithm Based on the Buffon Distance in Noisy Environment

WANG Xia^{1, 2
,},
WANG Yao-Min^1
,,
SHI Xin-Ling^1
,,
GAO Lian^1
,,
LI Peng^1
,

1.
School of Information Science and Engineering, Yunnan University, Kunming 650500
2.
School of Electrical and Information Engineering, Yunnan Minzu University, Kunming 650500

Funds: Supported by National Natural Science Foundation of China (61763046, 61762093, 61662089, 61761048), The 17th Batches of Young and Middle-aged Leaders in Academic and Technical Reserved Talents Project of Yunnan Province (2014HB019), The Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province, Yunnan Provincial Department of Education Science Research Fund Project (2017ZZX229), and Key Projects of Applied Basic Research of Yunnan Province (2018FA036)

More Information

Author Bio:
WANG Xia　Ph. D. candidate at Yunnan University. Her main research interest is intelligent optimization algorithm

WANG Yao-Min　Ph. D. candidate at Yunnan University. His research interest covers SDN, data center and intelligent optimization algorithm

SHI Xin-Ling　Professor at the School of Information Science and Engineering, Yunnan University. His research interest covers intelligent optimization algorithm, adaptive signal processing and information system and medical electronics. Corresponding author of this paper

GAO Lian　Lecturer at the School of Information Science and Engineering, Yunnan University. Her main research interest is biomedical signal processing

LI Peng　Associate professor at the School of Information Science and Engineering, Yunnan University. His research interest covers safety diagnosis, early warning and maintenance decision of power transmission and transformation system

摘要

摘要: 针对噪声环境下求解多个极值点的问题, 本文提出了噪声环境下基于蒲丰距离的依概率多峰优化算法(Probabilistic multimodal optimization algorithm based on the Button distance, PMB). 算法依据蒲丰投针原理提出噪声下的蒲丰距离和极值分辨度概念, 理论推导证明了二者与算法峰值检测率符合依概率关系. 在全局范围内依据蒲丰距离划分搜索空间, 可以使PMB算法保持较好的搜索多样性. 在局部范围内利用改进的斐波那契法进行探索, 减少了算法陷入噪声引起的局部最优的概率. 基于34个测试函数, 从依概率特性验证、寻优结果影响因素分析、多极值点寻优和多维函数寻优四个角度进行实验. 证明了蒲丰距离与算法的峰值检测率符合所推导的依概率关系. 对比噪声环境下的改进蝙蝠算法和粒子群算法, PMB算法在噪声环境中可以依定概率更精确地定位多峰函数的更多极值点, 从而证明了PMB算法原理的正确性和噪声条件下全局寻优的依概率性能, 具有理论意义和实用价值.
- 噪声环境 /
- 多峰优化 /
- 蒲丰距离 /
- 依概率
Abstract: To solve the problem of multiple extremum points optimization in noisy environment, a probabilistic multimodal optimization algorithm based on the Buffon distance (PMB) is proposed in this paper. Based on the principle of Buffon needles, the concepts of Buffon distance and resolution of extreme value under noisy environment are put forward. The theoretical derivation proves that the relationship between peak detection rate of PMB algorithm and Buffon distance conforms to a probabilistic relation. In global scope, the search space is divided by Buffon distance for diversity maintenance. The improved Fibonacci method is used in local search to reduce the probability of falling into the local optimum caused by noise. Based on 34 test functions, experiments are carried out from four aspects, including probabilistic property verification, analysis of influencing factors of optimization results, multiple extremum points optimization and multidimensional optimization. It is proved that the relationship of Buffon distance and peak detection rate of PMB algorithm is in line with the deduced probabilistic relationship. Compared with the improved bat algorithm and particle swarm optimization (PSO), PMB algorithm can locate extremum points of multimodal function by a steady probability in noise environment, and gain more extremum points accurately. Thus, the correctness of PMB algorithm principle and probabilistic performance of global optimization under noise condition are proved, which has theoretical and practical significance.
- Noisy environment /
- multimodal optimization /
- Buffon distance /
- probabilistic
注释:

1) 　收稿日期 2019-07-01 录用日期 2019-11-16　Manuscript received July 1, 2019; accepted November 16, 2019　国家自然科学基金(61763046, 61762093, 61662089, 61761048), 云南省第十七批中青年学术和技术带头人资助项目 (2014HB019), 云南省高校科技创新团队支持计划资助, 云南省教育厅科学研究基金(2017ZZX229), 云南省应用基础研究重点项目 (2018FA036) 资助　Supported by National Natural Science Foundation of China (61763046, 61762093, 61662089, 61761048), The 17th Batches of Young and Middle-aged Leaders in Academic and Technical Reserved Talents Project of Yunnan Province (2014HB019), The Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province, Yunnan Provincial Department of Education Science Research Fund Project (2017ZZX229),

2) and Key Projects of Applied Basic Research of Yunnan Province (2018FA036)　本文责任编委穆朝絮　Recommended by Associate Editor MU Chao-Xu　1. 云南大学信息学院昆明 650500 2. 云南民族大学电气信息工程学院昆明 650500　1. School of Information Science and Engineering, Yunnan University, Kunming 650500 2. School of Electrical and Informa-tion Engineering, Yunnan Minzu University, Kunming 650500

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图 1 极值分辨度

Fig. 1 Extreme value resolution

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图 2 函数波形与极值点

Fig. 2 Function waveform and extremum points

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图 3 蒲丰投针实验设计图

Fig. 3 Experimental design of Buffon needles

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图 4 式(9)证明示意图

Fig. 4 Diagram to prove (9)

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图 5 蒲丰距离与针长的关系示意图

Fig. 5 Diagram of the relationship between Buffon distance and needle length

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图 6 改进的斐波那契法的区域伸缩过程

Fig. 6 The process of regional scaling of the improved Fibonacci method

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图 7 斐波那契法的区域收缩过程

Fig. 7 The process of regional shrink of the Fibonacci method

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图 8 PMB算法的多级划分策略

Fig. 8 The multilevel partition strategy of PMB algorithm

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图 9 $a$ = 0.2时无噪声环境下PMB算法对TF1中的$f_1$的寻优结果

Fig. 9 Optimization results for $f_1$ of TF1 by PMB algorithm in noiseless environment with $a$ = 0.2

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图 14 $\sigma^2$ = 0.01, $a$ = 0.2时PMB算法对TF1中的$f_1$的寻优结果

Fig. 14 Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.01 and $a$ = 0.2

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图 11 $\sigma^2$ = 0.05, $a$ = 0.5时PMB算法对TF1中的$f_1$的寻优结果

Fig. 11 Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and$a$ = 0.5

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图 10 $a$ = 0.5时无噪声环境下PMB算法对TF1中的$f_1$的寻优结果

Fig. 10 Optimization results for $f_1$ of TF1 by PMB algorithm in noiseless environment with $a$ = 0.5

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图 12 $\sigma^2$ = 0.01,$a$ = 0.5时PMB算法对TF1中的$f_1$的寻优结果

Fig. 12 Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.01 and $a$ = 0.5

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图 13 $\sigma^2$ = 0.05,$a$ = 0.2时PMB算法对TF1中的$f_1$的寻优结果

Fig. 13 Optimization results for $f_1$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and $a$ = 0.2

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图 15 $a$ = 0.2, PMB算法在不同噪声环境下对TF1中的$f_1$求解所有极值点的定位误差

Fig. 15 Positioning errors of total-extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.2 under different noise environments

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图 16 $a$ = 1, $\sigma^2$ = 0.05, PMB算法对TF1中 $f_2$的寻优结果

Fig. 16 Optimization results for $f_2$ of TF1 by PMB algorithm in noise environment with $\sigma^2$ = 0.05 and $a$ = 1

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表 1 测试函数集1 (TF1)

Table 1 Test function set 1 (TF1)

函数	名称	定义域	全局最优值f (与之对应的最优解$\rm{x}_1 $,$ \rm{x}_2$)	表达式
$f_1$		[−1, 1]	3.5326 ($\pm $0.8781, $\pm $0.8781)	${{f}_{1}} = x_{1}^{2}+x_{2}^{2}-\cos(18{{x}_{1}})-\cos (18{{x}_{2}})$
$f_2$	Levy 05	[−10, 10]	−176.1375 (−1.3068, −1.4248)	$\begin{aligned} {f}_{2}=\;& \sum\nolimits_{i = 1}^{5}{i\cos [(i-1){{x}_{1}}}+i]\times \sum\nolimits_{j = 1}^{5}{j\cos [(j+1){{x}_{2}}+j]}+ \\&{{({{x}_{1}}+1.42513)}^{2}}+{{({{{x}}_{2}}+0.80032)}^{2}}\end{aligned}$
$f_3$	Michalewics	[0, ${\text{π}}$]	−1.8013 (2.2031, 1.5709)	${{f}_{3}} = -\displaystyle\sum\nolimits_{i = 1}^{2}{\sin ({{x}_{i}})}{{\left(\sin \left(i\times x_{i}^{2}/{\text{π}} \right)\right)}^{20}}$
$f_4$	Periodic	[0, ${\text{π}}$]	0.9 (0,0)	${{f}_{4}} = 1+{{\sin }^{2}}({{x}_{1}})+{{\sin }^{2}}({{x}_{2}})-0.1{{ \rm{e}}^{-(x_{1}^{2}+x_{2}^{2})}}$
$f_5$	Carrom Table	[−10, 10]	−24.1568 ($\pm $9.6466, 9.6466)	${ {f}_{5} } = -\left[{\left\{\cos ({ {x}_{1} })\cos({ {x}_{2} }){ { \rm{e} }^{\left\| 1-\sqrt{x_{1}^{2}+x_{2}^{2} }/{\text{π} } \right\|} }\right\} }^{2}\right]/30$
$f_6$	Holder Table 2	[−10, 10]	−19.2085 ($\pm $8.0550, $\pm $9.6646)	${ {f}_{6} } = -\left\| \sin ({ {x}_{1} })\cos ({ {x}_{2} }){ { \rm{e} }^{\left\| 1-\sqrt{ { {x}_{1} }+{ {x}_{2} } }/{\text{π} } \right\|} } \right\|$

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表 2 测试函数集2 (TF2) (维度为5)

Table 2 Test function set 2 (TF2) (Dimension is 5)

	函数	全局最优值$f_{i}^{*}$	函数名称
单峰函数	$f_1$	−1 400	Sphere function
	$f_2$	−1 300	Rotated high conditioned elliptic function
	$f_3$	−1 200	Rotated bent cigar function
	$f_4$	−1 100	Rotated discus function
	$f_5$	−1 000	Different powers function
多峰函数	$f_6$	−900	Rotated Rosenbrock＇s function
	$f_7$	−800	Rotated Schaffers F7 function
	$f_8$	−700	Rotated Ackley＇s function
	$f_9$	−600	Rotated Weierstrass function
	$f_{10}$	−500	Rotated Griewank＇s function
	$f_{11}$	−400	Rastrigin＇s function
	$f_{12}$	−300	Rotated Rastrigin＇s function
	$f_{13}$	−200	Non-continuous rotated Rastrigin＇s function
	$f_{14}$	−100	Schwefel＇s function
	$f_{15}$	100	Rotated Schwefel＇s function
	$f_{16}$	200	Rotated Katsuura function
	$f_{17}$	300	Lunacek bi_Rastrigin function
	$f_{18}$	400	Rotated lunacek bi_Rastrigin function
	$f_{19}$	500	Expanded Griewank＇s plus Rosenbrock＇s function
	$f_{20}$	600	Expanded Scaffer＇s F6 function
组合函数	$f_{21}$	700	Composition function 1 (n = 5, Rotated)
	$f_{22}$	800	Composition function 2 (n = 3, Unrotated)
	$f_{23}$	900	Composition function 3 (n = 3, Rotated)
	$f_{24}$	1 000	Composition function 4 (n = 3, Rotated)
	$f_{25}$	1 100	Composition function 5 (n = 3, Rotated)
	$f_{26}$	1 200	Composition function 6 (n = 5, Rotated)
	$f_{27}$	1 300	Composition function 7 (n = 5, Rotated)
	$f_{28}$	1 400	Composition function 8 (n = 5, Rotated)

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表 3 针对TF1中的$f_1$取不同蒲丰距离时PMB算法在不同噪声环境下的优化结果

Table 3 Optimization results of PMB algorithm for $f_1$ of TF1 in different noise environments with different Buffon distances

	$\sigma^2$	$a$	Num_mean	Num_max	Num_min	$\eta\_{\rm{mean}}$	$\eta\_{\rm{global}}$
50 次	0.01	0.5	29.62	33	26	82.28 %	100.00 %
	0.01	0.2	36.04	37	36	100.11 %	100.00 %
	0.05	0.5	31.32	35	27	87.00 %	100.00 %
	0.05	0.2	36.26	38	36	100.72 %	100.00 %
100 次	0.01	0.5	29.78	33	27	82.72 %	100.00 %
	0.01	0.2	36.02	37	36	100.06 %	100.00 %
	0.05	0.5	32.08	36	28	89.11 %	100.00 %
	0.05	0.2	36.37	38	36	101.03 %	100.00 %
200 次	0.01	0.5	29.715	34	23	82.54 %	100.00 %
	0.01	0.2	36.02	37	36	100.06 %	100.00 %
	0.05	0.5	32	35	26	88.89 %	100.00 %
	0.05	0.2	36.4	40	36	101.11 %	100.00 %

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表 4 $a$ = 0.2时PMB算法针对TF1中的 $f_1$在不同噪声环境下的全局极值点优化数值结果

Table 4 Numerical results of global extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.2 under different noise environments

极值点序号	P-MQHOA^[1] 无噪声			PMB
	P-MQHOA^[1] 无噪声			$\sigma^2$ = 0.01					$\sigma^2$ = 0.05
	$x_1$	$x_2$	$f$	$x_1$	$x_2$	$f$	$P_f$	$P_l$	$x_1$	$x_2$	$f$	$P_f$	$P_l$
1	−0.8781	−0.8781	3.5326	−0.8781	−0.878	3.8057	2.73 × 10⁻¹	1.42 × 10⁻⁴	−0.8776	−0.8773	4.1956	6.63 × 10⁻¹	9.37 × 10⁻⁴
2	−0.8781	0.8781	3.5326	−0.8789	0.8784	3.8027	2.70 × 10⁻¹	7.86 × 10⁻⁴	−0.8788	0.8782	4.1974	6.65 × 10⁻¹	8.19 × 10⁻⁴
3	0.8781	−0.8781	3.5326	0.8782	−0.8778	3.8086	2.76 × 10⁻¹	3.64 × 10⁻⁴	0.8788	−0.8784	4.2048	6.72 × 10⁻¹	7.36 × 10⁻⁴
4	0.8781	0.8781	3.5326	0.8783	0.8775	3.8049	2.72 × 10⁻¹	6.60 × 10⁻⁴	0.8774	0.8765	4.2050	6.72 × 10⁻¹	1.79 × 10⁻³

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表 5 $a$ = 0.5时PMB算法针对TF1中的 $f_1$在不同噪声环境下的全局极值点优化数值结果

Table 5 Numerical results of global extremum points for $f_1$ of TF1 searched by PMB algorithm with $a$ = 0.5 under different noise environments

极值点序号	P-MQHOA^[1] 无噪声			PMB
	P-MQHOA^[1] 无噪声			$\sigma^2$ = 0.01					$\sigma^2$ = 0.05
	$x_1$	$x_2$	$f$	$x_1$	$x_2$	$f$	$P_f$	$P_l$	$x_1$	$x_2$	$f$	$P_f$	$P_l$
1	−0.8781	−0.8781	3.5326	−0.8760	−0.8780	3.6923	1.60 × 10⁻¹	2.07 × 10⁻³	−0.8805	−0.8775	3.9568	4.24 × 10⁻¹	2.46 × 10⁻³
2	−0.8781	0.8781	3.5326	−0.8771	0.8782	3.6953	1.63 × 10⁻¹	1.03 × 10⁻³	−0.8770	0.8803	3.9510	4.18 × 10⁻¹	2.50 × 10⁻³
3	0.8781	−0.8781	3.5326	0.8775	−0.8794	3.6847	1.52 × 10⁻¹	1.05 × 10⁻³	0.8772	−0.8788	3.9467	4.14 × 10⁻¹	1.14 × 10⁻³
4	0.8781	0.8781	3.5326	0.8788	0.8784	3.6917	1.59 × 10⁻¹	8.18 × 10⁻⁴	0.8775	0.8756	3.9697	4.37 × 10⁻¹	2.52 × 10⁻³

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表 6 P-MQHOA^[1]算法与$a$ = 0.2时的PMB算法针对TF1中的 $f_1$在不同环境下的所有极值点求解结果对比

Table 6 Comparison of total-extremum points for $f_1$ of TF1 searched by P-MQHOA^[1] algorithm and PMB algorithm with $a$ = 0.2 under different noise environments

极值点序号	P-MQHOA^[1] 无噪声			PMB
	P-MQHOA^[1] 无噪声			$\sigma^2$ = 0.01			$\sigma^2$ = 0.09
	$x_1$	$x_2$	$f(x_1,x_2)$	$x_1$	$x_2$	$f(x_1,x_2)$	$x_1$	$x_2$	$f(x_1,x_2)$
1	−0.8781	−0.8781	3.5326	−0.8781	−0.8780	3.8057	−0.8762	−0.8777	4.4627
2	−0.8781	−0.5269	3.0421	−0.8778	−0.5263	3.3069	−0.8774	−0.5285	3.9294
3	−0.8781	−0.1756	2.7969	−0.8784	−0.1746	3.0482	−0.8778	−0.1818	3.6818
4	−0.8781	0.1756	2.7969	−0.8789	0.1763	3.0605	−0.8790	0.1814	3.7123
5	−0.8781	0.5269	3.0421	−0.8788	0.5277	3.3100	−0.8761	0.5258	3.9341
6	−0.8781	0.8781	3.5326	−0.8789	0.8784	3.8027	−0.8798	0.8774	4.4562
7	−0.5269	−0.8781	3.0421	−0.5276	−0.8780	3.3047	−0.5264	−0.8766	3.9330
8	−0.5269	−0.5269	2.5517	−0.5266	−0.5267	3.4294	−0.52621	−0.5269	2.7918
9	−0.5269	−0.1756	2.3065	−0.5264	−0.1756	2.5465	−0.5278	−0.1798	3.1998
10	−0.5269	0.1756	2.3065	−0.5263	0.1760	2.5499	−0.5289	0.1772	3.1915
11	−0.5269	0.5269	2.5517	−0.5268	0.5256	2.7924	−0.526	0.5278	3.4282
12	−0.5269	0.8781	3.0421	−0.5268	0.8775	3.3045	−0.5271	0.8770	3.9228
13	−0.1756	−0.8781	2.7969	−0.1770	−0.8786	3.0600	−0.1745	−0.8800	3.6858
14	−0.1756	−0.5269	2.3065	−0.1761	−0.5275	2.5472	−0.1792	−0.5261	3.1804
15	−0.1756	−0.1756	2.0613	−0.1772	−0.1755	2.2932	−0.1775	−0.1789	2.9170
16	−0.1756	0.1756	2.0613	−0.1765	0.1746	2.2879	−0.1755	0.1774	2.9344
17	−0.1756	0.5269	2.3065	−0.1763	0.5278	2.5418	−0.1757	0.5250	3.1700
18	−0.1756	0.8781	2.7969	−0.1756	0.8791	3.0500	−0.1784	0.8817	3.6725
19	0.1756	−0.8781	2.7969	0.1762	−0.8773	3.0574	0.1769	−0.8757	3.6846
20	0.1756	−0.5269	2.3065	0.1762	−0.5266	2.5532	0.1794	−0.5270	3.1708
21	0.1756	−0.1756	2.0613	0.1756	−0.1779	2.2964	0.1802	−0.1758	2.9292
22	0.1756	0.1756	2.0613	0.1756	0.1752	2.2921	0.1790	0.1766	2.9235
23	0.1756	0.5269	2.3065	0.1752	0.5269	2.5441	0.1820	0.5250	3.1637
24	0.1756	0.8781	2.7969	0.1756	0.8800	3.0573	0.1803	0.8784	3.6873
25	0.5269	−0.8781	3.0421	0.5253	−0.8767	3.2985	0.5250	−0.8808	3.9639
26	0.5269	−0.5269	2.5517	0.5254	−0.5261	2.8010	0.5256	−0.5244	3.4443
27	0.5269	−0.1756	2.3065	0.5275	−0.1752	2.5432	0.5266	−0.1785	3.1973
28	0.5269	0.1756	2.3065	0.5266	0.1775	2.5502	0.5291	0.1789	3.1769
29	0.5269	0.5269	2.5517	0.5273	0.5259	2.7941	0.5277	0.5257	3.4395
30	0.5269	0.8781	3.0421	0.5254	0.8776	3.3128	0.5242	0.8817	3.9135
31	0.8781	−0.8781	3.5326	0.8782	−0.8778	3.8086	0.8793	−0.8759	4.4479
32	0.8781	−0.5269	3.0421	0.8792	−0.5265	3.3119	0.8771	−0.5288	3.9199
33	0.8781	−0.1756	2.7969	0.8778	−0.1766	3.0594	0.8774	−0.1782	3.7063
34	0.8781	0.1756	2.7969	0.8780	0.1760	3.0549	0.8772	0.1784	3.6981
35	0.8781	0.5269	3.0421	0.8784	0.5265	3.3001	0.8788	0.5279	3.3046
36	0.8781	0.8781	3.5326	0.8783	0.8775	3.8049	0.8803	0.8792	4.4365

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表 7 PMB算法对TF1中的$f_2\sim f_6$在不同噪声环境下的全局极值点优化数值结果

Table 7 Numerical results of global extremum points for function $f_2\sim f_6$ of TF1searched by PMB algorithm under different noise environments

函数	全局极值	指标	IBA^[32]				PMB
函数	全局极值	指标	$\sigma^2$ = 0.01	$\sigma^2$ = 0.05	$\sigma^2$ = 0.07	$\sigma^2$ = 0.09	$\sigma^2$ = 0.01	$\sigma^2$ = 0.05	$\sigma^2$ = 0.07	$\sigma^2$ = 0.09
$f_2$	−176.1375	Mean	−176.1830	−176.3646	−176.5443	−176.5442	−176.1607	−176.0738	−176.0689	−175.9722
$f_2$	−176.1375	MAPE	0.0258	0.1289	0.1799	0.2309	0.0132	0.0362	0.0389	0.0938
$f_3$	−1.8013	Mean	−1.8468	−2.0283	−2.1206	−2.2095	−2.0007	−2.3048	−2.4433	−2.5351
$f_3$	−1.8013	MAPE	2.5281	12.6022	17.7255	22.6617	11.0675	27.9516	35.6393	40.7363
$f_4$	0.9	Mean	0.8556	0.6838	0.6053	0.5330	0.6534	0.2620	0.1312	0.0529
$f_4$	0.9	MAPE	4.9308	24.0261	32.7394	46.5648	27.4009	70.8921	85.4263	94.1181
$f_5$	−24.1568	Mean	−24.2016	−24.3806	−24.4665	−24.5567	−24.2903	−24.538	−24.5977	−24.6518
$f_5$	−24.1568	MAPE	0.1856	0.9266	1.2822	1.6556	0.5526	1.5781	1.8253	2.0490
$f_6$	−19.2085	Mean	−19.253	−19.4329	−19.5236	−19.6095	−19.3724	−19.6408	−19.7391	−19.8169
$f_6$	−19.2085	MAPE	0.2319	1.1683	1.6403	2.0875	0.8533	2.2504	2.7625	3.1673

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表 8 PMB算法对TF1中$f_2\sim f_4$在不同噪声环境下得到的全局极值点的定位误差

Table 8 Positioning errors of global extremum points for function $f_2\sim f_4$ of TF1 searched by PMB algorithm under different noise environments

函数	$\sigma^2$	IBA^[32]	PMB
函数	$\sigma^2$	$P_l$	$x_1$	$x_2$	$P_l$
$f_2$	0.01	0.00082301	−1.3062	−1.4246	0.0007
	0.05	0.0019	−1.3064	−1.4241	0.0009
	0.07	0.0022	−1.3079	−1.4244	0.0011
	0.09	0.0023	−1.3041	−1.4252	0.0027
$f_3$	0.01	0.0540	2.2088	1.5727	0.0059
	0.05	0.0132	2.1913	1.5723	0.0119
	0.07	0.0160	2.1882	1.5664	0.0156
	0.09	0.0201	2.1833	1.5710	0.0198
$f_4$	0.01	0.0282	0.0611	0.0698	0.0927
	0.05	0.0830	0.0836	0.0623	0.1042
	0.07	1.1067	0.0820	0.0655	0.1049
	0.09	2.2026	0.0579	0.0934	0.1099

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表 9 PMB算法对TF1中的$f_5\sim f_6$在不同噪声环境下的全局极值点优化数值结果

Table 9 Numerical results of global extremum points for function $f_5\sim f_6$ of TF1searched by PMB algorithm under different noise environments

函数	算法	IBA^[32]				PMB
	$\sigma^2$	0.01	0.05	0.07	0.09	0.01	0.05	0.07	0.09
	极值点序号	$P_l$				$(x_1, x_2), P_l$
$f_5$	1	0.0048	0.0151	0.0179	0.0206	(9.6453, −9.6439), 0.0030	(9.6498, −9.6444), 0.0039	(9.6398, −9.6475), 0.0068	(9.6559, −9.6482), 0.0094
	2	0.0057	0.0162	0.016	0.0158	(−9.6457, 9.6444), 0.0023	(−9.6473, 9.6442), 0.0025	(−9.6520, 9.6450), 0.0057	(−9.6473, 9.6401), 0.0065
	3	0.006	0.0112	0.0149	0.0153	(−9.6454, −9.6455), 0.0016	(−9.6460, −9.6413), 0.0053	(−9.6427, −9.6406), 0.0072	(−9.6407, −9.6419), 0.0076
	4	0.0044	0.0133	0.0151	0.0178	(9.6476, 9.6468), 0.0010	(9.6477, 9.6528), 0.0063	(9.6493, 9.6532), 0.0071	(9.6434, 9.6390), 0.0082
$f_6$	1	0.0107	0.0215	0.0238	0.0291	(8.0518, −9.6675), 0.0043	(8.0521, −9.6712), 0.0072	(8.0462, −9.6588), 0.0106	(8.0525, −9.6534), 0.0115
	2	0.0094	0.0207	0.0262	0.0289	(−8.0541, 9.6626), 0.0022	(−8.0532, 9.6629), 0.0025	(−8.0561, 9.6614), 0.0034	(−8.0586, 9.6567), 0.0087
	3	0.0088	0.0199	0.0242	0.0243	(−8.0513, −9.6634), 0.0039	(−8.0541, −9.6590), 0.0057	(−8.0595, −9.6698), 0.0069	(−8.0602, −9.6566), 0.0096
	4	0.0081	0.0241	0.021	0.0238	(8.0529, 9.6611), 0.0041	(8.0503, 9.6688), 0.0063	(8.0547, 9.6757), 0.0111	(8.0642, 9.6743), 0.0134

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表 10 PMB算法和PSO算法对TF2在无噪声环境下得到的全局极值点

Table 10 Global extremum points of TF2 obtained by PMB algorithm and PSO algorithm in noiseless environment

TF2	Opti- mum	PSO							PMB
TF2	Opti- mum	$f$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$t({\rm{s}})$	$f$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$t({\rm{s}})$
$f_1$	−1 400	−1 400.5153	−22.003	11.6561	−36.1216	69.3583	−37.5101	1.2932	−1 399.9755	−22.1013	11.5532	−35.9107	69.3869	−37.6351	4.024
$f_2$	−1 300	−608.3034	−23.2765	15.9243	−18.1464	70.6333	−35.3268	1.586	−1 284.0072	−22.1909	10.6542	−37.981	69.5946	−37.5306	13.084
$f_3$	−1 200	−1 191.9927	−90.9013	11.4835	−35.8577	68.8415	−40.1122	1.3798	−1 199.4865	−32.8088	11.5433	−35.9866	69.289	−38.0021	30.5866
$f_4$	−1 100	−1 100.2332	−22.0721	11.3036	−35.9249	69.1684	−37.6371	1.4573	−1 098.4189	−22.0954	11.9348	−36.1631	68.5683	−37.4961	52.0969
$f_5$	−1 000	−1 000.4002	−21.9647	11.3823	−36.0354	69.0788	−37.36	1.326	−999.9999	−21.9848	11.5549	−36.0044	69.3447	−37.6114	73.5208
$f_6$	−900	−900.1464	−48.0332	16.1924	−31.6003	85.6123	−64.8171	1.2566	−900.0000	−22.0309	11.5686	−36.0034	69.4094	−37.6539	6.7166
$f_7$	−800	−800.314	−47.6006	11.5431	−35.9756	69.3023	−37.8813	1.9918	−799.9792	−22.0032	11.5635	−36.004	69.3839	−37.5808	24.2937
$f_8$	−700	−680.2271	−89.9164	95.3877	55.5365	−63.3428	−20.1823	1.6314	−699.9191	−22.2758	11.5436	−36.0022	69.3631	−37.6026	44.2769
$f_9$	−600	−600.3439	75.0307	21.2084	−65.4178	−90.6598	25.5987	10.7053	−599.9286	−22.0921	11.5053	−35.998	69.3653	−37.6013	137.0404
$f_{10}$	−500	−500.3287	−22.7446	11.9779	−35.0425	69.3475	−37.3164	1.6048	−499.8497	−21.6232	11.8265	−36.1124	69.0713	−37.5719	164.8766
$f_{11}$	−400	−400.4644	−21.9891	11.5276	−36.014	69.3475	−37.5829	1.8479	−399.54	−21.6252	11.9614	−35.9928	69.484	−37.3817	203.6023
$f_{12}$	−300	−299.4175	−19.0411	5.6211	−50.5124	72.7959	−42.8024	1.9517	−299.2154	−22.224	11.6704	−35.3449	68.6809	−37.0768	244.3073
$f_{13}$	−200	−199.3823	−31.2247	9.7199	−34.7315	78.1449	−45.3504	1.8811	−199.7728	−22.0876	11.6491	−35.5865	69.2094	−37.2746	288.9675
$f_{14}$	−100	−100.0015	−6.2047	11.5466	−27.1106	69.3496	−32.6014	1.5652	−99.8081	−22.0000	11.4699	−36.0043	69.3659	−37.622	25.5247
$f_{15}$	100	156.5509	49.6019	63.3034	−72.268	−3.7613	−79.9879	1.6973	110.4386	−21.8498	11.9445	−35.3707	69.3721	−37.4593	45.1034
$f_{16}$	200	199.8538	−57.0813	58.5504	−31.0707	26.9993	−38.2225	3.9518	200.4084	−22.0509	11.5686	−36.0034	69.4094	−37.6539	31.5485
$f_{17} $	300	304.7026	8.0018	−16.5642	−5.9498	38.2561	−8.1092	1.5401	300.3612	−22.0466	11.5575	−37.5698	69.3452	−37.6059	114.3228
$f_{18}$	400	405.3374	8.8635	−22.2516	−5.391	38.826	−10.5181	1.6256	401.5735	−22.269	12.4883	−38.1444	68.9858	−37.9558	144.0145
$f_{19} $	500	499.8491	−36.1124	−7.1141	−60.5984	57.0958	−63.211	1.4022	500.0023	−22.1995	11.4599	−36.2687	68.9998	−37.9225	175.7896
$f_{20} $	600	599.7283	−43.4837	14.2779	−40.0212	72.9036	−43.5544	1.4423	600.0068	−22.0109	11.5786	−36.0034	69.4014	−37.6519	214.3663
$f_{21}$	700	999.5585	74.8671	7.56	60.3875	15.7259	31.6655	1.8884	709.2098	−22.1135	11.7078	−36.2284	69.2862	−37.5341	259.2839
$f_{22}$	800	899.5949	−48.5424	53.7715	13.7283	69.8239	−18.62 29	2.6328	821.3446	−22.0219	12.2169	−36.155	69.2986	−37.4481	338.1721
$f_{23}$	900	1 040.6959	−47.8633	59.6935	18.944	63.6605	−16.7946	2.5248	926.4912	−22.2323	11.3133	−36.2082	69.665	−37.8375	400.5942
$f_{24}$	1 000	1 104.3212	−43.5817	62.9002	24.6363	69.2317	−19.6422	11.4319	1 005.5802	−22.2796	11.6315	−36.0045	69.656	−37.0404	536.6315
$f_{25}$	1 100	1 199.5464	−48.5541	53.8377	13.7864	69.8039	−18.5822	11.4093	1 103.9827	−22.2144	11.3766	−35.7601	69.5939	−37.2318	674.0186
$f_{26}$	1 200	1 230.9793	−45.3432	36.5543	−61.2965	73.0248	−28.7484	11.8107	1 201.2061	−21.9038	11.8136	−36.0593	69.1101	−37.8049	824.5081
$f_{27}$	1 300	1 643.8584	39.9937	64.0666	74.3787	82.5407	88.9271	11.4554	1 316.2217	−22.0002	11.5006	−36.0034	69.4000	−37.6006	977.0531
$f_{28}$	1 400	1 699.6997	74.858	7.5684	60.4032	15.7134	31.6678	2.8537	1 404.1964	−21.9492	11.4562	−36.0421	69.5111	−37.5802	1 061.4819

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表 11 PSO算法对TF2在$\sigma^2$ = 0.01的噪声环境下得到的全局极值点

Table 11 Global extremum points of TF2 obtained by PSO algorithm in noise environment of $\sigma^2$ = 0.01

TF2	Optimum	$f$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	t (s)	$P_f$	$P_l$
$f_1$	−1 400	−1 400.4781	−22.0184	11.4865	−35.9794	69.3581	−37.6173	1.404	0.0372	0.2464
$f_2$	−1 300	−930.7041	−20.245	15.2674	−30.6074	67.6185	−38.8701	1.4961	322.4007	13.6581
$f_3$	−1 200	−1 199.0062	−50.622	11.5252	−35.947	69.152	−38.649	1.2854	7.0135	40.3071
$f_4$	−1 100	−1 100.2794	−21.9225	11.64	−36.0456	69.5782	−37.4029	1.4142	0.0462	0.6106
$f_5$	−1 000	−1 000.3961	−21.994	11.5758	−36.1201	69.1453	−37.9101	1.2586	0.0041	0.5937
$f_6$	−900	−900.3562	−31.8677	13.59	−34.0754	76.8813	−49.9714	1.218	0.2098	23.8924
$f_7$	−800	−800.3509	−-22.0771	11.5961	−35.183	69.3788	−37.6198	1.8579	0.0369	25.5374
$f_8$	−700	−680.3089	−71.3566	−56.4316	57.2977	−28.3447	−56.5965	1.615	0.0817	161.0824
$f_9$	−600	−599.2936	49.9955	−78.2402	0.1786	32.7815	56.5152	10.4697	1.0503	176.1058
$f_{10}$	−500	−500.2789	−22.3318	11.3539	−34.8006	70.2354	−36.8967	1.5857	0.0497	1.2581
$f_{11}$	−400	−400.4319	−21.9215	11.6111	−36.0095	69.3882	−37.6099	1.822	0.0325	0.1181
$f_{12}$	−300	−300.4235	−21.962	11.6181	−35.9548	69.2314	−37.4736	1.8597	1.006	17.2487
$f_{13}$	−200	−195.7745	−16.5439	0.7856	−63.1612	78.2956	−49.8104	1.8291	3.6078	33.5188
$f_{14}$	−100	−76.8796	93.8116	23.4253	−27.1196	69.3665	−37.609	1.5651	23.1219	100.8436
$f_{15}$	100	162.9651	63.763	73.9758	−59.6899	6.8698	−69.5651	1.6225	6.4142	26.3496
$f_{16}$	200	200.0371	−77.2249	85.4895	−99.6791	9.5552	0.5678	3.677	0.1834	87.4504
$f_{17}$	300	305.138	8.0174	−22.2004	−9.134	41.8141	−7.1107	1.5252	0.4354	7.4541
$f_{18}$	400	406.2551	10.8862	−24.2858	−0.1323	39.2559	−7.5586	1.5572	0.9177	6.6953
$f_{19}$	500	499.7671	−26.9034	2.4016	−45.2641	65.0955	−40.2783	1.3468	0.082	31.6291
$f_{20}$	600	599.6336	−71.294	14.1417	−38.0582	69.6662	−29.0219	1.4203	0.0947	31.6063
$f_{21} $	700	799.646	−48.5365	53.7646	13.7222	69.8263	−18.6266	1.8519	199.9126	158.1048
$f_{22}$	800	1 016.1259	95.0903	−70.2239	−51.7248	71.7396	−52.0955	2.4272	116.5309	203.5029
$f_{23}$	900	1 066.8577	−61.6704	86.9732	42.3647	41.522	−18.9232	2.4921	26.1619	44.4746
$f_{24}$	1 000	1 111.4644	−61.8977	82.0776	36.3207	76.7673	1.768	11.6881	7.1432	36.8097
$f_{25}$	1 100	1 199.6718	−48.5853	53.7148	13.7168	69.7682	−18.5628	11.7015	0.1254	0.1502
$f_{26}$	1 200	1 302.6409	−71.6478	58.3538	19.0587	68.5734	−24.1115	12.4385	71.6616	87.5525
$f_{27}$	1 300	1 599.7342	74.8682	7.5586	60.3873	15.7236	31.6624	12.1106	44.1241	111.1257
$f_{28}$	1 400	1 499.6147	−48.5389	53.7644	13.7202	69.8289	−18.6285	2.8779	200.085	158.1087

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表 12 PMB算法对TF2在$\sigma^2$ = 0.01的噪声环境下得到的全局极值点

Table 12 Global extremum points of TF2 obtained by PMB algorithm in noise environment of $\sigma^2$ = 0.01

TF2	Optimum	f	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$t({\rm{s}})$	$P_f$	$P_l$
f₁	−1 400	−1 400.0699	−21.8287	11.8038	−36.3299	69.5330	−37.6233	12.7743	0.0943	0.5782
f₂	−1 300	−608.3307	−23.1175	11.4652	−34.0409	70.2874	−36.6119	39.0930	675.6766	4.2854
f₃	−1 200	−1 017.2976	−32.7028	11.5533	−35.9896	69.2990	−38.0010	68.3085	182.1889	0.1070
f₄	−1 100	−1 054.7452	−22.0054	11.9548	−36.1731	68.5083	−37.3461	102.1388	43.6737	0.1863
f₅	−1 000	−1 000.2789	−21.9308	11.4701	−36.3134	69.5171	−37.2096	135.1712	0.2791	0.5448
f₆	−900	−900.0006	−22.1408	11.3658	−36.1004	69.2084	−37.6539	12.8520	0.0006	0.3210
f₇	−800	−800.0919	−21.9418	11.6135	−35.8100	69.3635	−37.6909	40.6402	0.1127	0.2376
f₈	−700	−699.7605	−22.1688	11.4203	−36.5141	69.3235	−37.5792	69.8338	0.1586	0.5393
f₉	−600	−600.1833	−21.8968	12.9478	−32.0218	69.5171	−37.6985	215.7291	0.2547	4.2381
f₁₀	−500	−499.9168	−21.3850	11.1694	−35.9969	69.3487	−37.4039	251.9397	0.0671	0.7791
f₁₁	−400	−400.0491	−21.9828	11.4866	−35.7514	69.3762	−37.4913	306.9611	0.5091	0.6597
f₁₂	−300	−299.9505	−21.8073	11.5243	−35.7096	69.1440	−37.0312	358.0750	0.7351	0.7379
f₁₃	−200	−200.0731	−21.8323	11.3919	−36.2855	69.2937	−37.4883	412.7622	0.3003	0.8202
f₁₄	−100	−99.4329	−22.1030	11.5419	−36.1002	69.3419	−37.6012	523.8452	0.3752	0.1612
f₁₅	100	104.4172	−21.7108	11.3782	−36.1444	69.3134	−37.7273	58.0593	6.0214	1.0070
f₁₆	200	200.8154	−22.1002	11.4383	−36.1730	69.4102	−37.6045	126.4925	0.4069	0.2250
f₁₇	300	301.8527	−22.1000	11.6151	−37.4783	69.4012	−37.5859	366.6641	1.4915	0.1345
f₁₈	400	404.3400	−22.2810	12.4912	−38.0133	68.8766	−37.9101	341.7595	2.7665	0.1771
f₁₉	500	500.1091	−22.1404	11.2668	−36.2592	68.9192	−37.9133	318.0817	0.1068	0.2178
f₂₀	600	599.9330	−22.0115	11.5642	−35.9012	69.4102	−37.6413	501.6370	0.0739	0.1041
f₂₁	700	704.7231	−22.1409	11.5481	−35.9884	69.3233	−37.5984	799.2632	4.4867	0.2989
f₂₂	800	843.9745	−22.0029	11.7082	−36.0243	69.3881	−38.1587	1 284.7595	22.6299	0.8884
f₂₃	900	923.2544	−21.9358	11.4092	−36.9182	69.2899	−37.7927	1 405.1109	3.2368	0.8625
f₂₄	1 000	1 002.7871	−22.0554	11.0213	−36.4846	69.5489	−37.5213	1 645.1116	2.7931	0.9464
f₂₅	1 100	1 100.8277	−21.8449	11.4803	−35.9915	69.3188	−37.5424	1 886.3396	3.1550	0.6107
f₂₆	1 200	1 215.0429	−21.8701	11.7852	−37.0089	69.1101	−37.1271	2 148.5756	13.8368	1.1675
f₂₇	1 300	1 379.5470	−22.2113	11.7141	−36.6134	69.4000	−37.2465	2 412.4953	63.3253	0.7665
f₂₈	1 400	1 401.3007	−21.9727	11.5578	−36.0055	69.3242	−37.6351	2 555.0831	2.8957	0.2240

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表 13 PMB算法对TF2中部分函数在无噪声环境和$\sigma^2$ = 0.01的噪声环境下的多极值点优化结果

Table 13 Optimization results of multiple extremum points for some functions of TF2 obtained by PMB algorithm in noiseless environment and noise environment of $\sigma^2$ = 0.01

TF2	Noiseless environment						$\sigma^2$ = 0.01
TF2	$f$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$f$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$P_f$	$P_l$
$f_6$	−900.0000	−22.0309	11.5686	−36.0034	69.4094	−37.6539	−900.0006	−22.1408	11.3658	−36.1004	69.2084	−37.6539	0.0006	0.3210
	−896.0639	−33.8126	44.7393	54.6931	77.1452	−50.0368	−896.0683	−34.4983	44.9367	54.8740	77.5853	−50.9570	0.0044	1.2579
	−899.9794	−28.6913	12.8088	−35.1466	74.6026	−46.3440	−900.2057	−28.8650	12.9076	−35.1465	74.7262	−46.3359	0.2264	0.2351
	−899.9959	−19.0610	10.9852	−36.4042	66.8717	−33.4840	−899.8525	−19.1580	11.2422	−36.3101	66.9289	−32.6954	0.1433	0.8423
	−899.9987	−20.6166	11.3082	−36.2410	68.2617	−35.8258	−899.9635	−20.4386	11.2333	−36.0719	68.0558	−35.3752	0.0353	0.5580
$f_7$	−799.9792	−22.0032	11.5635	−36.0040	69.3839	−37.5808	−800.0919	−21.9418	11.6135	−35.8100	69.3635	−37.6909	0.1127	0.2376
	−799.8756	−28.3647	11.5956	−35.8878	69.3717	−37.6959	−800.0568	−28.2097	11.5979	−35.9973	69.4689	−37.5283	0.1812	0.2712
	−799.7114	−40.2000	11.5118	−35.9099	69.3766	−37.8352	−799.8792	−40.1102	11.5317	−35.9960	69.3786	−37.8348	0.1678	0.1259
	−799.6562	−56.9976	11.5324	−35.9848	69.1641	−38.1372	−799.8114	−55.1614	11.5224	−35.9948	69.1645	−38.1372	0.1552	1.8363
	−799.5697	−66.1517	11.6099	−35.9001	69.4026	−37.9670	−799.6939	−65.2517	11.6094	−35.9078	69.3432	−37.9666	0.1242	0.9020
$f_8$	−699.9191	−22.2758	11.5436	−36.0022	69.3631	−37.6026	−699.7605	−22.1688	11.4203	−36.5141	69.3235	−37.5792	0.1586	0.5393
	−699.9042	−24.1426	11.5593	−36.0070	69.3609	−37.3899	−699.7605	−24.1231	11.5323	−35.9885	69.3697	−37.3910	0.1437	0.0391
	−699.6190	−45.4600	11.5691	−36.0014	84.1102	−20.9700	−699.7605	−45.4382	11.5681	−36.0127	84.1234	−20.9690	0.1415	0.0279
	−699.1928	−36.3628	11.5985	−35.9784	69.3447	−36.1163	−699.7605	−36.4531	11.5900	−35.9804	69.3459	−36.0956	0.5676	0.0931
	−699.4293	−30.7647	11.5746	−35.9425	69.3885	−37.6095	−699.7605	−30.7591	11.5895	−35.9425	69.4115	−37.5944	0.3312	0.0318
$f_{16}$	200.4084	−22.0509	11.5686	−36.0034	69.4094	−37.6539	200.8154	−22.1002	11.4383	−36.1730	69.4102	−37.6045	0.4069	0.2250
	200.0320	−52.1005	−61.6883	−32.3610	−46.5936	1.0443	200.3819	−52.1021	−61.7593	−32.3582	−46.5732	1.1053	0.3499	0.0958
	200.0353	−33.1330	−38.1129	2.5053	−47.4949	37.2437	200.1492	−33.1432	−38.1214	2.4901	−47.4899	37.2348	0.1139	0.0226
	200.0370	−25.6295	23.1822	−3.7438	−74.5210	−22.2897	200.1477	−25.6000	23.1812	−3.7542	−74.5312	−22.2982	0.1107	0.0340
	200.0511	31.4036	49.5090	46.7470	−62.4386	−15.8403	200.0922	31.4125	49.4947	46.7173	−62.4472	−15.8504	0.0411	0.0366
$f_{19}$	500.0023	−22.1995	11.4599	−36.2687	68.9998	−37.9225	500.1091	−22.1404	11.2668	−36.2592	68.9192	−37.9133	0.1068	0.2178
	500.0611	−25.0001	6.8588	−39.2834	63.8559	−42.1153	500.1292	−25.0018	6.8592	−39.2723	63.8337	−42.1010	0.0681	0.0287
	500.0907	−46.4426	−3.0700	−49.8269	56.0704	−50.5283	500.0979	−46.4534	−3.0612	−49.8173	56.1213	−50.5448	0.0072	0.0561
	500.1281	−31.3099	4.2889	−42.9418	58.0552	−46.9003	500.1475	−31.3063	4.2901	−42.9317	58.0661	−46.9169	0.0194	0.0226
	500.1030	−34.7069	−10.5994	−60.2413	54.6492	−60.9458	500.1952	−34.7672	−10.5528	−60.2362	54.6518	−60.9347	0.0922	0.0772
$f_{20}$	600.0068	−22.0109	11.5786	−36.0034	69.4014	−37.6519	599.9330	−22.0115	11.5642	−35.9012	69.4102	−37.6413	0.0739	0.1041
	600.0586	−100.0000	11.0005	−37.8204	70.8742	−25.9073	599.9468	−100.0000	11.1015	−37.8113	70.8513	−25.8984	0.1118	0.1043
	600.0591	−41.9112	12.0157	−38.3612	72.3213	−39.5142	600.0655	−41.8397	12.0277	−38.3525	72.3357	−39.5054	0.0063	0.0749
	600.0671	−25.7643	9.0700	−27.3033	69.5256	−36.3012	600.0348	−25.7528	9.0594	−27.3270	69.5188	−36.3140	0.0323	0.0319
	600.1270	−99.9565	11.5712	−36.7345	62.4002	−37.5823	600.2360	−100.0000	11.5834	−36.7231	62.4028	−37.5947	0.1089	0.0483

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参考文献(46)

[1]	陆志君, 安俊秀, 王鹏. 基于划分的多尺度量子谐振子算法多峰优化. 自动化学报, 2016, 42(2): 235-245 Lu Zhi-Jun, An Jun-Xiu, Wang Peng. Partition-based MQHOA for multimodal optimization. Acta Automatica Sinica, 2016, 42(2): 235-245
[2]	Rakshit P, Konar A. Realization of learning induced self-adaptive sampling in noisy optimization. Applied Soft Computing, 2018, 69: 288-315 doi: 10.1016/j.asoc.2018.04.052
[3]	Rakshit P, Konar A, Das S, Jain L C, Nagar A K. Uncertainty management in differential evolution induced multiobjective optimization in presence of measurement noise. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2014, 44(7): 922-937 doi: 10.1109/TSMC.2013.2282118
[4]	张勇, 巩敦卫, 胡滢, 张建化. 室内噪声环境下气味源的多机器人微粒群搜索方法. 电子学报, 2014, 42(1): 70-76 doi: 10.3969/j.issn.0372-2112.2014.01.011 Zhang Yong, Gong Dun-Wei, Hu Ying, Zhang Jian-Hua. A PSO-based multi-robot search method for odor source in indoor environment with noise. Acta Electronica Sinica, 2014, 42(1): 70-76 doi: 10.3969/j.issn.0372-2112.2014.01.011
[5]	Ariizumi R, Tesch M, Kato K, Choset H, Matsuno F. Multiobjective optimization based on expensive robotic experiments under heteroscedastic noise. IEEE Transactions on Robotics, 2017, 33(2): 468-483 doi: 10.1109/TRO.2016.2632739
[6]	Hong J H, Ryu K R. Simulation-based multimodal optimization of decoy system design using an archived noise-tolerant genetic algorithm. Engineering Applications of Artificial Intelligence, 2017, 65: 230-239 doi: 10.1016/j.engappai.2017.07.026
[7]	Yun R, Singh V P, Dong Z C. Long-term stochastic reservoir operation using a noisy genetic algorithm. Water Resources Management, 2010, 24(12): 3159-3172 doi: 10.1007/s11269-010-9600-5
[8]	de Lope J, Maravall D. Multi-objective dynamic optimization for automatic parallel parking. In: Proceedings of the 10th International Conference on Computer Aided Systems Theory. Las Palmas de Gran Canaria, Spain: Springer, 2005. 513−518
[9]	Hing J, Hart K, Goodman A. Towards autonomous weapons movement on an aircraft carrier: Autonomous swarm parking. In: Proceedings of the 20th International Conference on Human Interface and the Management of Information. Information in Applications and Services. Las Vegas, NV, USA: Springer, 2018. 403−418
[10]	Chang Y, Hao Y, Li C W. Phase dependent and independent frequency identification of weak signals based on duffing oscillator via Particle Swarm Optimization. Circuits, Systems, and Signal Processing, 2014, 33(1): 223-239 doi: 10.1007/s00034-013-9629-9
[11]	Pereira L A M, Papa J P, Coelho A L V, Lima C A M, Pereira D R, de Albuquerque V H C. Automatic identification of epileptic EEG signals through binary magnetic optimization algorithms. Neural Computing and Applications, 2019, 31(2): 1317-1329
[12]	Costa D M D, Paula T I, Silva P A P, Paiva A P. Normal boundary intersection method based on principal components and Taguchi$’$s signal-to-noise ratio applied to the multiobjective optimization of 12L14 free machining steel turning process. The International Journal of Advanced Manufacturing Technology, 2016, 87(1): 825-834
[13]	Nakama T. Theoretical analysis of genetic algorithms in noisy environments based on a Markov model. In: Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation. Atlanta, GA, USA: ACM, 2008. 1001−1008
[14]	李军华, 黎明. 噪声环境下遗传算法的收敛性和收敛速度估计. 电子学报, 2011, 39(8): 1898-1902 Li Jun-Hua, Li Ming. An analysis on convergence and convergence rate estimate of Genetic algorithms in noisy environments. Acta Electronica Sinica, 2011, 39(8): 1898-1902
[15]	Ma H P, Fei M R, Simon D, Chen Z X. Biogeography-based optimization in noisy environments. Transactions of the Institute of Measurement and Control, 2015, 37(2): 190-204 doi: 10.1177/0142331214537015
[16]	Beyer H G, Sendhoff B. Toward a steady-state analysis of an evolution strategy on a robust optimization problem with noise-induced multimodality. IEEE Transactions on Evolutionary Computation, 2017, 21(4): 629-643 doi: 10.1109/TEVC.2017.2668068
[17]	Zhan Z H, Zhang J, Li Y, Shi Y H. Orthogonal learning particle swarm optimization. IEEE Transactions on Evolutionary Computation, 2011, 15(6): 832-847 doi: 10.1109/TEVC.2010.2052054
[18]	李宝磊, 施心陵, 苟常兴, 吕丹桔, 安镇宙, 张榆锋. 多元优化算法及其收敛性分析. 自动化学报, 2015, 41(5): 949-959 Li Bao-Lei, Shi Xin-Ling, Gou Chang-Xing, Lv Dan-Ju, An Zhen-Zhou, Zhang Yu-Feng. Multivariant optimization algorithm and its convergence analysis. Acta Automatica Sinica, 2015, 41(5): 949-959
[19]	董易, 吕丹桔, 王霞, 王耀民, 李鹏, 施心陵. 斐波那契树优化算法全局随机性概率收敛分析. 控制与决策, 2018, 33(3): 439-446 Dong Yi, Lv Dan-Ju, Wang Xia, Wang Yao-Min, Li Peng, Shi Xin-Ling. A global randomness-based probability convergence analysis of Fibonacci tree optimization algorithm. Control and Decision, 2018, 33(3): 439-446
[20]	李军华, 黎明. 噪声环境下多模态函数优化的遗传算法. 电子学报, 2012, 40(2): 327-330 Li Jun-Hua, Li Ming. Genetic algorithm for multi-modal function optimization in noisy environments. Acta Electronica Sinica, 2012, 40(2): 327-330
[21]	黎明, 李军华. 噪声环境下遗传算法的性能评价. 电子学报, 2010, 38(9): 2090-2094 Li Ming, Li Jun-Hua. Performance evaluation of Genetic algorithm in noisy environments. Acta Electronica Sinica, 2010, 38(9): 2090-2094
[22]	李军华, 黎明, 陈昊, 伍家驹. 正态随机噪声环境下遗传算法的动态适应度评价. 电子学报, 2019, 47(3): 649-656 Li Jun-Hua, Li Ming, Chen Hao, Wu Jia-Ju. Dynamic fitness evaluation of Genetic algorithms in normal random noisy environments. Acta Electronica Sinica, 2019, 47(3): 649-656
[23]	Pan H, Wang L, Liu B. Particle swarm optimization for function optimization in noisy environment. Applied Mathematics and Computation, 2006, 181(2): 908-919 doi: 10.1016/j.amc.2006.01.066
[24]	Han L, He X S. A novel opposition-based particle swarm optimization for noisy problems. In: Proceedings of the 3rd International Conference on Natural Computation (ICNC 2007). Haikou, China: IEEE, 2007. 624−629
[25]	Zhang J Q, Xu L W, Li J, et al. Integrating Particle Swarm Optimization with Learning Automata to solve optimization problems in noisy environment. In: Proceedings of the 2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC). San Diego, CA, USA: IEEE, 2014. 1432−1437
[26]	Rada-Vilela J, Johnston M, Zhang M J. Population statistics for particle swarm optimization: Resampling methods in noisy optimization problems. Swarm and Evolutionary Computation, 2014, 17: 37-59 doi: 10.1016/j.swevo.2014.02.004
[27]	Taghiyeh S, Xu J. A new particle swarm optimization algorithm for noisy optimization problems. Swarm Intelligence, 2016, 10(3): 161-192 doi: 10.1007/s11721-016-0125-2
[28]	Zhang S, Xu J, Lee L H, Chew E P, Wong W P, Chen C H. Optimal computing budget allocation for Particle Swarm Optimization in stochastic optimization. IEEE Transactions on Evolutionary Computation, 2017, 21(2): 206-219 doi: 10.1109/TEVC.2016.2592185
[29]	Zhang J Q, Zhu X X, Wang Y H, Zhou M C. Dual-environmental particle swarm optimizer in noisy and noise-free environments. IEEE Transactions on Cybernetics, 2019, 49(6): 2011-2021 doi: 10.1109/TCYB.2018.2817020
[30]	肖辉辉, 万常选, 段艳明, 谭黔林. 基于引力搜索机制的花朵授粉算法. 自动化学报, 2017, 43(4): 576-594 Xiao Hui-Hui, Wan Chang-Xuan, Duan Yan-Ming, Tan Qian-Lin. Flower pollination algorithm based on gravity search mechanism. Acta Automatica Sinica, 2017, 43(4): 576-594
[31]	董易, 施心陵, 王霞, 王耀民, 吕丹桔, 张松海, 等. 斐波那契树优化算法求解多峰函数全局最优解的可达性分析. 自动化学报, 2018, 44(9): 1679-1689 Dong Yi, Shi Xin-Ling, Wang Xia, Wang Yao-Min, Lv Dan-Ju, Zhang Song-Hai, et al. On accessibility of Fibonacci tree optimization algorithm for global optima of multi-modal functions. Acta Automatica Sinica, 2018, 44(9): 1679-1689
[32]	Jamil M, Zepernick H J, Yang X S. Multimodal function optimization using an improved bat algorithm in noise-free and noisy environments. Nature-Inspired Computing and Optimization. Springer, 2017. 29−49
[33]	Rao R V, Kalyankar V D. Multi-pass turning process parameter optimization using teaching-learning-based optimization algorithm. Scientia Iranica, 2013, 20(3): 967-974
[34]	Goldberg D E, Deb K, Clark J H. Genetic algorithms, noise, and the sizing of populations. Complex System, 1992, 6: 333-362
[35]	刘漫丹. 一种新的启发式优化算法——五行环优化算法研究与分析. 自动化学报, 2020, 46(5): 957-970 Liu Man-Dan. Research and analysis of a novel heuristic algorithm: Five-elements cycle optimization algorithm. Acta Automatica Sinica, 2020, 46(5): 957-970
[36]	Cai X J, Gao X Z, Xue Y. Improved bat algorithm with optimal forage strategy and random disturbance strategy. International Journal of Bio-inspired Computation, 2016, 8(4): 205-214 doi: 10.1504/IJBIC.2016.078666
[37]	龙文, 伍铁斌, 唐明珠, 徐明, 蔡绍洪. 基于透镜成像学习策略的灰狼优化算法. 自动化学报, 2020, 46(10): 2148-2164 Long Wen, Wu Tie-Bin, Tang Ming-Zhu, Xu Ming, Cai Shao-Hong. Grey wolf optimizer algorithm based on lens imaging learning strategy. Acta Automatica Sinica, 2020, 46(10): 2148-2164
[38]	徐茂鑫, 张孝顺, 余涛. 迁移蜂群优化算法及其在无功优化中的应用. 自动化学报, 2017, 43(1): 83-93 Xu Mao-Xin, Zhang Xiao-Shun, Yu Tao. Transfer bees optimizer and its application on reactive power optimization. Acta Automatica Sinica, 2017, 43(1): 83-93
[39]	纪霞, 姚晟, 赵鹏. 相对邻域与剪枝策略优化的密度峰值聚类算法. 自动化学报, 2020, 46(3): 562-575 Ji Xia, Yao Shen, Zhao Peng. Relative neighborhood and pruning strategy optimized density peaks clustering algorithm. Acta Automatica Sinica, 2020, 46(3): 562-575
[40]	余伟伟, 谢承旺, 闭应洲, 夏学文, 李雄, 任柯燕, 等. 一种基于自适应模糊支配的高维多目标粒子群算法. 自动化学报, 2018, 44(12): 2278-2289 Yu Wei-Wei, Xie Cheng-Wang, Bi Ying-Zhou, Xia Xue-Wen, Li Xiong, Ren Ke-Yan, et al. Many-objective particle swarm optimization based on adaptive fuzzy dominance. Acta Automatica Sinica, 2018, 44(12): 2278-2289
[41]	Cui Z H, Zhang J J, Wang Y C, Cao Y, Cai X J, Zhang W S, et al. A pigeon-inspired optimization algorithm for many-objective optimization problems. Science China Information Sciences, 2019, 62(7): 70212 doi: 10.1007/s11432-018-9729-5
[42]	陈美蓉, 郭一楠, 巩敦卫, 杨振. 一类新型动态多目标鲁棒进化优化方法. 自动化学报, 2017, 43(11): 2014-2032 Chen Mei-Rong, Guo Yi-Nan, Gong Dun-Wei, Yang Zhen. A novel dynamic multi-objective robust evolutionary optimization method. Acta Automatica Sinica, 2017, 43(11): 2014-2032
[43]	Etminaniesfahani A, Ghanbarzadeh A, Marashi Z. Fibonacci indicator algorithm: A novel tool for complex optimization problems. Engineering Applications of Artificial Intelligence, 2018, 74: 1-9 doi: 10.1016/j.engappai.2018.04.012
[44]	Adam M, Assimakis N, Farina A. Golden section, Fibonacci sequence and the time invariant Kalman and Lainiotis filters. Applied Mathematics and Computation, 2015, 250: 817-831 doi: 10.1016/j.amc.2014.11.022
[45]	王芳芳. 面向单模和多模函数优化的多子群粒子群算法研究 [硕士学位论文], 南京农业大学, 中国, 2014 Wang Fang-Fang. A Study on Multi-Swarm Particle Swarm Optimization for Unimodal and Multi-Modal Function Optimization [Master thesis], Nanjing Agricultural University, China, 2014
[46]	Liang J J, Qu B Y, Suganthan P N, Hernández-Díaz A G. Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization, Technical Report 201212, Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China, Technical Report, Nanyang Technological University, Singapore, 2013.