带自适应精英扰动及惯性权重的反向粒子群优化算法

doi:10.11959/j.issn.1000-436x.2016224

摘要/Abstract

摘要：

针对反向粒子群优化算法存在的易陷入局部最优、计算开销大等问题，提出了一种带自适应精英粒子变异及非线性惯性权重的反向粒子群优化算法（OPSO-AEM＆NIW），来克服该算法的不足。OPSO-AEM＆NIW算法在一般性反向学习方法的基础上，利用粒子适应度比重等信息，引入了非线性的自适应惯性权重（NIW）调整各个粒子的活跃程度，继而加速算法的收敛过程。为避免粒子陷入局部最优解而导致搜索停滞现象的发生，提出了自适应精英变异策略（AEM）来增大搜索范围，结合精英粒子的反向搜索能力，达到跳出局部最优解的目的。上述2种机制的结合，可以有效克服反向粒子群算法的探索与开发的矛盾。实验结果表明，与主流反向粒子群优化算法相比，OPSO-AEM＆NIW算法无论是在计算精度还是计算开销上均具有较强的竞争能力。

关键词: 一般性反向学习, 粒子群优化, 自适应精英变异, 非线性惯性权重

Abstract:

An opposition-based particle swarm optimization with adaptive elite mutation and nonlinear inertia weight (OPSO-AEM＆NIW) was proposed to overcome the drawbacks, such as falling into local optimization, slow convergence speed of opposition-based particle swarm optimization. Two strategies were introduced to balance the contradiction be-tween exploration and exploitation during its iterations process. The first one was nonlinear adaptive inertia weight (NIW), which aim to accelerate the process of convergence of the algorithm by adjusting the active degree of each parti-cle using relative information such as particle fitness proportion. The second one was adaptive elite mutation strategy (AEM), which aim to avoid algorithm trap into local optimum by trigging particle's activity. Experimental results show OPSO-AEM＆NIW algorithm has stronger competitive ability compared with opposition-based particle swarm optimiza-tions and its varieties in both calculation accuracy and computation cost.

Key words: generalized opposition-based learning, particle swarm optimization, adaptive elite mutation, nonlinear inertia weight

董文永,康岚兰,刘宇航,李康顺. 带自适应精英扰动及惯性权重的反向粒子群优化算法[J]. 通信学报, 2016, 37(12): 1-10.

Wen-yong DONG,Lan-lan KANG,Yu-hang LIU,Kang-shun LI. Opposition-based particle swarm optimization with adaptive elite mutation and nonlinear inertia weight[J]. Journal on Communications, 2016, 37(12): 1-10.

图/表 11

图1

表1

表2

数值实验中使用的14个测试函数"

属性	测试函数	维度D	搜索区间	全局最优值	函数名
	$f_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$	30	[-100, 100]^D	0	Sphere
单峰函数	$f_{2} (x) = \sum_{i = 1}^{D} {(⌊ x_{i} + 0.5 ⌋)}^{2}$ $f_{3} (x) = \sum_{i = 1}^{D - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(1 - x_{i})}^{2}]$ $f_{4} (x) = \sum_{i = 1}^{D} {(\sum_{j = 1}^{i} x_{j})}^{2}$	303030	[-100, 100]^D[-30, 30]^D[-100, 100]^D	000	StepRosenbrockQuadric
	$f_{5} (x) = \sum_{i = 1}^{D} \| x_{i} \| + \prod_{i = 1}^{D} \| x_{i} \|$	30	[-10, 10]^D	0	S chwefel2.22
	$f_{6} (x) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10]$	30	[-5.12, 5.12]^D	0	Rastrigin
	$\begin{array}{l} f_{7} (x) = - 20 \exp (- 0.2 \sqrt{\frac{\sum_{i = 1}^{D} x_{i}^{2}}{D}} - \\ \exp \frac{\sum_{i = 1}^{D} \cos (2 π x_{i})}{D} + 20 + e \end{array}$	30	[-32, 32]^D	0	Ackley
多峰函数	$f_{8} (x) = \frac{1}{4000} \sum_{i = 1}^{D} x_{i}^{2} - \prod_{i = 1}^{D} \cos (\frac{x_{i}}{\sqrt{i}}) + 1$	30	[-600, 600]^D	0	Griewank
多峰函数	f₉ (x)= f₆ (z), z=xM	30	[-5.12, 5.12]^D	0	Rotated Rastrigin
	f₁₀ (x)= f₇ (z), z=xM	30	[-32, 32]^D	0	Rotated Ackley
	f₁₁ (x)= f₈ (z), z=xM	30	[-600, 600]^D	0	Rotated Griewank
	f₁₂ (x)= f₆ (z), z=x-o	30	[-5.12, 5.12]^D	0	Shifted Rastrigin
	f₁₃ (x)= f₇ (z), z=x-o	30	[-32, 32]^D	0	Shifted Ackley
	f₁₄ (x)= f₈ (z), z=x-o	30	[-600, 600]^D	0	Shifted Griewank
注：M为DXD正交矩阵，o为移位向量。

表2

表3

表4

表5

图2

表6

图3

表7

图4

参考文献 19

[1]	KENNEDY J , EBERHART R C . Particle swarm optimization[C]// IEEE International Conference on Neural Networks. Perth, Australia, 1995:1942-1948.
[2]	史霄波，张引，赵杉，等．基于离散多目标优化粒子群算法的多移动代理协作规划[J]．通信学报， 2016,37(6):29-37. SHI X B , ZHANG Y , ZHAO S , et al. Discrete multi-objective optimi-zation of particle swarm optimizer algorithm for multi-agents collabo-rative planning[J]. Journal on Communications, 2016,37(6):29-37.
[3]	INBARANI H H , AZAR A T , JOTHI G . Supervised hybrid feature selection based on PSO and rough sets for medical diagnosis[J]. Computer Methods and Programs in Biomedicine, 2014,113 (1):175-185.
[4]	ZAD B B , HASANVAND H , LOBRY J , et al. Optimal reactive power control of DGs for voltage regulation of MV distribution systems using sensitivity analysis method and PSO algorithm[J]. International Journal of Electrical Power and Energy System, 2015,68:52-60.
[5]	SHI Y , EBERHART R C . A modified particle swarm optimizer[C]// The IEEE Congress on Evolutionary Computation (CEC 1998). 1998:69-73.
[6]	TIZHOOSH H R . Opposition-based learning: a new scheme for ma-chine intelligence[C]// The IEEE International Conference of Intelli-gent for Modeling, Control and Automation. PiscatNIWay: Inst of Elec. and Elec Eng Computer Society. 2005:695-701.
[7]	WANG H , LI H , LIU Y , et al. Opposition-based particle swarm algo-rithm with Cauchy mutation[C]// The IEEE Congress on Evolutionary Computation. 2007:356-360.
[8]	WANG H , WU Z J , RAHNAMAYAN S , et al. Enhancing particle swarm optimization using generalized opposition-based learning[J]. Information Sciences, 2011,181:4699-4714.
[9]	周新宇，吴志健，王晖，等．一种精英反向学习的粒子群优化算法[J]．电子学报， 2013,41(8):1647-1652. ZHOU X Y , WU Z J , WANG H , et al. Elite opposition-based par-ticle swarm optimization[J]. Acta Electronica Sinica, 2013,41(8):1647-1652.
[10]	SHAHZAD F , BAIG A R , MASOOD S , et al. Opposition-based parti-cle swarm optimization with velocity clamping (OVCPSO)[J]. Ad-vances in Computational Intell, 2009,339:348-60.
[11]	KAUCIC M . A multi-start opposition-based particle swarm optimiza-tion algorithm with adaptive velocity for bound constrained global op-timization[J]. J Glob Optim, 2013,55:165-188.
[12]	PEHLIVANOGLU Y V . A new particle swarm optimization method enhanced with a periodic mutation strategy and neural networks[J]. IEEE Trans Evol Comput, 2013,17:436-452.
[13]	KARAFOTIAS G , HOOGENDOORN M , EIBEN A E . Parameter control in evolutionary algorithms: trends and challenges[J]. IEEE Transactions on Evolutionary Computation, 2015,19:167-187.
[14]	汪慎文，丁立新，谢承旺，等．应用精英反向学习策略的混合差分演化算法[M]．武汉大学学报(理学版)， 2013,59(2):111-116. WANG S W , DING L X , XIE C W , et al. A hybrid differential evolu-tion with elite opposition-based learning[J]. Journal of Wuhan Uni-versity, 2013,59(2):111-116.
[15]	OZCAN E , MOHAN C K . particle swarm optimization: surfing and waves[C]// Congress on Evolutionary Computation (CEC1999). 1999:1939-1944.
[16]	龚纯，王正林．精通MATLAB最优化计算[M]．电子工业出版社， 2012:283-285. GONG C , WANG Z L . Proficient optimization calculation in MAT-LAB[M]. Electronic Industry Press, 2012:283-285.
[17]	FRANS V D B . An analysis of particle swarm optimizers[D]. Depart-ment of Computer Science, University of Pretoria, South Africa, 2002.
[18]	TANG K , LI X D , SUGANTHAN P N , et al. Benchmark functions for the CEC'2010 special session and competition on large-scale global optimization[R]. Nature Inspired Computation and Applications Laboratory, USTC, China, 2009,21.
[19]	BERGH F , ENGELBRECHT A P . Effect of swarm size on cooperative particle swarm optimizers[C]// Genetic and Evolutionary Computation Conference. 2001:892-899.

c1	c2	w_max	w_min	jr	λ
1.496 18	1.496 18	0.5	0.2	0.3	30

测试函数		PSO		OPSO		GOPSO		OVCPSO		EOPSO		OPSO-AEM＆NIW
f₁		1.56×10^–3	+	4.59×10^–36	+	0	～	5.00×10^–1	+	0	～	0
f₂		7.18×10^–2	+	2.09×10^–35	+	3.02×10^–321	+	6.67×10^–2	+	1.97×10^–323	+	0
f₃		1.26	～	7.18	～	2.82×10¹	+	8.70×10¹	+	1.57×10^–24	－	6.94
f₄		5.51×10^–2	+	4.13×10⁴	+	1.32×10⁴	+	2.94×10¹	+	5.01×10^–3	+	0
f₅		–1.48×10^–3	+	–6.51×10^–12	+	-6.98×10^–162	+	–2.49	+	3.01×10^–162	+	–9.88×10^–324
f₆		2.06	+	1.51×10¹	+	0	～	7.57×10¹	+	2.09	+	0
f₇		4.45×10^–2	+	1.85	+	0	～	9.99×10^–1	+	1.86×10^–1	+	0
f₈		1.14×10^–3	+	3.83×10^–1	+	0	～	2.05×10^–2	+	1.26×10^–2	+	0
f₉		2.99	+	1.51×10¹	+	0	～	4.14×10¹	+	4.11	+	0
f₁₀		2.81×10^–2	+	2.98	+	0	－	1.97	+	3.41×10^–1	+	6.63×10^–15
f₁₁		7.95×10^–4	+	2.33×10^–2	+	0	～	6.14×10^–1	+	1.22×10^–2	+	0
f₁₂		4.68	+	1.35×10¹	+	0	～	4.07	+	1.76	+	0
f₁₃		4.25×10^–1	+	2.98	+	0	－	2.06×10¹	+	1.34×10^–1	+	3.55×10^–15
f₁₄		7.69	+	1.95×10^–2	+	0	～	1.02×10³	+	1.44×10^–2	+	0
	+	13		13		4		14		12		—
总计	～	1		1		8		0		1		—
	－	0		0		2		0		1		—

		GOPSO			OPSO-AEM＆NIW
测试函数
测试函数	标准方差	最佳值	最差值	标准方差	最佳值	最差值
f₁	0	0	7.41×10^–323	0	0	0
f₂	0	0	9.06×10^–320	0	0	0
f₃	1.36	2.46×10¹	2.90×10¹	1.10×10¹	2.92×10–3	2.89×10¹
f₃	2.02×10⁴	1.37×10^–100	6.01×10⁴	0	0	9.88×10^–324
f₅	2.91×10^–161	2.38×10^–175	1.56×10^–160	0	0	8.89×10^–323
f₆	0	0	0	0	0	0
f₇	0	0	0	0	0	0
f₈	0	0	0	0	0	0
f₉	0	0	0	0	0	0
f ₁₀	0	0	0	1.09×10^–14	0	4.62×10^–14
f ₁₁	0	0	0	0	0	0
f ₁₂	0	0	0	0	0	0
f ₁₃	0	0	0	1.83×10^–15	0	7.11×10^–15
f ₁₄	0	0	0	0	0	0

测试函数	GOPSO		OPSO-AEM＆NIW
测试函数	适应值	评估次数	适应值	评估次数
f₁	9.31×10^–17	47 556	4.27×10^–17	10 377
f₂	0	11 466	0	2 846
f₃	2.38×10¹	400 080	1.74	400 080
f₃	9.97×10^–17	254 778	3.18×10^–17	14 201
f₅	9.87×10^–17	85 230	5.96×10^–17	18 634
f₆	0	98 481	0	10 195
f₇	0	93 746	0	15 019
f₈	0	43 388	0	10 827
f₉	0	125 458	0	9 850
f ₁₀	0	79 714	3.55×10^–15	817 901
f ₁₁	0	47 401	0	10 965
f ₁₂	0	51 078	0	8 993
f ₁₃	0	89 212	3.55×10^–15	822 750
f ₁₄	0	47 886	0	10 083

测试函数	OPSO		OPSO-AEM＆NIW
测试函数	mean	MNLO	mean	MNLO
f₁	4.59×10^–36	9.09×10²	0	7.23
f₂	2.09×10^–35	3.37×10¹	0	7.43
f₃	7.18	8.29×10²	6.94	5.97
f₄	4.13×10⁴	2.67×10–1	0	1.07
f₅	–6.51×10^–12	7.41×10²	–9.88×10^–324	1.07
f₆	1.51×10¹	3.86×10²	0	6.33×10^–1
f₇	1.85	3.20×10²	0	3.60
f₈	3.83×10^–1	3.91×10²	0	9.87
f₉	1.51×10¹	3.91×10²	0	3.33×10^–1
f ₁₀	2.98	3.30×10²	6.63×10^–15	3.33
f ₁₁	2.33×10^–2	3.98×10²	0	1.21×10¹
f ₁₂	1.35×10¹	5.68×10²	0	2.12
f ₁₃	2.98	2.11×10²	3.55×10^–15	1.13
f ₁₄	1.95×10^–2	2.07×10²	0	3.41×10¹