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

Abstract

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

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.

Figures/Tables 11

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。

References 19

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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

Opposition-based particle swarm optimization with adaptive elite mutation and nonlinear inertia weight

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Figures/Tables 11

References 19

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测试函数	C=0.5	C=1.0	C=1.5	Mut
f₁	0.80	0.13	0.07	6
f₂	0.84	0.10	0.06	7
f₃	0.00	0.00	1.00	4
f₄	0.92	0.05	0.03	2
f₅	0.60	0.13	0.27	6
f₆	0.94	0.04	0.02	4
f₇	0.78	0.10	0.11	3
f₈	0.76	0.20	0.04	9
f₉	0.82	0.04	0.14	2
f ₁₀	0.71	0.17	0.12	5
f ₁₁	0.85	0.09	0.06	8
f ₁₂	0.89	0.06	0.06	1
f ₁₃	0.65	0.24	0.11	5
f ₁₄	0.90	0.06	0.04	9
总体平均概率	0.75	0.10	0.15	5

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