无惯性自适应精英变异反向粒子群忧化算法

doi:10.11959/j.issn.1000-436x.2017165

Abstract

Abstract:

Non-inertia1 opposition-based partic1e swarm optimization with adaptive e1ite mutation（NOPSO）was proposed to overcome the drawbacks，such as，s1ow convergence speed，fa11ing into 1oca1 optimization，of opposition-based partic1e swarm optimization.In addition to increasing the diversity of popu1ation，two mechanisms were introduced to ba1ance the contradiction between exp1oration and exp1oitation during its iterations process.The first one was non-inertia1 ve1ocity（NIV）equation，which aimed to acce1erate the process of convergence of the a1gorithm via better access to and use of environmenta1 information.The second one was adaptive e1ite mutation strategy（AEM），which aimed to avoid trap into 1oca1 optimum.Experimenta1 resu1ts show NOPSO a1gorithm has stronger competitive abi1ity compared with opposition-based partic1e swarm optimizations and its varieties in both ca1cu1ation accuracy and computation cost.

Key words: non-inertia1 ve1ocity equation, genera1ized opposition-based 1earning, adaptive e1ite mutation, partic1e swarm optimization

CLC Number:

TP181

Lan-lan KANG,Wen-yong DONG,Wan-juan SONG,Kang-shun LI. Non-inertial opposition-based particle swarm optimization with adaptive elite mutation[J]. Journal on Communications, 2017, 38(8): 66-78.

Figures/Tables 17

	测试函数	搜索空间	函数名称
	$f_{1} (x) = \sum_{i = 1}^{D} x_{i}^{2}$	-100，100]^D	Sphere
	$f_{2} (x) = \sum_{I = 1}^{D} {(⌊ x_{i} + 0.5 ⌋)}^{2}$	-100，100]^D	Step
单峰函数	$f_{3} (x) = \sum_{i = 1}^{D - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(1 - x_{i})}^{2}]$	-30，30]^D	Rosenbrock
	$f_{4} (x) = \sum_{i = 1}^{D} {(\sum_{j = 1}^{i} x_{j})}^{2}$	-100，100]^D	Quadric
	$f_{5} (x) = \sum_{i = 1}^{D} \| x_{i} \| + \prod_{i = 1}^{D} \| x_{i} \|$	-10，10]^D	Schwefel2.22
	$f_{6} (x) = \sum_{i = 1}^{D} {(10^{6})}^{\frac{i - 1}{D - 1}} x_{i}^{2}$	-100，100]^D	Elliptic
	$f_{7} (x) = f_{6} (z), z = x M$	-5.12，5.12]^D	Rotated Elliptic
	$f_{8} (x) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10]$	-5.12，5.12]^D	Rastrigin
多峰函数	$f_{9} (x) = - 20 \exp (- 0.2 \sqrt{\frac{\sum_{i = 1}^{D} x_{i}^{2}}{D}}) - \exp (\frac{\sum_{i = 1}^{D} \cos (2 π x_{i})}{D}) + 20 + e$	-32，32]^D	Ackley
	$f_{10} (x) = \frac{1}{4 000} \sum_{i = 1}^{D} x_{i}^{2} - \prod_{i = 1}^{D} \cos (\frac{x_{i}}{\sqrt{i}}) + 1$	-600，600]^D	Griewank
	$f_{11} (x) = f_{8} (z), z = x M$	-32，32]^D	Rotated Rastrigin
	$f_{12} (x) = f_{9} (z), z = x M$	-600，600]^D	Rotated Ackley
	$f_{13} (x) = f_{10} (z), z = x M$	-32，32]^D	Rotated Griewank

测试函数	更新式	Mean	Worst	Best	Std Dev	Fes
	NIV-D	4.71×10^?17	9.77×10^?17	2.38×10^?18	2.50×10^?17	74 700
f₁	NIV-U	3.73×10^?17	9.65×10^?17	2.99×10^?20	2.65×10^?17	6 671（最优）
	NIV-R	6.23×10^?9	1.79×10^?8	4.53×10^?10	4.45×10^?9	810 081
	NIV-D	0	0	0	0	2 065
f₂	NIV-U	0	0	0	0	1 811（最优）
	NIV-R	0	0	0	0	1 683
	NIV-D	2.89×10¹	2.90×10¹	2.87×10¹	5.54×10^?2	810 081
f₃	NIV-U	2.84×10¹	2.90×10¹	1.54×10¹	2.43	810 081
	NIV-R	2.17×10^?5（最优）	9.16×10^?5（最优）	1.97×10^?6（最优）	2.05×10^?5（最优）	810 081
	NIV-D	5.25×10^?17	9.42×10^?17	7.18×10^?20	3.07×10^?17	11 126
f₄	NIV-U	5.17×10^?17	9.97×10^?17	3.15×10^?19	3.66×10^?17	8 701（最优）
	NIV-R	5.59×10^?4	5.04×10^?3	4.25×10^?5	8.64×10^?4	810 081
	NIV-D	6.57×10^?17	9.83×10^?17	1.74×10^?17	2.28×10^?17	12 713
f₅	NIV-U	6.36×10^?17	9.95×10^?17	4.71×10^?18	2.62×10^?17	10 985（最优）
	NIV-R	3.83×10^?4	7.59×10^?4	7.43×10^?5	1.49×10^?4	810 081
	NIV-D	4.93×10^?17	9.81×10^?17	8.20×10^?19	3.37×10^?17	8 685
f₆	NIV-U	4.49×10^?17	9.93×10^?17	8.82×10^?19	3.19×10^?17	7 748（最优）
	NIV-R	3.89×10^?4	9.02×10^?4	8.42×10^?5	2.10×10^?4	810 081
	NIV-D	5.29×10^?17	1.00×10^?15	4.85×10^?19	3.17×10^?17	9 576
f₇	NIV-U	4.36×10^?17	9.52×10^?17	2.79×10^?19	3.31×10^?17	8 207（最优）
	NIV-R	2.51×10^?4	7.35×10^?4	3.34×10^?5	1.75×10^?4	810 081
	NIV-D	0	0	0	0	8 156
f₈	NIV-U	0	0	0	0	6 808（最优）
	NIV-R	8.57×10^?7	1.87×10^?6	1.53×10^?7	4.68×10^?7	810 081
	NIV-D	0	0	0	0	12 411
f₉	NIV-U	0	0	0	0	10 999（最优）
	NIV-R	4.88×10^?5	9.52×10^?5	1.10×10^?5	1.84×10^?5	810 081
	NIV-D	0	0	0	0	7 678
f₁₀	NIV-U	0	0	0	0	6 625（最优）
	NIV-R	3.68×10^?10	1.66×10^?9	2.34×10^?11	3.02×10^?10	810 081
	NIV-D	0	0	0	0	8 067
f₁₁	NIV-U	0	0	0	0	6 628（最优）
	NIV-R	1.17×10^?6	3.98×10^?6	1.69×10^?7	9.46×10^?7	810 081
	NIV-D	0	0	0	0	12 576
f₁₂	NIV-U	0	0	0	0	10 710（最优）
	NIV-R	4.80×10^?5	9.36×10^?5	8.19×10^?6	2.03×10^?5	810 081
	NIV-D	0	0	0	0	7 640
f₁₃	NIV-U	0	0	0	0	6 582（最优）
	NIV-R	2.13×10^?10	8.46×10^?10	1.04×10^?11	2.06×10^?10	810 081

References 16

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c₁	c₂	s	jr	λ	N	D
1.496 18	1.496 18	0.2	0.3	10	40	30

测试函数		PSO	OPSO	GOPSO	OVCPSO	EOPSO	NOPSO
	f₁	1.56×10^?3（+）	4.59×10^?36（+）	0（最优）（～）	5.00×10^?1（+）	0.00（～）	0（最优）
	f₂	7.18×10^?2（+）	2.09×10^?35（+）	3.02×10^?321（+）	6.67×10^?2（+）	1.97×10^?323（+）	0（最优）
单峰函数	f₃	1.26（-）	7.18（-）	2.82×10¹（+）	8.70×10¹（+）	1.57×10^?24（-）	3.09
	f₄	5.51×10^?2（+）	4.13×10⁴（+）	1.32×10⁴（+）	2.94×10¹（+）	5.01×10^?3（+）	0（最优）
	f₅	-1.48×10^?3（+）	-6.51×10^?12（+）	-6.98×10^?162（+）	-2.49（+）	3.01×10^?162（+）	0（最优）
	f₆	8.84（+）	1.43×10^?32（+）	4.21×10^?316（最优）（+）	1.39×10^?4（+）	9.88×10^?324（+）	0（最优）
	f₇	4.07×10²（+）	9.82×10⁶（+）	1.96×10⁶（最优）（+）	3.06（+）	5.08×10²（+）	0（最优）
	f₈	2.06（+）	1.51×10¹（+）	0（最优）（～）	7.57×10^?1（+）	2.09（+）	0（最优）
多峰函数	f₉	4.45×10^?2（+）	1.85（+）	0（最优）（～）	9.99×10^?1（+）	1.86×10^?1（+）	0（最优）
	f₁₀	1.14×10^?3（+）	3.83×10¹（+）	0（最优）（～）	2.05×10^?2（+）	1.26×10^?2（+）	0（最优）
	f₁₁	2.99（+）	1.51×10¹（+）	0（最优）（～）	4.14×10¹（+）	4.11（+）	0（最优）
	f₁₂	2.81×10^?2（+）	2.98（+）	0（最优）（～）	1.97（+）	3.41×10^?1（+）	0（最优）
	f₁₃	7.95×10^?4（+）	2.33×10^?2（+）	0（最优）（～）	6.14×10^?1（+）	1.22×10^?2（+）	0（最优）
	+	12	12	6	13	11	—
总计	～	0	0	7	0	1	—
	-	1	1	0	0	1	—

性能指标	GOPSO	NOPSO
Mean	2.82×10¹	3.09（最优）
Std Dev	1.36（最优）	8.61
Best	2.46×10¹	2.25×10^?4
Worst	2.90×10¹	2.89×10¹

测试函数	算法	Mean	Worst	Best	Std Dev
	PSO	1.77×10^?2	5.32×10^?1	0	9.55×10^?2
	PSOM	8.66×10^?3	2.60×10^?1	0	4.67×10^?2
f₁	PSOL	5.37×10³	1.57×10⁴	1.64×10³	2.68×10³
	DPSO	7.13×10^?3	2.14×10^?1	0	3.84×10^?2
	NOPSO	0	0	0	0
	PSO	6.67×10^?2	2.00	0	3.59×10^?1
	PSOM	3.33×10^?2	1.00	0	1.80×10^?1
f₂	PSOL	4.98×10³	1.32×10⁴	9.26×10²	2.65×10³
	DPSO	3.33×10^?2	1.00	0	1.80×10^?1
	NOPSO	0	0	0	0
	PSO	4.81	1.44×10²	0	2.59×10¹
	PSOM	4.10	1.23×10²	0	2.21×10¹
f₃	PSOL	2.06×10⁶	9.47×10⁶	2.13×10⁵	2.32×10⁶
	DPSO	2.51	7.54×10¹	0	1.35×10¹
	NOPSO	2.89×10¹	2.90×10¹	2.88×10¹	4.96×10^?2
	PSO	2.25×10^?1	6.76	0	1.21
	PSOM	1.96×10^?1	5.89	0	1.06
f₄	PSOL	1.83×10⁴	4.76×10⁴	3.10×10³	1.05×10⁴
	DPSO	2.80×10^?1	8.39	0	1.51
	NOPSO	0	0	0	0
	PSO	1.09×10^?1	3.27	0	5.86×10^?1
	PSOM	1.12×10^?1	3.35	0	6.02×10^?1
f₅	PSOL	2.27×10¹	5.39×10¹	9.20	1.05×10¹
	DPSO	1.67×10^?1	5.01	0	8.99×10^?1
	NOPSO	0	0	0	0
	PSO	8.84	2.65×10²	0	4.76×10¹
	PSOM	7.34×10¹	2.20×10³	0	3.95×10²
f₆	PSOL	2.90×10⁷	7.91×10⁷	5.94×10⁶	1.85×10⁷
	DPSO	5.30	1.59×10²	0	2.86×10¹
	NOPSO	0	0	0	0
	PSO	4.07×10²	1.22×10⁴	0	2.19×10³
	PSOM	5.32×10¹	1.60×10³	0	2.87×10²
f₇	PSOL	3.70×10⁷	1.13×10⁸	3.74×10⁶	2.76×10⁷
	DPSO	3.36×10²	1.01×10⁴	0	1.81×10³
	NOPSO	0	0	0	0
	PSO	2.93	8.78×10¹	0	1.58×10¹
	PSOM	3.07	9.22×10¹	0	1.66×10¹
f₈	PSOL	1.47×10²	2.04×10²	9.80×10¹	2.74×10¹
	DPSO	2.50	7.51×10¹	0	1.35×10¹
	NOPSO	0	0	0	0
	PSO	5.47×10^?2	1.64	0	2.95×10^?1
	PSOM	6.25×10^?2	1.88	0	3.37×10^?1
f₉	PSOL	1.21×10¹	1.63×10¹	8.93	1.93
	DPSO	5.63×10^?2	1.69	0	3.03×10^?1
	NOPSO	0	0	0	0

测试函数	算法	Mean	Worst	Best	Std Dev
	PSO	1.05×10^?3	3.14×10^?2	0	5.64×10^?3
	PSOM	1.62×10^?3	4.86×10^?2	0	8.72×10^?3
f₁₀	PSOL	3.62×10¹	7.14×10¹	9.25	1.90×10¹
	DPSO	9.11×10^?4	2.73×10^?2	0	4.91×10^?3
	NOPSO	0	0	0	0
	PSO	2.47	7.41×10¹	0	1.33×10¹
	PSOM	2.58	7.74×10¹	0	1.39×10¹
f₁₁	PSOL	1.42×10²	1.82×10²	8.07×10¹	2.49×10¹
	DPSO	3.10	9.29×10¹	0	1.67×10¹
	NOPSO	0	0	0	0
	PSO	4.70×10^?2	1.41	0	2.53×10^?1
	PSOM	4.78×10^?2	1.43	0	2.58×10^?1
f₁₂	PSOL	1.24×10¹	1.64×10¹	6.83	2.15
	DPSO	5.27×10^?2	1.58	0	2.84×10^?1
	NOPSO	0	0	0	0
	PSO	1.06×10^?3	3.16×10^?2	0	5.68×10^?3
	PSOM	1.08×10^?3	3.22×10^?2	0	5.79×10^?3
f₁₃	PSOL	3.73×10¹	8.90×10¹	1.33×10¹	1.95×10¹
	DPSO	1.03×10^?3	3.10×10^?2	0	5.56×10^?3
	NOPSO	0	0	0	0

Non-inertial opposition-based particle swarm optimization with adaptive elite mutation

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测试函数	C=0.5	C=1.0	C=1.5	变异位占优次数
f₁	0.82	0.11	0.07	6
f₂	0.97	0.03	0.00	5
f₃	0.00	0.00	1.00	4
f₄	0.92	0.05	0.03	2
f₅	0.83	0.07	0.09	2
f₆	0.84	0.08	0.08	5
f₇	0.79	0.09	0.12	6
f₈	0.94	0.04	0.02	1
f₉	0.78	0.10	0.11	3
f₁₀	0.83	0.09	0.08	9
f₁₁	0.92	0.04	0.04	2
f₁₂	0.78	0 .10	0.12	2
f₁₃	0.85	0.09	0.06	8
总体平均概率	0.79	0.07	0.14	4.23

组合	c₁	c₂	s	平均秩	排名
1	1.0	1.0	0.10	1.83	1
2	1.0	1.5	0.15	2.25	2
3	1.0	2.0	0.20	5.92	6
4	1.5	1.0	0.15	3.83	3
5	1.5	1.5	0.20	4.08	4
6	1.5	2.0	0.10	7.92	8
7	2.0	1.0	0.20	4.53	5
8	2.0	1.5	0.10	6.75	7
9	2.0	2.0	0.15	9.00	9

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