一種正交反向?qū)W習(xí)螢火蟲(chóng)算法

周凌云; 丁立新; 馬懋德; 唐菀

doi:10.11999/JEIT180187

一種正交反向?qū)W習(xí)螢火蟲(chóng)算法

doi: 10.11999/JEIT180187 cstr: 32379.14.JEIT180187

周凌云^{1, 2},
丁立新^1, ,,
馬懋德^{2, 3},
唐菀²

1.
武漢大學(xué)計(jì)算機(jī)學(xué)院 ??武漢 ??430072
2.
中南民族大學(xué)計(jì)算機(jī)科學(xué)學(xué)院 ??武漢 ??430074
3.
南洋理工大學(xué)電氣電子工程學(xué)院 ??新加坡 ??639798

基金項(xiàng)目: 國(guó)家自然科學(xué)基金(61379059)，中南民族大學(xué)中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金(CZY18012)

詳細(xì)信息

作者簡(jiǎn)介:
周凌云：女，1979年生，講師，研究方向?yàn)橛?jì)算智能

丁立新：男，1967年生，教授，研究方向?yàn)橛?jì)算智能與機(jī)器學(xué)習(xí)

馬懋德：男，1957年生，教授，研究方向主要為無(wú)線網(wǎng)絡(luò)

唐菀：女，1974年生，教授，研究方向主要為數(shù)據(jù)中心網(wǎng)絡(luò)

通訊作者:
丁立新　lxding@whu.edu.cn

中圖分類號(hào): TP301.6
計(jì)量
- 文章訪問(wèn)數(shù): 1904
- HTML全文瀏覽量: 1005
- PDF下載量: 119
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2018-02-10
- 修回日期: 2018-08-23
- 網(wǎng)絡(luò)出版日期: 2018-08-29
- 刊出日期: 2019-01-01

Orthogonal Opposition Based Firefly Algorithm

Lingyun ZHOU^{1, 2},
Lixin DING^{1
, ,},
Maode MA^{2, 3},
Wan TANG²

1.
Computer School, Wuhan University, Wuhan 430072, China
2.
College of Computer Science, South-Central University for Nationalities, Wuhan 430074, China
3.
School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore

Funds: The National Natural Science Foundation of China (61379059), The Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (CZY18012)

摘要

摘要:
針對(duì)螢火蟲(chóng)算法求解復(fù)雜優(yōu)化問(wèn)題時(shí)收斂精度較低的問(wèn)題，該文提出一種正交反向?qū)W習(xí)策略，嵌入螢火蟲(chóng)算法，得到一種正交反向?qū)W習(xí)螢火蟲(chóng)算法。正交反向?qū)W習(xí)策略中，采用重心反向計(jì)算，利用群體搜索經(jīng)驗(yàn)的同時(shí)避免搜索依賴坐標(biāo)；采用正交試驗(yàn)設(shè)計(jì)，構(gòu)建部分維上取反向值的正交反向候選解，充分挖掘個(gè)體和反向個(gè)體在不同維度上的有利信息。在標(biāo)準(zhǔn)測(cè)試集上進(jìn)行驗(yàn)證，實(shí)驗(yàn)結(jié)果說(shuō)明了正交反向?qū)W習(xí)策略的有效性。與多種新近的改進(jìn)螢火蟲(chóng)算法相比，該算法在大多數(shù)函數(shù)上獲得更高的求解精度。
- 螢火蟲(chóng)算法 /
- 反向?qū)W習(xí) /
- 優(yōu)化 /
- 正交試驗(yàn)設(shè)計(jì) /
- 收斂精度
Abstract:
Firefly Algorithm (FA) may suffer from the defect of low convergence accuracy depending on the complexity of the optimization problem. To overcome the drawback, a novel learning strategy named Orthogonal Opposition Based Learning (OOBL) is proposed and integrated into FA. In OOBL, first, the opposite is calculated by the centroid opposition, making full use of the population search experience and avoiding depending on the system of coordinates. Second, the orthogonal opposite candidate solutions are constructed by orthogonal experiment design, combining the useful information from the individual and its opposite. The proposed algorithm is tested on the standard benchmark suite and compared with some recently introduced FA variants. The experimental results verify the effectiveness of OOBL and show the outstanding convergence accuracy of the proposed algorithm on most of the test functions.
- Firefly algorithm /
- Opposition-based learning /
- Optimization /
- Orthogonal experimental design /
- Convergence accuracy

HTML全文

圖 1 試驗(yàn)解的構(gòu)建

下載: 全尺寸圖片幻燈片

圖 2 FA與OOFA在函數(shù)f₁, f₄, f₉和f₁₂上的收斂曲線

下載: 全尺寸圖片幻燈片

表 1 算法1：OOBL策略

輸入：種群X，一個(gè)個(gè)體的索引ind和正交表L；
輸出：新種群X。
步驟：
(1) 根據(jù)式(5)計(jì)算當(dāng)前種群重心G；
(2) 根據(jù)式(6)計(jì)算指定個(gè)體的反向個(gè)體ox；
(3) 根據(jù)式(7)更新群體邊界；根據(jù)式(8)對(duì)ox進(jìn)行邊界檢查；
(4) for i=1: L的行數(shù)M
(5) 　for j=1: 問(wèn)題維數(shù)D
(6)　　 if L(i, j)==1
(7)　　　oox(i, j)=X(ind, j)；
(8) 　　else
(9)　　　 oox(i, j)=ox( j)；
(10)　　end if
(11)　end for
(12) end for
(13) 評(píng)估正交反向候選解，評(píng)估次數(shù)FEs=FEs+M–1；
(14) 從X和正交反向候選解中選出適應(yīng)值最優(yōu)的N個(gè)個(gè)體。

下載: 導(dǎo)出CSV

表 2 算法2：OOFA算法

輸入：目標(biāo)函數(shù)；
輸出：全局最優(yōu)位置及適應(yīng)值。
步驟：
(1) 隨機(jī)初始化有N個(gè)個(gè)體的種群X；
(2) 評(píng)估初始種群f(X)，當(dāng)前函數(shù)評(píng)估次數(shù)FEs=N；
(3) 根據(jù)種群適應(yīng)值排序；
(4) 根據(jù)函數(shù)維數(shù)D，生成2水平D因素的正交表L；
(5) while 未達(dá)迭代終止條件
(6)　　 for i=1:N
(7)　　　 for j=1: i
(8) 　　　　根據(jù)式(1)和式(2)，第i個(gè)個(gè)體向第j個(gè)個(gè)體移位；
(9) 　　　end for
(10) 　end for
(11) 　對(duì)種群進(jìn)行邊界檢查；
(12) 　評(píng)估種群，函數(shù)評(píng)估次數(shù)FEs=FEs+N；
(13) 　隨機(jī)選擇群體中一個(gè)個(gè)體，執(zhí)行OOBL；
(14)　根據(jù)式(3)更新步長(zhǎng)因子；
(15) end while

下載: 導(dǎo)出CSV

表 3 FA與OOFA的比較結(jié)果

函數(shù)	FA				OOFA				加速比R
函數(shù)	Mean	SD	FEs	T (s)	Mean	SD	FEs	T (s)	加速比R
f₁	5.16E+04	6.35E+03	80232	3.85	1.70E–10	8.44E–10	1091	0.03	73.54
f₂	7.30E+08	2.15E+08	72314	3.82	3.39E+07	1.04E+07	1704	0.06	42.44
f₃	4.74E+16	1.43E+17	16662	0.94	1.61E+09	8.63E+08	616	0.02	27.05
f₄	9.36E+04	1.29E+04	37973	1.94	7.25E+04	1.22E+04	5669	0.17	6.70
f₅	1.58E+04	3.90E+03	44544	2.25	1.06E+02	3.32E+01	1421	0.04	31.35
f₆	8.71E+03	2.01E+03	36463	1.85	6.90E+01	2.47E+01	904	0.03	40.34
f₇	1.08E+05	1.13E+05	32322	2.11	4.11E+01	1.12E+01	1082	0.05	29.87
f₈	2.10E+01	5.29E–02	82741	4.87	2.10E+01	6.09E–02	85977	3.23	0.96
f₉	4.28E+01	1.42E+00	64643	18.07	2.14E+01	4.18E+00	4178	1.08	15.47
f₁₀	6.79E+03	1.12E+03	72373	3.95	1.67E+01	1.19E+01	1094	0.04	66.15
f₁₁	8.38E+02	9.77E+01	52407	2.85	1.13E+01	3.35E+00	989	0.03	52.99
f₁₂	8.43E+02	9.43E+01	56517	3.51	1.05E+02	3.38E+01	1133	0.05	49.88
f₁₃	8.17E+02	9.91E+01	60595	3.89	8.64E+01	3.04E+01	1304	0.06	46.47
f₁₄	8.10E+03	3.23E+02	80418	4.48	1.55E+03	5.45E+02	5359	0.19	15.01
f₁₅	8.07E+03	3.59E+02	72859	4.24	4.41E+03	7.63E+02	5598	0.21	13.02
f₁₆	3.19E+00	5.04E–01	57793	11.01	1.79E+00	5.38E–01	10998	1.86	5.25
f₁₇	1.40E+03	1.91E+02	56622	2.95	4.09E+01	3.16E+00	1955	0.06	28.96
f₁₈	1.44E+03	1.68E+02	52612	3.01	1.06E+02	2.98E+01	1484	0.05	35.45
f₁₉	1.04E+06	4.38E+05	52422	2.75	3.34E+00	7.17E–01	1230	0.04	42.62
f₂₀	1.50E+01	1.43E–05	8888	0.51	1.50E+01	3.14E–05	962	0.04	9.24
f₂₁	3.57E+03	1.80E+02	64395	5.24	2.92E+02	2.75E+01	1901	0.12	33.87
f₂₂	8.87E+03	3.07E+02	61069	4.75	1.91E+03	1.25E+03	4629	0.27	13.19
f₂₃	9.05E+03	4.02E+02	49134	4.21	5.61E+03	1.21E+03	4437	0.29	11.07
f₂₄	4.15E+02	2.78E+01	32295	10.01	2.17E+02	6.37E+00	652	0.19	49.53
f₂₅	4.09E+02	1.66E+01	44404	13.74	2.70E+02	1.38E+01	728	0.21	60.99
f₂₆	3.08E+02	4.83E+01	64359	20.94	2.95E+02	2.03E+01	41667	12.63	1.54
f₂₇	1.64E+03	5.93E+01	32762	10.50	4.51E+02	6.27E+01	955	0.29	34.31
f₂₈	6.25E+03	5.82E+02	48551	4.94	3.00E+02	6.06E–03	1499	0.12	32.39

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表 4 各FA變種算法結(jié)果的比較(Mean±SD)

函數(shù)	MFA	VSSFA	OFA	RaFA	ODFA	OOFA
f₁	3.58E+04±5.83E+03	3.45E+04±4.31E+03	2.42E+04±5.28E+03	4.03E+02±7.35E+02	1.32E+04±5.63E+03	1.70E–10±8.44E–10
f₂	5.03E+08±1.67E+08	3.73E+08±7.44E+07	3.34E+08±1.65E+08	3.71E+07±2.17E+07	2.19E+08±8.17E+07	3.39E+07±1.04E+07
f₃	1.47E+15±2.87E+15	1.68E+13±2.15E+13	2.01E+14±5.77E+14	2.62E+10±1.55E+10	6.04E+14±2.98E+15	1.61E+09±8.63E+08
f₄	9.27E+04±1.08E+04	8.04E+04±8.81E+03	8.68E+04±1.27E+04	1.04E+05±3.33E+04	8.21E+04±2.27E+04	7.25E+04±1.22E+04
f₅	9.85E+03±4.47E+03	8.56E+03±1.64E+03	5.70E+03±1.94E+03	4.99E+02±1.01E+03	1.14E+03±6.20E+02	1.06E+02±3.32E+01
f₆	5.84E+03±1.72E+03	4.49E+03±5.99E+02	3.48E+03±1.19E+03	1.71E+02±7.17E+01	1.64E+03±1.07E+03	6.90E+01±2.47E+01
f₇	2.52E+04±2.64E+04	2.61E+03±1.28E+03	6.42E+03±1.09E+04	3.03E+04±4.31E+04	2.40E+04±5.81E+04	4.11E+01±1.12E+01
f₈	2.10E+01±6.57E–02	2.10E+01±5.85E–02	2.10E+01±6.54E–02	2.11E+01±5.93E–02	2.10E+01±5.39E–02	2.10E+01±6.09E–02
f₉	4.16E+01±1.81E+00	4.05E+01±9.64E–01	3.83E+01±3.17E+00	3.96E+01±2.48E+00	3.72E+01±3.35E+00	2.14E+01±4.18E+00
f₁₀	5.14E+03±1.10E+03	4.59E+03±5.23E+02	3.29E+03±8.44E+02	1.49E+02±1.23E+02	1.97E+03±9.37E+02	1.67E+01±1.19E+01
f₁₁	6.54E+02±1.11E+02	6.50E+02±4.33E+01	4.60E+02±9.00E+01	1.52E+02±5.41E+01	4.65E+02±9.63E+01	1.13E+01±3.35E+00
f₁₂	7.40E+02±8.81E+01	6.52E+02±4.22E+01	5.13E+02±9.01E+01	8.69E+02±1.54E+02	6.73E+02±1.39E+02	1.05E+02±3.38E+01
f₁₃	7.42E+02±9.14E+01	6.50E+02±5.49E+01	5.48E+02±7.61E+01	8.81E+02±1.31E+02	6.86E+02±9.57E+01	8.64E+01±3.04E+01
f₁₄	7.44E+03±4.96E+02	7.66E+03±2.56E+02	5.83E+03±6.32E+02	1.62E+03±3.60E+02	7.27E+03±6.99E+02	1.55E+03±5.45E+02
f₁₅	7.56E+03±5.92E+02	7.65E+03±3.13E+02	5.72E+03±7.14E+02	4.89E+03±1.01E+03	7.26E+03±4.44E+02	4.41E+03±7.63E+02
f₁₆	2.70E+00±4.78E–01	2.76E+00±3.05E–01	1.47E+00±4.45E–01	1.95E+00±5.61E–01	2.46E+00±4.73E–01	1.79E+00±5.38E–01
f₁₇	1.26E+03±1.41E+02	1.18E+03±8.23E+01	6.52E+02±1.04E+02	9.87E+01±4.59E+01	8.98E+02±1.68E+02	4.09E+01±3.16E+00
f₁₈	1.28E+03±2.00E+02	1.16E+03±9.17E+01	7.18E+02±1.52E+02	1.55E+03±2.47E+02	1.03E+03±1.92E+02	1.06E+02±2.98E+01
f₁₉	2.75E+05±2.01E+05	2.39E+05±6.70E+04	5.00E+04±3.86E+04	7.96E+01±1.68E+02	2.68E+04±5.41E+04	3.34E+00±7.17E–01
f₂₀	1.50E+01±3.72E–02	1.50E+01±2.29E–02	1.50E+01±6.32E–08	1.49E+01±1.78E–01	1.37E+01±8.18E–01	1.50E+01±3.14E–05
f₂₁	3.03E+03±2.98E+02	3.10E+03±1.43E+02	2.47E+03±1.64E+02	4.57E+02±1.71E+02	2.16E+03±3.59E+02	2.92E+02±2.75E+01
f₂₂	8.43E+03±4.74E+02	8.29E+03±2.74E+02	6.77E+03±1.01E+03	2.07E+03±5.70E+02	7.63E+03±9.75E+02	1.91E+03±1.25E+03
f₂₃	8.16E+03±4.96E+02	8.28E+03±2.43E+02	6.82E+03±8.01E+02	6.38E+03±9.46E+02	7.58E+03±6.23E+02	5.61E+03±1.21E+03
f₂₄	3.83E+02±2.26E+01	3.57E+02±8.70E+00	3.45E+02±1.59E+01	3.69E+02±2.90E+01	3.48E+02±1.61E+01	2.17E+02±6.37E+00
f₂₅	4.02E+02±1.31E+01	3.76E+02±6.32E+00	3.85E+02±1.81E+01	3.94E+02±1.78E+01	3.74E+02±1.73E+01	2.70E+02±1.38E+01
f₂₆	2.73E+02±4.82E+01	2.50E+02±1.53E+01	2.62E+02±5.77E+01	3.31E+02±1.08E+02	3.04E+02±9.55E+01	2.95E+02±2.03E+01
f₂₇	1.56E+03±6.16E+01	1.47E+03±3.89E+01	1.41E+03±4.78E+01	1.60E+03±1.07E+02	1.46E+03±9.06E+01	4.51E+02±6.27E+01
f₂₈	5.60E+03±5.24E+02	4.96E+03±2.87E+02	4.56E+03±5.53E+02	6.00E+03±1.08E+03	4.45E+03±6.04E+02	3.00E+02±6.06E–03
–/≈/+	25/2/1	25/2/1	24/2/2	27/0/1	26/1/1	—
P-value	1.18E–5	1.18E–5	1.33E–5	4.46E–6	7.03E–6	—
Friedman	5.30	4.30	3.13	3.61	3.32	1.34

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參考文獻(xiàn)(23)

YANG Xinshe. Firefly algorithms for multimodal optimization[C]. International Symposium on Stochastic Algorithms, Berlin, Germany, 2009: 169–178.

HASSANZADEH T and KANAN H R. Fuzzy FA: A modified firefly algorithm[J]. Applied Artificial Intelligence, 2014, 28(1): 47–65. doi: 10.1080/08839514.2014.862773

HAJI V H and MONJE C A. Fractional-order PID control of a chopper-fed DC motor drive using a novel firefly algorithm with dynamic control mechanism[J]. Soft Computing, 2018, 22(18): 6135–6146. doi: 10.1007/s00500-017-2677-5

ZHANG Yong, SONG Xianfang, and GONG Dunwei. A return-cost-based binary firefly algorithm for feature selection[J]. Information Sciences, 2017, 418: 561–574. doi: 10.1016/j.ins.2017.08.047

FISTER I, FISTER Jr I, YANG X S, et al. A comprehensive review of firefly algorithms[J]. Swarm and Evolutionary Computation, 2013, 13: 34–46. doi: 10.1016/j.swevo.2013.06.001

YU Shuhao, ZHU Shenglong, MA Yanyu, et al. A variable step size firefly algorithm for numerical optimization[J]. Applied Mathematics and Computation, 2015, 263: 214–220. doi: 10.1016/j.amc.2015.04.065

WANG Hui, ZHOU Xinyu, SUN Hui, et al. Firefly algorithm with adaptive control parameters[J]. Soft Computing, 2017, 21(17): 5091–5102. doi: 10.1007/s00500-016-2104-3

WANG Hui, WANG Wenjun, SUN Hui, et al. Firefly algorithm with random attraction[J]. International Journal of Bio-Inspired Computation, 2016, 8(1): 33–41. doi: 10.1504/ijbic.2016.074630

VERMA O P, AGGARWAL D, PATODI T, et al. Opposition and dimensional based modified firefly algorithm[J]. Expert Systems with Applications, 2016, 44: 168–176. doi: 10.1016/j.eswa.2015.08.054

GANDOMI A H, YANG X S, TALATAHARI S, et al. Firefly algorithm with chaos[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(1): 89–98. doi: 10.1016/j.cnsns.2012.06.009

TIZHOOSH H R. Opposition-based learning: A new scheme for machine intelligence[C]. International Conference on Computational Intelligence for Modelling, Control and Automation, Vienna, Austria, 2005: 695–701.

RAHNAMAYAN H, TIZHOOSH H R, and SALAMA M. Opposition-based differential evolution[J]. IEEE Transactions on Evolutionary Computation, 2008, 12(1): 64–79. doi: 10.1109/TEVC.2007.894200

WANG Hui, WU Zhijian, RAHNAMAYAN S, et al. Enhancing particle swarm optimization using generalized opposition-based learning[J]. Information Sciences, 2011, 181(20): 4699–4714. doi: 10.1016/j.ins.2011.03.016

YU Shuhao, ZHU Shenglong, MA Yan, et al. Enhancing firefly algorithm using generalized opposition-based learning[J]. Computing, Springer Vienna, 2015, 97(7): 741–754. doi: 10.1007/s00607-015-0456-7

PARK S Y and LEE J J. Stochastic opposition-based learning using a beta distribution in differential evolution[J]. IEEE Transactions on Cybernetics, 2016, 46(10): 2184–2194. doi: 10.1109/TCYB.2015.2469722

YANG Xinshe. Cuckoo Search and Firefly Algorithm[M]. London, UK: Springer, 2014: 1–26. doi: 10.1007/978-3-319-02141-6.

RAHNAMAYAN S, JESUTHASAN J, BOURENNANI F, et al. Computing opposition by involving entire population[C]. IEEE Congress on Evolutionary Computation, Beijin, China, 2014: 1800–1807.

方開(kāi)泰, 劉民千, 周永道. 試驗(yàn)設(shè)計(jì)與建模[M]. 北京: 高等教育出版社, 2011: 81–101.

FANG Kaitai, LIU Minqian, and ZHOU Yongdao. Design and Modeling of Experiments[M]. Beijing: Higher Education Press, 2011: 81–101.

ZHAN Zhihui, ZHANG Jun, LI Yun, et al. Orthogonal learning particle swarm optimization[J]. IEEE Transactions on Evolutionary Computation, 2011, 15(6): 832–847. doi: 10.1109/TEVC.2010.2052054

SUGANTHAN P N, HANSEN N, LIANG J J, et al. Problem definitions and evaluation criteria for the CEC 2013 special session on real-parameter optimization[R]. Computational Intelligence Laboratory, Zhengzhou University, China and Nanyang Technological, Singapore, Technical Report 201212, 2013.

TILAHUN S L and ONG H C. Modified firefly algorithm[J]. Journal of Applied Mathematics, 2012, 12: 2428–2439. doi: 10.1155/2012/467631

DERRAC J, CARCIA S, MOLINA D, et al. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms[J]. Swarm and Evolutionary Computation, 2011, 1(11): 3–18. doi: 10.1016/j.swevo.2011.02.002

周凌云, 丁立新, 彭虎, 等. 一種鄰域重心反向?qū)W習(xí)的粒子群優(yōu)化算法[J]. 電子學(xué)報(bào), 2017, 45(11): 2815–2824. doi: 10.3969/j.issn.0372-2112.2017.11.032

ZHOU Lingyun, DING Lixin, PENG Hu, et al. Neighborhood centroid opposition-based particle swarm optimization[J]. Acta Electronica Sinica, 2017, 45(11): 2815–2824. doi: 10.3969/j.issn.0372-2112.2017.11.032

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