Data-driven evolutionary sampling optimization for expensive problems

doi:10.23919/JSEE.2021.000027

Journal of Systems Engineering and Electronics ›› 2021, Vol. 32 ›› Issue (2): 318-330.doi: 10.23919/JSEE.2021.000027

• INTELLIGENT OPTIMIZATION AND SCHEDULING • Previous Articles Next Articles

Data-driven evolutionary sampling optimization for expensive problems

Huixiang ZHEN¹(), Wenyin GONG^1,*(), Ling WANG²()

¹ School of Computer Science, China University of Geosciences, Wuhan 430074, China
² Department of Automation, Tsinghua University, Beijing 100084, China

Received:2020-09-01 Online:2021-04-29 Published:2021-04-29
Contact: Wenyin GONG E-mail:zhenhuixiang@cug.edu.cn;wygong@cug.edu.cn;wangling@tsinghua.edu.cn
About author:|ZHEN Huixiang was born in 1995. He is now pursuing his Ph.D. degree in the School of Computer Science, China University of Geosciences, Wuhan, China. His research interests include the evolutionary computation and its applications. E-mail: zhenhuixiang@cug.edu.cn||GONG Wenyin was born in 1979. He is currently a professor and Ph.D. student supervisor in the School of Computer Science, China University of Geosciences, Wuhan, China. He has published over 50 research papers in journals and international conferences. His research interests include differential evolution, memetic algorithms, multi-objective optimization, and their applications. E-mail: wygong@cug.edu.cn||WANG Ling was born in 1972. He is currently a professor and Ph.D. student supervisor in the Department of Automation, Tsinghua University, Beijing, China. He has authored five academic books and over 260 refereed papers. His research interests include intelligent optimization and production scheduling. E-mail: wangling@tsinghua.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (62076225; 62073300), the Natural Science Foundation for Distinguished Young Scholars of Hubei (2019CFA081), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUGGC03);This work was supported by the National Natural Science Foundation of China (62076225; 62073300), the Natural Science Foundation for Distinguished Young Scholars of Hubei (2019CFA081), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUGGC03)

Abstract

Abstract:

Surrogate models have shown to be effective in assisting evolutionary algorithms (EAs) for solving computationally expensive complex optimization problems. However, the effectiveness of the existing surrogate-assisted evolutionary algorithms still needs to be improved. A data-driven evolutionary sampling optimization (DESO) framework is proposed, where at each generation it randomly employs one of two evolutionary sampling strategies, surrogate screening and surrogate local search based on historical data, to effectively balance global and local search. In DESO, the radial basis function (RBF) is used as the surrogate model in the sampling strategy, and different degrees of the evolutionary process are used to sample candidate points. The sampled points by sampling strategies are evaluated, and then added into the database for the updating surrogate model and population in the next sampling. To get the insight of DESO, extensive experiments and analysis of DESO have been performed. The proposed algorithm presents superior computational efficiency and robustness compared with five state-of-the-art algorithms on benchmark problems from 20 to 200 dimensions. Besides, DESO is applied to an airfoil design problem to show its effectiveness.

Key words: evolutionary algorithm (EA), surrogate model, data-driven, evolutionary sampling, airfoil design

Huixiang ZHEN, Wenyin GONG, Ling WANG. Data-driven evolutionary sampling optimization for expensive problems[J]. Journal of Systems Engineering and Electronics, 2021, 32(2): 318-330.

Figures/Tables 16

Fig.1

Table 1

Table 2

Fig.2

Fig.3

Fig.4

Fig.5

Table 3

Comparison of DESO using different sampling strategies"

Problem	DESO-SS	DESO-SLS	DESO
Problem	Mean (Standard)	Mean (Standard)	Mean (Standard)
E20	3.6664E-07(1.9910E-07)	3.2815E-01(3.5169E-01)	3.7414E-15(4.5651E-15)
E30	7.4239E-03(1.0365E-02)	9.4273E-01(5.4385E-01)	9.0387E-08(9.1891E-08)
E50	3.4855E+00(2.3903E+00)	6.2837E+00(2.2834E+00)	7.6666E-03(5.8049E-03)
E100	2.9328E+02(6.2192E+01)	7.2900E+01(2.8170E+01)	1.2283E+01(4.2566E+00)
R20	1.6134E+01(1.3308E+00)	2.6261E+01(8.6217E+00)	1.3039E+01(1.1425E+00)
R30	2.7951E+01(9.6690E-01)	3.4825E+01(7.0368E+00)	2.4713E+01(1.2670E+00)
R50	6.6163E+01(2.0960E+01)	5.9539E+01(6.9527E+00)	4.6602E+01(9.8297E-01)
R100	3.7220E+02(6.3760E+01)	1.5384E+02(2.1127E+01)	1.0127E+02(2.3479E+00)
A20	1.4194E+00(4.9019E-01)	6.0124E-01(5.2485E-01)	3.9556E-06(3.0972E-06)
A30	3.1331E+00(5.5586E-01)	4.8188E-01(3.1148E-01)	3.2057E-05(1.8247E-05)
A50	4.1422E+00(6.2899E-01)	1.3383E+00(3.4250E-01)	2.6203E-01(4.7966E-01)
A100	7.6050E+00(9.1124E-01)	3.4714E+00(4.7813E-01)	2.8883E+00(4.7124E-01)
G20	5.8210E-01(2.1501E-01)	2.4515E-01(4.0269E-01)	5.6784E-02(1.5159E-01)
G30	5.3728E-01(1.8834E-01)	3.8896E-02(1.9702E-02)	7.9600E-04(2.6309E-03)
G50	1.0964E+00(1.2913E-01)	1.9459E-01(6.1384E-02)	9.5999E-03(1.0159E-02)
G100	1.8652E+01(4.9693E+00)	2.5451E+00(6.6982E-01)	1.0250E+00(5.6619E-02)
SRR20	?1.5860E+02(1.4435E+01)	?1.3860E+02(1.8281E+01)	?1.5919E+02(2.3821E+01)
SRR30	?5.6265E+01(2.6909E+01)	?3.8143E+01(3.4480E+01)	?7.2471E+01(2.3485E+01)
SRR50	2.2778E+02(5.4075E+01)	2.5261E+02(4.4738E+01)	2.3005E+02(4.6423E+01)
SRR100	1.1417E+03(6.8809E+01)	1.3542E+03(1.0738E+02)	1.2021E+03(1.1833E+02)
RHC20	1.1807E+03(5.0535E+01)	1.2103E+03(6.3443E+01)	1.1926E+03(3.5859E+01)
RHC30	9.6940E+02(2.3313E+01)	1.0127E+03(3.8582E+01)	9.6599E+02(1.8853E+01)
RHC50	1.0631E+03(2.1495E+01)	1.0988E+03(3.6477E+01)	1.0411E+03(1.7546E+01)
RHC100	1.3871E+03(3.8007E+01)	1.3784E+03(3.3023E+01)	1.3541E+03(4.9898E+01)

Table 3

Table 4

Table 5

Comparison of DESO with other algorithms"

Problem	SHADE		GPEME		SA-COSO		SHPSO		ESAO		DESO
Problem	Mean	Standard	Mean	Standard	Mean	Standard	Mean	Standard	Mean	Standard	Mean	Standard
E20	121.994	54.925	1.300E-05	2.180E-05	N/A	N/A	N/A	N/A	1.81E-04	4.68E-04	3.741E-15	4.565E-15
E30	475.636	121.912	0.076	4.010E-02	N/A	N/A	0.212	0.152	0.027	0.070	9.039E-08	9.189E-08
E50	1698.723	391.560	221.080	81.612	51.475	16.246	4.028	2.060	0.740	0.555	7.667E-03	5.805E-03
E100	9140.519	1495.311	N/A	N/A	1033.200	317.180	76.106	21.447	1282.900	134.390	12.283	4.257
E200	44123.612	6920.598	N/A	N/A	16382.000	2981.100	N/A	N/A	1.76E+04	1.17E+03	479.920	103.740
R20	363.972	128.456	22.428	18.795	N/A	N/A	N/A	N/A	15.162	1.629	13.039	1.143
R30	1058.046	352.854	46.177	25.520	N/A	N/A	28.566	0.404	25.036	1.570	24.713	1.267
R50	2257.047	932.045	258.280	80.188	252.580	40.744	50.800	3.031	47.391	1.712	46.602	0.983
R100	6676.190	1502.416	N/A	N/A	2714.200	117.020	165.590	26.366	578.840	44.767	101.270	2.348
R200	16730.022	3760.863	N/A	N/A	16411.000	4096.500	N/A	N/A	4.31E+03	2.84E+02	348.720	36.776
A20	15.800	1.184	0.199	0.577	N/A	N/A	N/A	N/A	6.865	3.259	3.956E-06	3.097E-06
A30	16.507	1.205	3.011	0.925	N/A	N/A	1.442	0.774	2.521	0.840	3.206E-05	1.825E-05
A50	17.817	0.841	13.233	1.585	8.932	1.067	1.839	5.637	1.431	0.249	0.262	0.480
A100	18.532	0.411	N/A	N/A	15.756	0.503	4.113	0.592	10.364	0.211	2.888	0.471
A200	18.865	0.323	N/A	N/A	17.868	0.022	N/A	N/A	14.696	0.2193	5.943	0.400
G20	54.934	17.333	0.031	0.068	N/A	N/A	N/A	N/A	0.972	0.039	0.057	0.152
G30	127.121	47.575	0.997	0.108	N/A	N/A	0.921	0.088	0.953	0.050	7.960E-04	2.631E-03
G50	280.808	43.906	36.646	13.176	6.006	1.104	0.945	0.061	0.940	0.042	9.600E-03	1.016E-02
G100	726.981	103.158	N/A	N/A	63.353	19.021	1.070	0.020	57.342	5.839	1.025	0.057
G200	1528.380	190.728	N/A	N/A	577.760	101.400	N/A	N/A	572.903	36.043	16.141	3.381
SRR20	?8.473	59.846	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	?1.59E+02	2.38E+01
SRR30	179.346	79.508	?21.861	36.449	N/A	N/A	?92.830	22.544	6.325	26.477	?72.471	23.485
SRR50	633.467	89.457	N/A	N/A	197.160	30.599	134.420	32.256	198.610	45.825	230.050	46.423
SRR100	2235.075	267.703	N/A	N/A	1273.100	117.190	801.730	72.252	713.470	26.454	1202.100	118.330
SRR200	6480.683	252.517	N/A	N/A	3927.500	272.540	N/A	N/A	5.39E+03	156.8544	4853.100	151.620
RHC20	1395.179	81.068	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	1.19E+03	35.9
RHC30	1229.265	49.472	958.590	25.695	N/A	N/A	939.610	9.018	931.670	8.942	965.990	18.853
RHC50	1304.984	33.006	N/A	N/A	1080.900	32.859	996.600	22.145	975.320	37.110	1041.100	17.546
RHC100	1567.296	48.421	N/A	N/A	1365.700	30.867	1419.800	38.238	1372.400	27.539	1354.100	49.898
RHC200	1571.078	38.177	N/A	N/A	1347.300	24.665	N/A	N/A	1.456E+03	20.432	1345.500	74.897

Table 5

Fig.6

Table 6

Table 7

Fig.7

Fig.8

Fig.9

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Abbreviation	Problem	d	Search space	Optimum	Property
E	Ellipsoid	20,30,50,100,200	[?5.12, 5.12]^d	0	Unimodal
R	Rosenbrock	20,30,50,100,200	[?2.048, 2.048]^d	0	Multimodal
A	Ackley	20,30,50,100,200	[?32.768, 32.768]^d	0	Multimodal
G	Griewank	20,30,50,100,200	[?600, 600]^d	0	Multimodal
SRR	Shifted rotated rastrigin (f10 in CEC05 [48])	20,30,50,100,200	[?5, 5]^d	?330	Very complicated multimodal
RHC	Rotated hybrid composition function (f19 in CEC05 [48])	20,30,50,100,200	[?5, 5]^d	10	Very complicated multimodal

Problem	Best	Worst	Mean	Median	Standard
E20	1.64E-16	1.95E-14	3.74E-15	2.55E-15	4.57E-15
E30	1.00E-09	3.96E-07	9.04E-08	8.36E-08	9.19E-08
E50	1.12E-03	2.70E-02	7.67E-03	5.69E-03	5.80E-03
E100	6.38E+00	1.96E+01	1.23E+01	1.15E+01	4.26E+00
E200	3.12E+02	6.65E+02	4.80E+02	4.62E+02	1.04E+02
R20	1.16E+01	1.65E+01	1.30E+01	1.30E+01	1.14E+00
R30	2.35E+01	2.84E+01	2.47E+01	2.45E+01	1.27E+00
R50	4.50E+01	4.92E+01	4.66E+01	4.66E+01	9.83E-01
R100	9.78E+01	1.07E+02	1.01E+02	1.00E+02	2.35E+00
R200	2.72E+02	4.25E+02	3.49E+02	3.49E+02	3.68E+01
A20	8.52E-07	1.42E-05	3.96E-06	2.92E-06	3.10E-06
A30	7.64E-06	8.95E-05	3.21E-05	2.56E-05	1.82E-05
A50	4.24E-04	1.37E+00	2.62E-01	1.82E-03	4.80E-01
A100	2.04E+00	3.75E+00	2.89E+00	2.89E+00	4.71E-01
A200	5.10E+00	6.56E+00	5.94E+00	5.94E+00	4.00E-01
G20	2.04E-10	5.01E-01	5.68E-02	1.46E-06	1.52E-01
G30	1.60E-06	1.19E-02	7.96E-04	4.02E-05	2.63E-03
G50	1.47E-03	3.89E-02	9.60E-03	6.03E-03	1.02E-02
G100	9.20E-01	1.10E+00	1.03E+00	1.03E+00	5.66E-02
G200	9.45E+00	2.39E+01	1.61E+01	1.55E+01	3.38E+00
SRR20	?2.12E+02	?1.17E+02	?1.59E+02	?1.55E+02	2.38E+01
SRR30	?1.11E+02	?3.04E+01	?7.25E+01	?7.72E+01	2.35E+01
SRR50	1.63E+02	3.46E+02	2.30E+02	2.25E+02	4.64E+01
SRR100	9.99E+02	1.39E+03	1.20E+03	1.19E+03	1.18E+02
SRR200	4.55E+03	5.14E+03	4.85E+03	4.83E+03	1.52E+02
RHC20	1.11E+03	1.25E+03	1.19E+03	1.20E+03	3.59E+01
RHC30	9.41E+02	1.02E+03	9.66E+02	9.64E+02	1.89E+01
RHC50	1.02E+03	1.07E+03	1.04E+03	1.04E+03	1.75E+01
RHC100	1.19E+03	1.40E+03	1.35E+03	1.37E+03	4.99E+01
RHC200	1.19E+03	1.50E+03	1.35E+03	1.35E+03	7.49E+01

Problem	DESO-DE	DESO-PSO	DESO-JADE
Problem	Mean (Standard)	Mean (Standard)	Mean (Standard)
E30	9.0387E-08(9.1891E-08)	1.0431E-08 (1.2405E-08)	2.0660E-08(3.5522E-08)
E50	7.6666E-03(5.8049E-03)	2.9575E-03(2.5580E-03)	2.0126E-03(2.1218E-03)
E100	1.2283E+01(4.2566E+00)	3.1564E+00(1.1696E+00)	2.9194E+00(1.4148E+00)
R30	2.4713E+01(1.2670E+00)	2.3802E+01(1.0361E+00)	2.4079E+01(5.9837E-01)
R50	4.6602E+01(9.8297E-01)	4.5721E+01(6.5097E-01)	4.5523E+01(7.1813E-01)
R100	1.0127E+02(2.3479E+00)	9.7918E+01(4.5785E-01)	9.8597E+01(6.2668E-01)
A30	3.2057E-05(1.8247E-05)	4.9287E-01(6.3449E-01)	5.7762E-02(2.5830E-01)
A50	2.6203E-01(4.7966E-01)	7.8886E-01(7.0006E-01)	6.0062E-01(6.9700E-01)
A100	2.8883E+00(4.7124E-01)	2.8424E+00(3.9646E-01)	3.0345E+00(2.9678E-01)
G30	7.9600E-04(2.6309E-03)	1.3273E-02(4.9904E-02)	5.0524E-05(1.5774E-04)
G50	9.5999E-03(1.0159E-02)	2.7314E-03(5.1458E-03)	2.4616E-03(3.0528E-03)
G100	1.0250E+00(5.6619E-02)	4.1510E-01(7.4805E-02)	5.2188E-01(1.2095E-01)

Design parameter	Original value	Search range
(R_le) Leading edge radius	0.021 6	0.001 5
(X_up) Position of upper crest	0.344 5	0.025
(Y_up) Upper crest point	0.079 12	0.015
(YXX_up) Upper crest curvature	?0.644 8	0.01
(X_lo) Position of lower crest	0.17	0.02
(Y_lo) Lower crest point	?0.033 797	0.015
(YXX_lo) Lower crest curvature	0.674 8	0.075
(T_TE) Trailing edge thickness	0	0
(T_off) Trailing edge offset	0	0
(α_TE) Trailing edge direction angle	0	0.175
(β_TE)) Trailing edge wedge angle	0	0.05

Algorithm	Best	Worst	Mean	Median	Standard
DESO	0.958 81	0.958 62	0.958 76	0.958 78	0.000 049
DE	0.958 5	0.952 14	0.956 51	0.956 61	0.001 502
SHPSO	0.958 78	0.957 32	0.958 53	0.958 63	0.000 33
ESAO	0.958 68	0.957 91	0.958 38	0.958 4	0.000 221

Data-driven evolutionary sampling optimization for expensive problems

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[1]	Alireza MOHSENI, Vincent DUCHAINE, Tony WONG. Experimental study of path planning problem using EMCOA for a holonomic mobile robot [J]. Journal of Systems Engineering and Electronics, 2021, 32(6): 1450-1462.
[2]	Rongling Lang, Zheping Xu, and Fei Gao. Data-driven fault diagnosis method for analog circuits based on robust competitive agglomeration [J]. Journal of Systems Engineering and Electronics, 2013, 24(4): 706-712.