Integrating Variable Reduction Strategy With Evolutionary Algorithms for Solving Nonlinear Equations Systems

Aijuan Song; Guohua Wu; Witold Pedrycz; Ling Wang

doi:10.1109/JAS.2021.1004278

Volume 9 Issue 1

Jan. 2022

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2022 > 9(1): 75-89

A. J. Song, G. H. Wu, W. Pedrycz, and L. Wang, “Integrating variable reduction strategy with evolutionary algorithms for solving nonlinear equations systems,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 1, pp. 75–89, Jan. 2022. doi: 10.1109/JAS.2021.1004278

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A. J. Song, G. H. Wu, W. Pedrycz, and L. Wang, “Integrating variable reduction strategy with evolutionary algorithms for solving nonlinear equations systems,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 1, pp. 75–89, Jan. 2022. doi: 10.1109/JAS.2021.1004278

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Integrating Variable Reduction Strategy With Evolutionary Algorithms for Solving Nonlinear Equations Systems

doi: 10.1109/JAS.2021.1004278

Aijuan Song^1
,,
Guohua Wu^{1
,
,},
Witold Pedrycz^{2, 3, 4
,},
Ling Wang^5
,

1.
School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China
2.
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada
3.
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4.
Systems Research Institute, Polish Academy of Sciences, Warsaw 01447, Poland
5.
Department of Automation, Tsinghua University, Beijing 100084, China

Funds: This work was supported by the National Natural Science Foundation of China (62073341), and in part by the Natural Science Fund for Distinguished Young Scholars of Hunan Province (2019JJ20026)

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Author Bio:
Aijuan Song received the B.E. degree from Central South University, China, in 2019. Currently, she is a master student at the School of Traffic & Transportation Engineering, Central South University. Her research interests include computational intelligence, multiobjective evolutionary algorithms, planning and scheduling

Guohua Wu received the B.S. degree in information systems and the Ph.D degree in operations research from National University of Defense Technology, China, in 2008 and 2014, respectively. During 2012 and 2014, he was a Visiting Ph.D Student at University of Alberta, Edmonton, Canada. He is currently a Professor at the School of Traffic and Transportation Engineering, Central South University, Changsha, China. His current research interests include planning and scheduling, computational intelligence and machine learning. He has authored more than 80 referred papers including those published in IEEE TCYB, IEEE TSMCA and IEEE TITS. He serves as an Associate Editor of Swarm and Evolutionary Computation Journal, an editorial board member of International Journal of Bio-Inspired Computation, and a Guest Editor of Information Sciences and Memetic Computing. He is a regular reviewer of more than 20 journals including IEEE TEVC, IEEE TCYB, IEEE TSMCA and Information Sciences

Witold Pedrycz received the M.Sc. degree in computer science, the Ph.D. degree in computer engineering, and the D.Sci. degree in systems science from the Silesian University of Technology, Gliwice, Poland, in 1977, 1980, and 1984, respectively. He is a Professor and the Canada Research Chair (CRC-Computational Intelligence) with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, and also with the Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He is also with the Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland. In 2012, he was elected a Fellow of the Royal Society of Canada. His current research interests include computational intelligence, knowledge discovery and data mining, pattern recognition, knowledge-based neural networks, and software engineering. He has published numerous papers in the above areas. He is an Editor-in-Chief of Information Sciences. He currently serves as an Associate Editor of IEEE Transactions on Fuzzy Systems and IEEE Transactions on Systems, Man and Cybernetics: System and is a member of a number of editorial boards of other international journals

Ling Wang received the B.Sc. degree in automation and the Ph.D. degree in control theory and control engineering from Tsinghua University, Beijing, China, in 1995 and 1999, respectively. Since 1999, he has been with the Department of Automation, Tsinghua University, where he became a Full Professor in 2008. He has authored five academic books and over 260 refereed papers. His current research interests include intelligent optimization and production scheduling. He was a Recipient of the National Natural Science Fund for Distinguished Young Scholars of China, the National Natural Science Award (Second Place) in 2014, the Science and Technology Award of Beijing City in 2008, the Natural Science Award (First Place in 2003 and Second Place in 2007) nominated by the Ministry of Education of China. He is currently the Editor-in-Chief of the International Journal of Automation and Control and an Associate Editor of the IEEE Transactions on Evolutionary Computation
Corresponding author: Guohua Wu, e-mail: guohuawu@csu.edu.cn
¹ The statistical tests reported in this paper are calculated by the KEEL3.0 software [32].
² http://faculty.csu.edu.cn/guohuawu/zh_CN/zdylm/193832/list/index.htm
³ Except for VR-DR-JADE, DR-JADE, and MONES, the data of other seven compared methods are from the literature [20] and the detailed results of the eleven methods are reported in the supplementary file².
⁴ The date of DR-JADE comes from [20].
Received Date: 2021-03-27
Accepted Date: 2021-05-21

Available Online: 2023-03-29

Abstract

Abstract

Nonlinear equations systems (NESs) are widely used in real-world problems and they are difficult to solve due to their nonlinearity and multiple roots. Evolutionary algorithms (EAs) are one of the methods for solving NESs, given their global search capabilities and ability to locate multiple roots of a NES simultaneously within one run. Currently, the majority of research on using EAs to solve NESs focuses on transformation techniques and improving the performance of the used EAs. By contrast, problem domain knowledge of NESs is investigated in this study, where we propose the incorporation of a variable reduction strategy (VRS) into EAs to solve NESs. The VRS makes full use of the systems of expressing a NES and uses some variables (i.e., core variable) to represent other variables (i.e., reduced variables) through variable relationships that exist in the equation systems. It enables the reduction of partial variables and equations and shrinks the decision space, thereby reducing the complexity of the problem and improving the search efficiency of the EAs. To test the effectiveness of VRS in dealing with NESs, this paper mainly integrates the VRS into two existing state-of-the-art EA methods (i.e., MONES and DR-JADE) according to the integration framework of the VRS and EA, respectively. Experimental results show that, with the assistance of the VRS, the EA methods can produce better results than the original methods and other compared methods. Furthermore, extensive experiments regarding the influence of different reduction schemes and EAs substantiate that a better EA for solving a NES with more reduced variables tends to provide better performance.
- Evolutionary algorithm (EA),
- nonlinear equations systems (ENSs),
- problem domain knowledge,
- variable reduction strategy (VRS)

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¹ The statistical tests reported in this paper are calculated by the KEEL3.0 software [32].
² http://faculty.csu.edu.cn/guohuawu/zh_CN/zdylm/193832/list/index.htm
³ Except for VR-DR-JADE, DR-JADE, and MONES, the data of other seven compared methods are from the literature [20] and the detailed results of the eleven methods are reported in the supplementary file².
⁴ The date of DR-JADE comes from [20].

References(43)

References

[1]	D. Mehta and C. Grosan, “A collection of challenging optimization problems in science, engineering and economics,” in Proc. IEEE Congress on Evolutionary Computation (CEC), IEEE, 2015, pp. 2697–2704.
[2]	A. Holstad, “Numerical solution of nonlinear equations in chemical speciation calculations,” Computational geosciences, vol. 3, no. 3–4, pp. 229–257, 1999.
[3]	C. L. Collins, “Forward kinematics of planar parallel manipulators in the Clifford algebra of p2,” Mechanism and Machine Theory, vol. 37, no. 8, pp. 799–813, 2002. doi: 10.1016/S0094-114X(02)00023-X
[4]	F. Facchinei and C. Kanzow, “Generalized nash equilibrium problems,” 4or, vol. 5, no. 3, pp. 173–210, 2007. doi: 10.1007/s10288-007-0054-4
[5]	N. I. Chaudhary, M. S. Aslam, and M. A. Z. Raja, “Modified volterra lms algorithm to fractional order for identification of hammerstein non-linear system,” IET Signal Processing, vol. 11, no. 8, pp. 975–985, 2017. doi: 10.1049/iet-spr.2016.0578
[6]	G. Yuan and X. Lu, “A new backtracking inexact bfgs method for symmetric nonlinear equations,” Computers &Mathematics with Applications, vol. 55, no. 1, pp. 116–129, 2008.
[7]	C. L. Karr, B. Weck, and L. M. Freeman, “Solutions to systems of nonlinear equations via a genetic algorithm,” Engineering Applications of Artificial Intelligence, vol. 11, no. 3, pp. 369–375, 1998. doi: 10.1016/S0952-1976(97)00067-5
[8]	C. Grosan and A. Abraham, “A new approach for solving nonlinear equations systems,” IEEE Trans. Systems,Man,and Cybernetics-Part A:Systems and Humans, vol. 38, no. 3, pp. 698–714, 2008. doi: 10.1109/TSMCA.2008.918599
[9]	D. J. Bates, A. J. Sommese, J. D. Hauenstein, and C. W. Wampler, Numerically Solving Polynomial Systems with Bertini. Philadelphia, USA: SIAM, 2013.
[10]	J. Denis and H. Wolkowicz, “Least change secant methods, sizing, and shifting,” SIAM Journal of Numerical Analisys, vol. 30, pp. 1291–1314, 1993. doi: 10.1137/0730067
[11]	J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, USA: SIAM, 2000.
[12]	L. Wang, Intelligent Optimization Algorithms with Applications. Beijing, China: Tsinghua University & Springer Press, 2001.
[13]	H.-T. Geng, Y.-J. Sun, Q.-X. Song, and T.-T. Wu, “Research of ranking method in evolution strategy for solving nonlinear system of equations,” in Proc. First Int. Conf. Information Science and Engineering, IEEE, 2009, pp. 348–351.
[14]	A. Ouyang, Y. Zhou, and Q. Luo, “Hybrid particle swarm optimization algorithm for solving systems of nonlinear equations,” in Proc. IEEE Int. Conf. Granular Computing, 2009, pp. 460–465.
[15]	O. E. Turgut, M. S. Turgut, and M. T. Coban, “Chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations,” Computers &Mathematics with Applications, vol. 68, no. 4, pp. 508–530, 2014.
[16]	W. Gong, Y. Wang, Z. Cai, and L. Wang, “Finding multiple roots of nonlinear equation systems via a repulsion-based adaptive differential evolution,” IEEE Trans. Systems,Man,and Cybernetics:Systems, vol. 50, no. 4, pp. 1499–1513, 2020. doi: 10.1109/TSMC.2018.2828018
[17]	H. Ren, L. Wu, W. Bi, and I. K. Argyros, “Solving nonlinear equations system via an efficient genetic algorithm with symmetric and harmonious individuals,” Applied Mathematics and Computation, vol. 219, no. 23, pp. 10967–10973, 2013. doi: 10.1016/j.amc.2013.04.041
[18]	G. Joshi and M. B. Krishna, “Solving system of non-linear equations using genetic algorithm,” in Proc. Int. Conf. Advances in Computing, Communications and Informatics (ICACCI), IEEE, 2014, pp. 1302–1308.
[19]	H.-G. Beyer and H.-P. Schwefel, “Evolution strategies – A comprehensive introduction,” Natural Computing, vol. 1, no. 1, pp. 3–52, 2002. doi: 10.1023/A:1015059928466
[20]	Z. Liao, W. Gong, X. Yan, L. Wang, and C. Hu, “Solving nonlinear equations system with dynamic repulsion-based evolutionary algorithms,” IEEE Trans. Systems,Man,and Cybernetics:Systems, vol. 50, no. 4, pp. 1590–1601, 2020. doi: 10.1109/TSMC.2018.2852798
[21]	G. Wu, W. Pedrycz, P. N. Suganthan, and R. Mallipeddi, “A variable reduction strategy for evolutionary algorithms handling equality constraints,” Applied Soft Computing, vol. 37, pp. 774–786, 2015. doi: 10.1016/j.asoc.2015.09.007
[22]	G. Wu, W. Pedrycz, P. N. Suganthan, and H. Li, “Using variable reduction strategy to accelerate evolutionary optimization,” Applied Soft Computing, vol. 61, pp. 283–293, 2017. doi: 10.1016/j.asoc.2017.08.012
[23]	W. Song, Y. Wang, H.-X. Li, and Z. Cai, “Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization,” IEEE Trans. Evolutionary Computation, vol. 19, no. 3, pp. 414–431, 2014.
[24]	S. Qin, S. Zeng, W. Dong, and X. Li, “Nonlinear equation systems solved by many-objective hype,” in Proc. IEEE Congress Evolutionary Computation (CEC), 2015, pp. 2691–2696.
[25]	M. J. Hirsch, P. M. Pardalos, and M. G. Resende, “Solving systems of nonlinear equations with continuous grasp,” Nonlinear Analysis:Real World Applications, vol. 10, no. 4, pp. 2000–2006, 2009. doi: 10.1016/j.nonrwa.2008.03.006
[26]	Y. Wang, H.-X. Li, G. G. Yen, and W. Song, “Mommop: Multiobjective optimization for locating multiple optimal solutions of multimodal optimization problems,” IEEE Trans. Cybernetics, vol. 45, no. 4, pp. 830–843, 2014.
[27]	A. Mousa and I. El-Desoky, “GENLS: Co-evolutionary algorithm for nonlinear system of equations,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 633–642, 2008. doi: 10.1016/j.amc.2007.08.088
[28]	A. F. Kuri-Morales, R. H. No, and D. México, “Solution of simultaneous non-linear equations using genetic algorithms,” WSEAS Trans. Systems, vol. 2, no. 1, pp. 44–51, 2003.
[29]	K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evolutionary Computation, vol. 6, no. 2, pp. 182–197, 2002. doi: 10.1109/4235.996017
[30]	J. Zhang and A. C. Sanderson, “JADE: adaptive differential evolution with optional external archive,” IEEE Trans. Evolutionary Computation, vol. 13, no. 5, pp. 945–958, 2009. doi: 10.1109/TEVC.2009.2014613
[31]	Y. Wang, J. Xiang, and Z. Cai, “A regularity model-based multiobjective estimation of distribution algorithm with reducing redundant cluster operator,” Applied Soft Computing, vol. 12, no. 11, pp. 3526–3538, 2012. doi: 10.1016/j.asoc.2012.06.008
[32]	I. Triguero, S. González, J. M. Moyano, S. García, J. Alcala-Fdez, J. Luengo, A. Fernández, M. J. Del Jesus, L. Sanchez, and F. Herrera, “Keel 3.0: An open source software for multi-stage analysis in data mining,” Int. Journal of Computational Intelligence Systems, vol. 10, no. 1, pp. 1238–1249, 2017. doi: 10.2991/ijcis.10.1.82
[33]	R. Thomsen, “Multimodal optimization using crowding-based differential evolution,” in Proc. Congress Evolutionary Computation (IEEE Cat. No. 04TH8753), IEEE, 2004, vol. 2, pp. 1382–1389.
[34]	X. Li, A. Engelbrecht, and M. G. Epitropakis, “Benchmark functions for cec2013 special session and competition on niching methods for multimodal function optimization,” Tech. Rep., Royal Melbourne Institute of Technology University, Evolutionary Computation and Machine Learning Group, Australia, 2013.
[35]	W. Gong, Y. Wang, Z. Cai, and S. Yang, “A weighted biobjective transformation technique for locating multiple optimal solutions of nonlinear equation systems,” IEEE Trans. Evolutionary Computation, vol. 21, no. 5, pp. 697–713, 2017. doi: 10.1109/TEVC.2017.2670779
[36]	G. C. Ramadas, E. M. Fernandes, and A. M. A. Rocha, “Multiple roots of systems of equations by repulsion merit functions,” in Proc. Int. Conf. Computational Science and Its Applications, Springer, 2014, pp. 126–139.
[37]	B.-Y. Qu, P. N. Suganthan, and J.-J. Liang, “Differential evolution with neighborhood mutation for multimodal optimization,” IEEE Trans. Evolutionary Computation, vol. 16, no. 5, pp. 601–614, 2012. doi: 10.1109/TEVC.2011.2161873
[38]	M. A. Z. Raja, A. K. Kiani, A. Shehzad, and A. Zameer, “Memetic computing through bio-inspired heuristics integration with sequential quadratic programming for nonlinear systems arising in different physical models,” SpringerPlus, vol. 5, no. 1, Article No. 2063, 2016. doi: 10.1186/s40064-016-3750-8
[39]	M. A. Z. Raja, A. Zameer, A. K. Kiani, A. Shehzad, and M. A. R. Khan, “Nature-inspired computational intelligence integration with nelder– mead method to solve nonlinear benchmark models,” Neural Computing and Applications, vol. 29, no. 4, pp. 1169–1193, 2018. doi: 10.1007/s00521-016-2523-1
[40]	X. Zhang, Q. Wan, and Y. Fan, “Applying modified cuckoo search algorithm for solving systems of nonlinear equations,” Neural Computing and Applications, vol. 31, no. 2, pp. 553–576, 2019. doi: 10.1007/s00521-017-3088-3
[41]	J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar, “Comprehensive learning particle swarm optimizer for global optimization of multimodal functions,” IEEE Trans. Evolutionary Computation, vol. 10, no. 3, pp. 281–295, 2006. doi: 10.1109/TEVC.2005.857610
[42]	G. Wu, R. Mallipeddi, and P. N. Suganthan, “Ensemble strategies for population-based optimization algorithms – A survey,” Swarm and Evolutionary Computation, vol. 44, pp. 695–711, 2019. doi: 10.1016/j.swevo.2018.08.015
[43]	H. Chen, G. Wu, W. Pedrycz, P. N. Suganthan, L. Xing, and X. Zhu, “An adaptive resource allocation strategy for objective space partition-based multiobjective optimization,” IEEE Trans. Systems,Man,and Cybernetics:Systems, vol. 51, no. 3, pp. 1507–1522, 2021.

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Variable reduction strategy was proposed to reduce nonlinear equations systems
A framework of variable reduction strategy and evolutionary algorithms was presented
Variable reduction strategy enables a better performance of an original algorithm
A better algorithm and a reduction scheme with more reduced variables perform better

Integrating Variable Reduction Strategy With Evolutionary Algorithms for Solving Nonlinear Equations Systems

doi: 10.1109/JAS.2021.1004278

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