JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (10): 22-29.doi: 10.6040/j.issn.1671-9352.0.2023.236

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MNSGA-Ⅱ algorithm based on bi-objective for solving nonlinear equation systems

Zhenai LI(),Hui WEI*(),Xin CHEN   

  1. School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, Anhui, China
  • Received:2023-05-24 Online:2024-10-20 Published:2024-10-10
  • Contact: Hui WEI E-mail:1936596383@qq.com;weihui@aust.edu.cn

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Abstract:

MONES transformation technique is applied to transform the problem of solving nonlinear equation systems into a bi-objective optimization problem, and a dynamic crowding distance strategy of MNSGA-Ⅱ algorithm is included to dynamically calculate individual crowding distance in the process of population selection, which improves the diversity of Pareto front. In order to verify the performance of algorithm, thirty nonlinear equation systems are selected for testing NSGA-Ⅱ, dynamic NSGA-Ⅱ and MNSGA-Ⅱ algorithm based on MONES transformation technique. Experimental results show that MNSGA-Ⅱ algorithm based on MONES transformation technique has a better root-found ratio and success rate. Finally, the Pareto front of three algorithms mentioned above is compared, and the uniformity and convergence of Pareto front of the proposed algorithm performs better than others'.

Key words: nonlinear equation system, MONES transformation technique, dynamic crowding distance, non-dominated sorting genetic algorithm

CLC Number: 

  • O241

Fig.1

Flow diagram of MNSGA-Ⅱ algorithm"

Table 1

Characteristics of thirty NES"

方程组 n① Range② LE③ NE④ R⑤ max_gen⑥
F1 20 [-1, 1]n 0 2 2 500
F2 2 [-1, 1]n 1 1 11 500
F3 2 [-1, 1]n 0 2 15 500
F4 2 [-10, 10]n 0 2 13 500
F5 10 [-2, 2]n 0 10 1 500
F6 2 [-1, 1]n 4 1 8 500
F7 2 [0, 1], [-10, 0] 0 2 2 500
F8 2 [0, 1]n 0 2 7 500
F9 5 [-10, 10]n 0 1 3 500
F10 3 [-5, 5], [-1, 3], [-5, 5] 0 3 2 500
F11 2 [-1, 1], [-10, 10] 0 2 4 500
F12 2 [-1, 2]n 0 2 10 500
F13 3 [-0.6, 6], [-0.6, 0.6], [-5, 5] 0 3 12 500
F14 2 [-5, 5]n 0 2 9 500
F15 2 [0.25, 1], [1.5, 2π] 0 2 2 500
F16 2 [0, 2π]n 0 2 13 500
F17 8 [-1, 1]n 1 7 16 500
F18 2 [-2, 2]n 0 2 6 500
F19 20 [-2, 2]n 19 1 2 500
F20 3 [-1, 1]n 0 3 7 500
F21 2 [-2, 2]n 0 2 4 500
F22 2 [-2, 2]n 0 2 6 500
F23 3 [-20, 20]n 0 3 16 500
F24 3 [0, 1]n 0 3 8 500
F25 3 [-3, 3]n 0 3 2 500
F26 2 [-1, -0.1], [-2, 2] 0 2 2 500
F27 2 [-5, 1.5], [0, 5] 0 2 3 500
F28 2 [0, 2],[10, 30] 0 2 2 500
F29 3 [0, 2], [-10, 10], [-1, 1] 0 3 5 500
F30 2 [-2, 2], [0, 1.1] 0 2 4 500

Table 2

Root-found ratio RR and success rate RS of different algorithms"

方程组MONES_NSGA-Ⅱ MONES_DNSGA-Ⅱ MONES_MNSGA-Ⅱ
RR RS RR RS RR RS
F1 0.983 3 0.966 7   0.000 0 0.000 0   0.000 0 0.000 0
F2 0.942 4 0.566 7 0.969 6 0.700 0 0.993 9 0.933 3
F3 0.937 8 0.333 3 0.940 0 0.366 7 0.962 2 0.533 3
F4 0.589 7 0.000 0 0.610 3 0.000 0 0.612 8 0.000 0
F5 0.933 3 0.933 3 1.000 0 1.000 0 1.000 0 1.000 0
F6 1.000 0 1.000 0 0.933 3 0.933 3 1.000 0 1.000 0
F7 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F8 0.895 2 0.266 7 0.909 5 0.366 7 0.928 6 0.500 0
F9 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0
F10 0.383 3 0.000 0 0.500 0 0.133 3 0.516 7 0.100 0
F11 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F12 0.806 7 0.166 7 0.820 0 0.233 3 0.846 7 0.333 3
F13 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F14 0.177 8 0.000 0 0.1852 0.000 0 0.207 4 0.000 0
F15 0.500 0 0.000 0 0.500 0 0.000 0 0.500 0 0.000 0
F16 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F17 0.133 3 0.000 0 0.158 3 0.033 3 0.166 7 0.033 3
F18 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F19 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0
F20 0.685 7 0.033 3 0.719 0 0.066 7 0.742 9 0.133 3
F21 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F22 0.977 8 0.933 3 1.000 0 1.000 0 1.000 0 1.000 0
F23 0.006 3 0.000 0 0.033 3 0.000 0 0.035 4 0.000 0
F24 0.729 2 0.033 3 0.766 7 0.000 0 0.779 2 0.066 7
F25 0.016 7 0.000 0 0.050 0 0.000 0 0.050 0 0.000 0
F26 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F27 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
F28 0.150 0 0.000 0 0.216 7 0.000 0 0.283 3 0.000 0
F29 0.5133 0.000 0 0.566 7 0.066 7 0.640 0 0.033 3
F30 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
均值 0.678 7 0.474 4   0.696 0 0.496 7   0.708 9 0.522 2

Table 3

Friedman test"

算法 RR RS
MONES_NSGA-Ⅱ 2.550 0 2.333 3
MONES_DNSGA-Ⅱ 1.966 7 1.866 7
MONES_MNSGA-Ⅱ 1.483 3 1.716 7

Table 4

Wilcoxon test"

MONES_MNSGA-Ⅱ VS RR   RS
R+① R-② p-value③ R+① R-② p-value⑤
MONES_NSGA-Ⅱ 465 0 0.563 0   465 0 0.466 8
MONES_DNSGA-Ⅱ 465 0 0.806 0 420 45 0.809 6

Table 5

ZDT test function and characteristics"

测试函数 公式 决策空间 PF特征
ZDT1 $\begin{aligned}& f_1(\boldsymbol{X})=x_1 \\& f_2(\boldsymbol{X})=g(\boldsymbol{X})\left(1-\sqrt{f_1 / g(\boldsymbol{X})}\right) \\& g(\boldsymbol{X})=1+9 \cdot\left(\sum_{i=2}^n x_i\right) /(n-1)\end{aligned}$ n=30 xi∈[0, 1] 凸型连续
ZDT2 $\begin{aligned}& f_1(\boldsymbol{X})=x_1 \\& f_2(\boldsymbol{X})=g(\boldsymbol{X})\left(1-\left(f_1 / g(\boldsymbol{X})\right)^2\right) \\& g(\boldsymbol{X})=1+9 \cdot\left(\sum_{i=2}^n x_i\right) /(n-1)\end{aligned}$ n=30 xi∈[0, 1] 凹型连续
ZDT3 $\begin{aligned}& f_1(\boldsymbol{X})=x_1 \\& f_2(\boldsymbol{X})=g(\boldsymbol{X})\left(1-\sqrt{f_1 / g(\boldsymbol{X})}-f_1 / g(\boldsymbol{X}) \cdot \sin \left(10 \pi x_1\right)\right) \\& g(X)=1+9 \cdot\left(\sum_{i=2}^n x_i\right) /(n-1)\end{aligned}$ n=30 xi∈[0, 1] 凸型不连续

Fig.2

PF obtained from solving ZDT1 test function using three algorithms"

Fig.3

PF obtained from solving ZDT2 test function using three algorithms"

Fig.4

PF obtained from solving ZDT3 test function using three algorithms"

1 FUJITA H , CIMR D . Computer aided detection for fibrillations and flutters using deep convolutional neural network[J]. Information Sciences, 2019, 486, 231- 239.
doi: 10.1016/j.ins.2019.02.065
2 ZHAO J , XU Y T , FUJITA H . An improved non-parallel Universum support vector machine and its safe sample screening rule[J]. Knowledge-Based Systems, 2019, 170, 79- 88.
doi: 10.1016/j.knosys.2019.01.031
3 WU S , WANG H J . A modified Newton-like method for nonlinear equations[J]. Computational and Applied Mathematics, 2020, 39 (3): 238.
doi: 10.1007/s40314-020-01283-8
4 WANG X F , JIN Y F H , ZHAO Y L . Derivative-free iterative methods with some Kurchatov-type accelerating parameters for solving nonlinear systems[J]. Symmetry, 2021, 13 (6): 943.
doi: 10.3390/sym13060943
5 YUAN G L , LI T T , HU W J . A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems[J]. Applied Numerical Mathematics, 2020, 147, 129- 141.
doi: 10.1016/j.apnum.2019.08.022
6 WANG K , GONG W Y , LIAO Z W , et al. Hybrid niching-based differential evolution with two archives for nonlinear equation system[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2022, 52 (12): 7469- 7481.
doi: 10.1109/TSMC.2022.3157816
7 ALGELANY A M , EL-SHORBAGY M A . Chaotic enhanced genetic algorithm for solving the nonlinear system of equations[J]. Computational Intelligence and Neuroscience, 2022, 2022, 1376479.
8 PAN L Q , ZHAO Y , LI L H . Neighborhood-based particle swarm optimization with discrete crossover for nonlinear equation systems[J]. Swarm and Evolutionary Computation, 2022, 69, 101019.
doi: 10.1016/j.swevo.2021.101019
9 NOBAHARI H , NASROLLAHI S . A terminal guidance algorithm based on ant colony optimization[J]. Computers and Electrical Engineering, 2019, 77, 128- 146.
doi: 10.1016/j.compeleceng.2019.05.012
10 GAO W F , LUO Y T , XU J W , et al. Evolutionary algorithm with multiobjective optimization technique for solving nonlinear equation systems[J]. Information Sciences, 2020, 541, 345- 361.
doi: 10.1016/j.ins.2020.06.042
11 SONG W , WANG Y , LI H X , et al. Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization[J]. IEEE Transactions on Evolutionary Computation, 2014, 19 (3): 414- 431.
12 DEB K , PRATAP A , AGRAWAL S , et al. A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. IEEE Transactions on Evolutionary Computation, 2002, 6 (2): 182- 197.
doi: 10.1109/4235.996017
13 GONG W Y , WANG Y , CAI Z H , et al. A weighted biobjective transformation technique for locating multiple optimal solutions of nonlinear equation systems[J]. IEEE Transactions on Evolutionary Computation, 2017, 21 (5): 697- 713.
doi: 10.1109/TEVC.2017.2670779
14 JI J Y , WONG M L . Decomposition-based multiobjective optimization for nonlinear equation systems with many and infinitely many roots[J]. Information Sciences, 2022, 610, 605- 623.
doi: 10.1016/j.ins.2022.07.187
15 LIANG Z P , WU T C , MA X L , et al. A dynamic multiobjective evolutionary algorithm based on decision variable classification[J]. IEEE Transactions on Cybernetics, 2022, 52 (3): 1602- 1615.
doi: 10.1109/TCYB.2020.2986600
16 ZHANG K , SHEN C N , LIU X M , et al. Multiobjective evolution strategy for dynamic multiobjective optimization[J]. IEEE Transactions on Evolutionary Computation, 2020, 24 (5): 974- 988.
doi: 10.1109/TEVC.2020.2985323
17 WANG J H , LIANG G X , ZHANG J . Cooperative differential evolution framework for constrained multiobjective optimization[J]. IEEE Transactions on Cybernetics, 2018, 49 (6): 2060- 2072.
18 LUO B, ZHENG J H, XIE J L, et al. Dynamic crowding distance: a new diversity maintenance strategy for MOEAs[C]//2008 Fourth International Conference on Natural Computation. New York: IEEE, 2008: 580-585.
19 LIAO Z W , GONG W Y , WANG L , et al. A decomposition-based differential evolution with reinitialization for nonlinear equations systems[J]. Knowledge-Based Systems, 2020, 191, 105312.
doi: 10.1016/j.knosys.2019.105312
20 廖作文, 龚文引, 王凌. 基于改进环拓扑混合群体智能算法的非线性方程组多根联解[J]. 中国科学: 信息科学, 2020, 50 (3): 396- 407.
LIAO Zuowen , GONG Wenyin , WANG Ling . A hybrid swarm intelligence with improved ring topology for nonlinear equations[J]. Scientia Sinica Informationis, 2022, 50 (3): 396- 407.
21 GONG W Y , WANG Y , CAI Z H , et al. Finding multiple roots of nonlinear equation systems via a repulsion-based adaptive differential evolution[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2018, 50 (4): 1499- 1513.
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