Design of normal distribution-based algorithm for solving systems of nonlinear equations

Document Type : Research Paper


Department of Industrial Engineering, School of Engineering, Damghan University, Damghan, Iran.


In this paper, a completely new statistical-based approach is developed for solving the system of nonlinear equations. The developed approach utilizes the characteristics of the normal distribution to search the solution space. The normal distribution is generally introduced by two parameters, i.e., mean and standard deviation. In the developed algorithm, large values of standard deviation enable the algorithm to escape from a local optimum, and small values of standard deviation help the algorithm to find the global optimum. In the following, six benchmark tests and thirteen benchmark case problems are investigated to evaluate the performance of the Normal Distribution-based Algorithm (NDA). The obtained statistical results of NDA are compared with those of PSO, ICA, CS, and ACO. Based on the obtained results, NDA is the least time-consuming algorithm that gets high-quality solutions. Furthermore, few input parameters and simple structure introduce NDA as a user friendly and easy-to-understand algorithm. 


  • [1]          A.  Abdelhalim,  K.   Nakata,   M.   El-Alem,   and   A.   Eltawil,   A   hybrid   evolutionary-simplex   search   method to solve nonlinear constrained optimization problems, Soft Computing, 23(22) (2019), 12001-12015, Doi: 10.1080/0305215x.2017.1340945.
  • [2]          M. Abdollahi, A. Bouyer, and D. Abdollahi, Improved cuckoo optimization algorithm for solving  systems  of  nonlinear equations, The Journal of Supercomputing, 72(3) (2016), 1246-1269, Doi:10.1007/s11227-016-1660-8.
  • [3]          M.   Abdollahi,   A.   Isazadeh,   and   D.   Abdollahi,   Imperialist    competitive    algorithm    for    solving    sys-  tems of nonlinear equations, Computers and Mathematics with Applications, 65(12) (2013), 1894-1908, Doi:10.1016/j.camwa.2013.04.018.
  • [4]          E.  Atashpaz-Gargari  and  C.  Lucas,  Imperialist  competitive  algorithm:  an   algorithm   for   optimization   in- spired by imperialistic competition, IEEE congress on evolutionary computation, (2007), 4661-4667, Doi: 10.1109/CEC.2007.4425083.
  • [5]          X. Chen and C. T. Kelley, Convergence of the EDIIS algorithm for nonlinear equations, SIAM Journal on Scientific Computing, 41(1) (2019), A365-A379, Doi:
  • [6]          I.M. El-Emary and M.A. El-Kareem, Towards using genetic algorithm for solving nonlinear  equation  systems,  World Applied Sciences Journal, 5(3) (2008), 282-289.
  • [7]          C.  Grosan  and  A.  Abraham,  A  new  approach  for   solving   nonlinear   equations   systems,   IEEE   Transac- tions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 38(3) (2008), 698-714, Doi: 10.1109/TSMCA.2008.918599.
  • [8]          M.J. Hirsch, P.M. Pardalos, and M. G. Resende, Solving systems of nonlinear equations with continuous GRASP, Nonlinear Analysis: Real World Applications, 10(4) (2009), 2000-2006, Doi: 10.1016/j.nonrwa.2008.03.006.
  • [9]          S. Hosseini and A. Al  Khaled,  A  survey  on  the  imperialist  competitive  algorithm  metaheuristic:  implementa-  tion in engineering domain and directions for future research, Applied Soft Computing, 24 (2014), 1078-1094, Doi:10.1016/j.asoc.2014.08.024.
  • [10]        A. M. Ibrahim and M. A. Tawhid, A hybridization of differential evolution and monarch butterfly optimization for solving systems of nonlinear equations, Journal of Computational Design and Engineering, 6(3) (2019a), 354-367, Doi:10.1016/j.jcde.2018.10.006.
  • [11]        A. M. Ibrahim and M. A. Tawhid, A hybridization of cuckoo search and particle swarm optimization for solving nonlinear systems, Evolutionary Intelligence, 12(4) (2019b), 541-561, Doi:10.1016/j.camwa.2006.12.081.
  •  [12]       A. F. Izmailov, A. S Kurennoy, and M. V. Solodov, Critical solutions of nonlinear equations: stability issues, Mathematical Programming, (2016), 1-33, Doi: 10.1007/s10107-016-1047-x.
  • [13]        M. Jaberipour, E. Khorram, and B. Karimi, Particle swarm algorithm for solving systems of nonlinear equations, Computers and Mathematics with Applications, 62(2) (2011), 566-576, Doi: 10.1016/j.camwa.2011.05.031.
  • [14]        M. Juneja and S. K. Nagar, Particle swarm optimization algorithm and its parameters: A review, International Conference on Control, Computing, Communication and Materials (ICCCCM), IEEE, (2016), 1-5, Doi: 10.1109/ICCCCM.2016.7918233.
  • [15]        JA. Koupaei and S. Hosseini, A new hybrid algorithm based on chaotic maps for solving systems of nonlinear equations, Chaos Solitons Fractals, 81 (2015), 233245, Doi: 10.1016/j.chaos.2015.09.027
  • [16]        G. Li and Z. Zeng, A neural-network algorithm for solving nonlinear equation systems, International Conference     on Computational Intelligence and Security (IEEE), 1 (2008), 20-23, Doi: 10.1109/CIS.2008.65.
  • [17]        MD. Li, H. Zhao, XW. Weng, and T. Han, A  novel  nature-inspired  algorithm  for  optimization:  virus  colony  search, Advances in Engineering Software, 92 (2016), 6588, Doi: 10.1016/j.advengsoft.2015.11.004.
  • [18]        L. Liu, M. X. Liu, N. Wang, and P. Zou, Modified cuckoo search algorithm with variational parameters and logistic  map, Algorithms, 11(3) (2018), 30, Doi: 10.3390/a11030030.
  • [19]        M.A. Luersen and R. Le Riche, Globalized NelderMead method for engineering optimization, Computers and structures, 82(23) (2004), 2251-2260, Doi: 10.1016/j.compstruc.2004.03.072.
  • [20]        A. Majd, M. Abdollahi, G. Sahebi, D. Abdollahi, M. Daneshtalab, J.  Plosila,  and  H.  Tenhunen, Multi-population parallel imperialist competitive algorithm for solving systems  of  nonlinear  equations,  In- ternational Conference on High Performance Computing and Simulation (HPCS), IEEE, (2016), 767-775, Doi:10.1109/HPCSim.2016.7568412.
  • [21]        A. Majd, G. Sahebi, M. Daneshtalab, J. Plosila, S. Lotfi, and H. Tenhunen, Parallel imperialist competitive algorithms, Concurrency and Computation: Practice and Experience, 30(7) (2018), e4393, Doi: 10.1002/cpe.4393.
  • [22]        E.L. Melnick and A. Tenenbein, Misspecifications of the normal distribution, The American Statistician, 36(4) (1982), 372-373.
  • [23]        Y.  Mo,  H.  Liu,  and  Q.  Wang,  Conjugate   direction   particle   swarm   optimization   solving   systems   of  nonlinear equations, Computers and Mathematics with Applications, 57(11) (2009), 1877-1882, Doi: 10.1016/j.camwa.2008.10.005.
  • [24]        H. Mhlenbein, M. Schomisch, and J. Born, The parallel genetic algorithm as function optimizer, Parallel comput- ing, 17(6-7) (1991), 619-632, Doi:10.1016/S0167-8191(05)80052-3.
  • [25]        H.A. Oliveira and A. Petraglia, Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing, Applied Soft Computing, 13(11) (2013), 4349-4357, Doi: 10.1016/j.asoc.2013.06.018.
  • [26]        J. Pei,  Z. Drai,  M. Drai,  N. Mladenovi,  and P.M.  Pardalos,  Continuous variable neighborhood  search  (C-VNS)  for solving systems of nonlinear equations, INFORMS Journal on Computing, 31(2) (2019), 235-250, Doi: 10.1287/ijoc.2018.0876.
  • [27]        E. Pourjafari and H.  Mojallali,  Solving  nonlinear  equations  systems  with  a  new  approach  based  on  invasive  weed optimization algorithm and clustering, Swarm and Evolutionary Computation, 4 (2012), 33-43, Doi: 10.1016/j.swevo.2011.12.001.
  • [28]        M. A. Z. Raja, M. A. Zameer, A.K. Kiani, A. Shehzad, and M. A. R. Khan, Nature-inspired computational intelligence integration with NelderMead method to solve nonlinear benchmark models, Neural Computing and Applications, 29(4) (2018), 1169-1193, Doi: 10.1007/s00521-016-2523-1.
  • [29]        J. R. Sharma, I. K. Argyros, and D. Kumar, On a general class of optimal order multipoint methods for solving nonlinear equations, Journal of Mathematical Analysis and Applications, (2016), Doi: 10.1016/j.jmaa.2016.12.051.
  • [30]        J. R. Sharma and H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations, Applied Mathematics and Computation, 222 (2013), 497-506, Doi: 10.1016/j.amc.2013.07.066.
  • [31]        M. Shehab, A.T. Khader, and M. A. Al-Betar, A survey on applications and  variants  of  the  cuckoo  search  algorithm, Applied Soft Computing, 61 (2017), 1041-1059, Doi: 10.1016/j.asoc.2017.02.034.
  • [32]        N. Singh and S. B. Singh, Hybrid algorithm of particle swarm optimization and grey wolf optimizer for improving convergence performance, Journal of Applied Mathematics, (2017), Doi: 10.1016/j.asoc.2017.02.034.
  • [33]        OE. Turgut, MS. Turgut, and MT. Coban, Chaotic quantum behaved particle swarm optimization  algorithm  for solving nonlinear system of equations, Computers and Mathematics with Applications, 68(4) (2014), 508530, Doi: 10.1016/j.camwa.2014.06.013.
  • [34]        C. Wang, R. Luo, K. Wu, and B. Han, A new filled function method for  an  unconstrained  nonlinear  equation, Journal of Computational and Applied Mathematics, 235(6) (2011), 1689-1699, Doi: 10.1016/
  • [35]        X. Wang and N. Zhou, Pattern Search Firefly Algorithm for Solving Systems of Nonlinear Equations, Seventh International Symposium on Computational Intelligence and Design, IEEE, 2 (2014), 228-231, Doi: 10.1109/IS- CID.2014.222.
  • [36]        X. S. Yang and S. Deb, Cuckoo search via Lvy flights, World congress on nature and biologically inspired computing (NaBIC), IEEE, (2009), 210-214, Doi: 10.1109/NABIC.2009.5393690.
  • [37]        X. Zhang, Q. Wan, and Y. Fan, Applying modified cuckoo search algorithm for  solving  systems  of  nonlinear  equations, Neural Computing and Applications, 31(2) (2019), 553-576, Doi: 10.1007/s00521-017-3088-3.