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Soft Computing

, Volume 23, Issue 22, pp 12001–12015 | Cite as

A hybrid evolutionary-simplex search method to solve nonlinear constrained optimization problems

  • Alyaa AbdelhalimEmail author
  • Kazuhide Nakata
  • Mahmoud El-Alem
  • Amr Eltawil
Methodologies and Application

Abstract

This research article presents a novel design of a hybrid evolutionary-simplex search method to solve the class of general nonlinear constrained optimization problems. In this article, the particle swarm optimization (PSO) method and the Nelder–Mead (NM) simplex search algorithm are utilized in a unified way to enhance the overall performance of the proposed solution method. The NM algorithm is used as an integrative step in the PSO method to reinforce the convergence of the PSO method and overcome the global search weakness in the NM algorithm. On the other hand, a penalty function technique is embedded in the proposed method to solve constrained optimization problems. Two levels of numerical experiments were conducted to evaluate the proposed method. First, a comparison is conducted with well-known benchmark problems. Second, the proposed method is tested in solving three engineering design optimization problems. In addition, the results of the proposed method were compared to optimization methods published in the literature in three main criteria: effectiveness, efficiency and robustness. The results show the competitive performance of the proposed method in this article.

Keywords

Particle swarm optimization Nonlinear optimization Constrained optimization problem Simplex search algorithm 

Notes

Acknowledgements

The Egyptian Ministry of Higher Education (MOHE) grant and the Japanese International Cooperation Agency (JICA) in the scope of the Egypt Japan University of Science and Technology (E-JUST) sponsored this research.

Compliance with ethical standards

Conflict of interest

The Authors listed in this article declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Abdelhalim A, Nakata K, El-Alem M, Eltawil A (2017) Guided particle swarm optimization method to solve general nonlinear optimization problems. Eng Optim.  https://doi.org/10.1080/0305215x.2017.1340945 CrossRefGoogle Scholar
  2. Arora JS (2004) Introduction to optimum design. Elsevier, AmsterdamCrossRefGoogle Scholar
  3. Baudin M (2010) Nelder–Mead user’s manual. Consortium Scilab, Digiteo. https://www.scilab.org/sites/default/files/neldermead.pdf
  4. Bertsekas DP (2014) Constrained optimization and lagrange multiplier methods. Elsevier, AmsterdamzbMATHGoogle Scholar
  5. Carroll CW (1961) The created response surface technique for optimizing nonlinear, restrained systems. Oper Res 9:169–184.  https://doi.org/10.1287/opre.9.2.169 MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chelouah R, Siarry P (2000) A continuous genetic algorithm designed for the global optimization of multimodal functions. J Heuristics 6:191–213.  https://doi.org/10.1023/A:1009626110229 CrossRefzbMATHGoogle Scholar
  7. Chelouah R, Siarry P (2003) Genetic and Nelder–Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions. Eur J Oper Res 148:335–348.  https://doi.org/10.1016/S0377-2217(02)00401-0 MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chelouah R, Siarry P (2005) A hybrid method combining continuous tabu search and Nelder–Mead simplex algorithms for the global optimization of multiminima functions. Eur J Oper Res 161:636–654.  https://doi.org/10.1016/j.ejor.2003.08.053 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Coello CA (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41:113–127.  https://doi.org/10.1016/S0166-3615(99)00046-9 CrossRefGoogle Scholar
  10. Coello CA (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191:1245–1287.  https://doi.org/10.1016/S0045-7825(01)00323-1 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Coello CA, Becerra RL (2004) Efficient evolutionary optimization through the use of a cultural algorithm. Eng Optim 36:219–236.  https://doi.org/10.1080/03052150410001647966 CrossRefGoogle Scholar
  12. Coello C, Carlos A (1999) A survey of constraint handling techniques used with evolutionary algorithms. Lania-RI-99-04, Lab NacGoogle Scholar
  13. Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc 49:1–23MathSciNetCrossRefGoogle Scholar
  14. Deng W, Chen R, He B et al (2012) A novel two-stage hybrid swarm intelligence optimization algorithm and application. Soft Comput 16:1707–1722.  https://doi.org/10.1007/s00500-012-0855-z CrossRefGoogle Scholar
  15. Deng W, Zhao H, Liu J et al (2014) An improved CACO algorithm based on adaptive method and multi-variant strategies. Soft Comput 19:701–713.  https://doi.org/10.1007/s00500-014-1294-9 CrossRefGoogle Scholar
  16. Deng W, Yao R, Zhao H et al (2017a) A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm. Soft Comput.  https://doi.org/10.1007/s00500-017-2940-9 CrossRefGoogle Scholar
  17. Deng W, Zhao H, Yang X et al (2017b) Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl Soft Comput J 59:288–302.  https://doi.org/10.1016/j.asoc.2017.06.004 CrossRefGoogle Scholar
  18. Deng W, Zhao H, Zou L et al (2017c) A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput 21:4387–4398.  https://doi.org/10.1007/s00500-016-2071-8 CrossRefGoogle Scholar
  19. Deng W, Zhang S, Zhao H, Yang X (2018) A novel fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing. IEEE Access 6:35042–35056.  https://doi.org/10.1109/ACCESS.2018.2834540 CrossRefGoogle Scholar
  20. Dennis JE Jr, Woods DJ (1987) Optimization on microcomputers: the Nelder–Mead simplex algorithm. New computing environements: microcomputers in large scale computing. SIAM, PhiladelphiaGoogle Scholar
  21. Fan S-KS, Zahara E (2007) A hybrid simplex search and particle swarm optimization for unconstrained optimization. Eur J Oper Res 181:527–548.  https://doi.org/10.1016/j.ejor.2006.06.034 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Fiacco AV, McCormick GP (1966) Extensions of SUMT for nonlinear programming: equality constraints and extrapolation. Manag Sci 12:816–828.  https://doi.org/10.1287/mnsc.12.11.816 MathSciNetCrossRefzbMATHGoogle Scholar
  23. He Q, Wang L (2007a) A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Appl Math Comput 186:1407–1422.  https://doi.org/10.1016/j.amc.2006.07.134 MathSciNetCrossRefzbMATHGoogle Scholar
  24. He Q, Wang L (2007b) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20:89–99.  https://doi.org/10.1016/j.engappai.2006.03.003 CrossRefGoogle Scholar
  25. Hedar A, Fukushima M (2003) Minimizing multimodal functions by simplex coding genetic algorithm. Optim Methods Softw 18:265–282MathSciNetCrossRefGoogle Scholar
  26. Kannan BK, Kramer SN (1994) An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J Mech Des 116:405.  https://doi.org/10.1115/1.2919393 CrossRefGoogle Scholar
  27. Kayhan AH, Ceylan H, Tamer Ayvaz M, Gurarslan G (2010) PSOLVER: a new hybrid particle swarm optimization algorithm for solving continuous optimization problems. Expert Syst Appl 37:6798–6808.  https://doi.org/10.1016/j.eswa.2010.03.046 CrossRefGoogle Scholar
  28. Kelley CT (1999) Detection and remediation of stagnation in the Nelder–Mead algorithm using a sufficient dcrease conditions. SIAM J Optim 10:43–55MathSciNetCrossRefGoogle Scholar
  29. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN’95—international conference on neural networks. IEEE, pp 1942–1948Google Scholar
  30. Kennedy J, Eberhart R, Shi Y (2001) Swarm intelligence. Morgan Kaufmann Publishers, San FranciscoGoogle Scholar
  31. Kou X, Liu S, Zhang J, Zheng W (2009) Co-evolutionary particle swarm optimization to solve constrained optimization problems. Comput Math Appl 57:1776–1784CrossRefGoogle Scholar
  32. Liang JJ, Runarsson TP, Clerc M et al (2006) Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization. Evol Comput 251–256. http://www3.ntu.edu.sg/home/epnsugan/index_files/CEC10-Const/TR-April-2010.pdf
  33. Mazhoud I, Hadj-Hamou K, Bigeon J, Joyeux P (2013) Particle swarm optimization for solving engineering problems: a new constraint-handling mechanism. Eng Appl Artif Intell 26:1263–1273.  https://doi.org/10.1016/j.engappai.2013.02.002 CrossRefGoogle Scholar
  34. Michalewicz Z (1995) A survey of constraint handling techniques in evolutionary computation methods. Evol Program 4:135–155Google Scholar
  35. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313.  https://doi.org/10.1093/comjnl/7.4.308 MathSciNetCrossRefzbMATHGoogle Scholar
  36. Petalas YG, Parsopoulos KE, Vrahatis MN (2007) Memetic particle swarm optimization. Ann Oper Res 156:99–127.  https://doi.org/10.1007/s10479-007-0224-y MathSciNetCrossRefzbMATHGoogle Scholar
  37. Pulido GT, Coello CaC (2004) A constraint-handling mechanism for particle swarm optimization. In: Proceedings of 2004 congress on evolutionary computation (IEEE Cat No04TH8753) 2, pp 1396–1403.  https://doi.org/10.1109/cec.2004.1331060
  38. Ropke S (2005) Heuristic and exact algorithms for vehicle routing problems. Unpubl Ph.D. thesis, Comput Sci Dep Univ Copenhagen 256Google Scholar
  39. Runarsson TP (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4:284–294.  https://doi.org/10.1109/4235.873238 CrossRefGoogle Scholar
  40. Siarry P, Berthiau G, Durdin F, Haussy J (1997) Enhanced simulated annealing for globally minimizing functions of many-continuous variables. ACM Trans Math Softw 23:209–228.  https://doi.org/10.1145/264029.264043 MathSciNetCrossRefzbMATHGoogle Scholar
  41. Sun C, Zeng J, Pan J (2011) An improved vector particle swarm optimization for constrained optimization problems. Inf Sci (Ny) 181:1153–1163.  https://doi.org/10.1016/j.ins.2010.11.033 CrossRefGoogle Scholar
  42. Wang PC, Shoup TE (2011) Parameter sensitivity study of the Nelder–Mead simplex method. Adv Eng Softw 42:529–533.  https://doi.org/10.1016/j.advengsoft.2011.04.004 CrossRefzbMATHGoogle Scholar
  43. Wang Y, Cai Z, Zhou Y (2009) Accelerating adaptive trade-off model using shrinking space technique for constrained evolutionary optimization. Int J Numer Methods Eng 77:1501–1534.  https://doi.org/10.1002/nme.2451 MathSciNetCrossRefzbMATHGoogle Scholar
  44. Yadav A, Deep K (2014) An efficient co-swarm particle swarm optimization for non-linear constrained optimization. J Comput Sci 5:258–268.  https://doi.org/10.1016/j.jocs.2013.05.011 CrossRefGoogle Scholar
  45. Yang X-S (2010) Engineering optimization: an introduction with metaheuristic applications. Wiley, HobokenCrossRefGoogle Scholar
  46. Yang X-S, Deb S, Fong S (2011) Accelerated particle swarm optimization and support vector machine for business optimization and applications. In: Fong S (ed) Networked Digital Technologies. NDT 2011. Communications in Computer and Information Science, vol 136. Springer, Berlin, HeidelbergCrossRefGoogle Scholar
  47. Zahara E, Kao Y-T (2009) Hybrid Nelder–Mead simplex search and particle swarm optimization for constrained engineering design problems. Expert Syst Appl 36:3880–3886.  https://doi.org/10.1016/j.eswa.2008.02.039 CrossRefGoogle Scholar
  48. Zhao H, Sun M, Deng W, Yang X (2017) A new feature extraction method based on EEMD and multi-scale fuzzy entropy for motor bearing. Entropy.  https://doi.org/10.3390/e19010014 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Alyaa Abdelhalim
    • 1
    Email author
  • Kazuhide Nakata
    • 2
  • Mahmoud El-Alem
    • 3
  • Amr Eltawil
    • 4
  1. 1.Production Engineering DepartmentAlexandria UniversityAlexandriaEgypt
  2. 2.Department of Industrial Engineering and EconomicsTokyo Institute of TechnologyTokyoJapan
  3. 3.Department of Mathematics, Faculty of ScienceAlexandria UniversityAlexandriaEgypt
  4. 4.Department of Industrial Engineering and Systems ManagementEgypt Japan University of Science and TechnologyNew Borg Elarab City, AlexandriaEgypt

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