Triangular Gaussian mutation to differential evolution


Differential evolution (DE) has been a popular algorithm for its simple structure and few control parameters. However, there are some open issues in DE regrading its mutation strategies. An interesting one is how to balance the exploration and exploitation behaviour when performing mutation, and this has attracted a growing number of research interests over a decade. To address this issue, this paper presents a triangular Gaussian mutation strategy. This strategy utilizes the physical positions and the fitness differences of the vertices in the triangular structure. Based on this strategy, a triangular Gaussian mutation to DE and its improved version (ITGDE) are suggested. Empirical studies are carried out on the 20 benchmark functions and show that, in comparison with several state-of-the-art DE variants, ITGDE obtains significantly better or at least comparable results, suggesting the proposed mutation strategy is promising for DE.

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  1. Brest J, Greiner S, Bošković B, Mernik M, Zumer V (2006) Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 10(6):646–657

    Article  Google Scholar 

  2. Brest J, Maučec MS (2009) Population size reduction for the differential evolution algorithm. Appl Intell 29(3):228–247

    Article  Google Scholar 

  3. Darwin C (2009) On the origin of the species by means of natural selection or the preservation of favoured races in the struggle for life. Penguin Classics, London

    Google Scholar 

  4. Das S, Abraham A, Chakraborty UK, Konar A (2009) Differential evolution using a neighborhood-based mutation operator. IEEE Trans Evol Comput. 13(3):526–553

    Article  Google Scholar 

  5. Das S, Suganthan PN (2011) Differential evolution: a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31

    Article  Google Scholar 

  6. Das S, Mullick SS, Suganthan PN (2016) Recent advances in differential evolution—an updated survey. Swarm Evol Comput 27(1):1–30

    Article  Google Scholar 

  7. Epitropakis MG, Tasoulis DK, Pavlidis NG, Vrahatis MN (2011) Enhancing differential evolution utilizing proximity-based mutation operators. IEEE Trans Evol Comput 15(1):99–119

    Article  Google Scholar 

  8. Fan HY, Lampinen J (2003) A trigonometric mutation operation to differential evolution. J Glob Optim 27(1):105–129

    MathSciNet  MATH  Article  Google Scholar 

  9. Gamperle R, Muller SD, Koumoutsakos P (2002), A parameter study for differential evolution. In: Proceedings of internationaal conference on advances in intelligent systems, fuzzy systems, evolutionary computation, pp 293–298

  10. García S, Molina D, Lozano M, Herrera F (2009) A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J Heuristics 15(6):617–644

    MATH  Article  Google Scholar 

  11. García S, Fernández A, Luengo J, Herrera F (2010) Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power. Inf Sci 180(10):2044–2064

    Article  Google Scholar 

  12. Ghosh A, Das S, Chowdhury A, Giri R (2011) An improved differential evolution algorithm with fitness-based adaptation of the control parameters. Inf Sci 181(18):3749–3765

    MathSciNet  Article  Google Scholar 

  13. Gong W, Cai Z (2013) Differential evolution with ranking-based mutation operators. IEEE Trans Cybern 43(6):2066–2081

    Article  Google Scholar 

  14. Guo SM, Yang CC, Hsu PH, Tsai JC (2014) Improving differential evolution with successful-parent-selecting framework. IEEE Trans Evol Comput 19(5):717–730

    Article  Google Scholar 

  15. Kennedy J (2003) Bare bones particle swarms. In: Proceedings of IEEE swarm intelligence symposium , pp 80–87

  16. Mallipeddi R, Suganthan PN, Pan QK, Tasgetiren MF (2011) Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl Soft Comput 11(2):1679–1696

    Article  Google Scholar 

  17. Price K, Storn RM, Lampinen JA (2006) Differential evolution: a practical approach to global optimization. Springer, Berlin

    MATH  Google Scholar 

  18. Qin AK, Huang VL, Suganthan PN (2009) Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans Evol Comput 13(2):398–417

    Article  Google Scholar 

  19. Rahnamayan S, Tizhoosh HR, Salama M (2008) Opposition-based differential evolution. IEEE Trans Evol Comput 12(1):64–79

    Article  Google Scholar 

  20. Storn R, Price K (1995) Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces, vol 3. ICSI, Berkeley

    MATH  Google Scholar 

  21. Tanabe R, Fukunaga AS (2013) Success-history based parameter adaptation for differential evolution. In: Proceedings of 2013 IEEE congress on evolutionary computation, pp 71–78

  22. Tanabe R, Fukunaga AS (2014) Improving the search performance of shade using linear population size reduction. In: Proceedings of 2014 IEEE congress on evolutionary computation, pp 1658–1665

  23. Wang H, Wu ZJ, Liu Y, Jiang DZ, Chen LL (2009) Space transformation search: a new evolutionary technique. In: Proceedings of 1st ACM/SIGEVO summit on genetic and evolutionary computation, pp 537–544

  24. Wang H, Rahnamayan S, Sun H, Omran MG (2013) Gaussian bare-bones differential evolution. IEEE Trans Cybern 43(2):634–647

    Article  Google Scholar 

  25. Wang Y, Cai Z, Zhang Q (2011) Differential evolution with composite trial vector generation strategies and control parameters. IEEE Trans Evol Comput 15(1):55–66

    Article  Google Scholar 

  26. Wang Y, Cai ZX, Zhang Q (2012) Enhancing the search ability of differential evolution through orthogonal crossover. Inf Sci 185(1):153–177

    MathSciNet  Article  Google Scholar 

  27. Wang Y, Liu ZZ, Li J, Li HX, Yen GG (2016) Utilizing cumulative population distribution information in differential evolution. Appl Soft Comput 48(1):329–346

    Article  Google Scholar 

  28. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102

    Article  Google Scholar 

  29. Yildiz AR (2013) A new hybrid differential evolution algorithm for the selection of optimal machining parameters in milling operations. Appl Soft Comput 13(3):1561–1566

  30. Yu WJ, Shen M, Chen WN, Zhan ZH, Gong YJ, Lin Y, Liu O, Zhang J (2014) Differential evolution with two-level parameter adaptation. IEEE Trans Cybern 44(7):1080–1099

    Article  Google Scholar 

  31. Zhang J, Sanderson AC (2009) JADE: adaptive differential evolution with optional external archive. IEEE Trans Evol Comput 13(5):945–958

    Article  Google Scholar 

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This work was supported by National Natural Science Foundation of China (61501198), Wuhan Youth Science and Technology Chenguang program (2014072704011248), Natural Science Foundation of Hubei Province (2014CFB461).

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Correspondence to Yong Wu.

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Guo, J., Wu, Y., Xie, W. et al. Triangular Gaussian mutation to differential evolution. Soft Comput 24, 9307–9320 (2020).

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  • Differential evolution
  • Gaussian distribution
  • Triangular structure
  • Global optimum