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Adaptive α-stable differential evolution in numerical optimization

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An Erratum to this article was published on 28 March 2017

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Abstract

Although Differential Evolution (DE) is a simple yet powerful evolutionary algorithm, it requires an adaptive parameter control to achieve its optimal performance. In this paper, DE with an adaptive parameter control using the \(\alpha\)-stable distribution is proposed. First, the proposed algorithm allocated a carefully calculated stable distribution, evaluated by an adaptation manner, to each individual. After that, each individual adjusts its own control parameters by using the assigned stable distribution. Thus, we propose a parameter control scheme that adapts the stability parameter of the \(\alpha\)-stable distribution to allocate proper stable distributions to each individual, used for tuning control parameters. We compared the optimization performances of the proposed algorithm with conventional DE and state-of-the-art DE variants at 30 and 100 dimensions of conventional benchmark problems. Also, we evaluated the optimization performances at high dimensional problems i.e., 100, 200, and 300 dimensions of CEC2008 benchmark problems. Our experiment results showed that the proposed algorithm is able to discover better final solutions than the compared DE algorithms and has the robust performance at both lower and higher dimensions.

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  • 28 March 2017

    An erratum to this article has been published.

References

  • Abbass HA, Sarker R (2002) The pareto differential evolution algorithm. Int J Artif Intell Tools 11(04):531–552

    Article  Google Scholar 

  • Ali M, Pant M (2011) Improving the performance of differential evolution algorithm using cauchy mutation. Soft Comput 15(5):991–1007

    Article  Google Scholar 

  • Angeline PJ (1995) Adaptive and self-adaptive evolutionary computations. In: Computational intelligence: a dynamic systems perspective. Citeseer

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

    Article  Google Scholar 

  • Brest J, Maučec MS (2008) Population size reduction for the differential evolution algorithm. Appl Intell 29(3):228–247

    Article  Google Scholar 

  • Brest J, Zamuda A, Boskovic B, Maucec MS, Zumer V (2008) High-dimensional real-parameter optimization using self-adaptive differential evolution algorithm with population size reduction. In: IEEE congress on evolutionary computation, 2008. CEC 2008. (IEEE World Congress on Computational Intelligence), pp 2032–2039. IEEE

  • Chambers JM, Mallows CL, Stuck B (1976) A method for simulating stable random variables. J Am Stat Assoc 71(354):340–344

    Article  MATH  MathSciNet  Google Scholar 

  • Choi TJ, Ahn CW, An J (2013) An adaptive cauchy differential evolution algorithm for global numerical optimization. Sci World J 2013

  • Choi TJ, Ahn CW (2015a) An adaptive cauchy differential evolution algorithm with population size reduction and modified multiple mutation strategies. In: Proceedings of the 18th Asia Pacific symposium on intelligent and evolutionary systems, vol 2. Springer, p 13–26

  • Choi TJ, Ahn CW (2015b) An adaptive population resizing scheme for differential evolution in numerical optimization. J Comput Theor Nanosci 12(7):1336–1350

    Article  Google Scholar 

  • Choi TJ, Ahn CW (2014) An adaptive cauchy differential evolution algorithm with bias strategy adaptation mechanism for global numerical optimization. J Comput 9(9):2139–2145

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Eiben AE, Hinterding R, Michalewicz Z (1999) Parameter control in evolutionary algorithms. IEEE Trans Evolut Comput 3(2):124–141

    Article  Google Scholar 

  • Eiben AE, Smith JE (2003) Introduction to evolutionary computing. Springer, New York

    Book  MATH  Google Scholar 

  • Fogel LJ, Owens AJ, Walsh MJ (1998) Artificial intelligence through simulated evolution. In: Fogel DB (ed) Evolutionary computation: the fossil record, pp 227–296. Wiley/IEEE Press. doi:10.1109/9780470544600.ch7

  • Gao Y, Wang YJ (2007) A memetic differential evolutionary algorithm for high dimensional functions’ optimization. In: Third international conference on natural computation, 2007. ICNC 2007, vol. 4. p 188–192. IEEE

  • Gong W, Cai Z, Wang Y (2014) Repairing the crossover rate in adaptive differential evolution. Appl Soft Comput 15:149–168

    Article  Google Scholar 

  • Gong W, Cai Z, Liang D (2015a) Adaptive ranking mutation operator based differential evolution for constrained optimization. IEEE Trans Cybern 45(4):716–727

    Article  Google Scholar 

  • Gong W, Zhou A, Cai Z (2015b) A multioperator search strategy based on cheap surrogate models for evolutionary optimization. IEEE Trans Evolut Comput 19(5):746–758

    Article  Google Scholar 

  • Hall P (1981) A comedy of errors: the canonical form for a stable characteristic function. Bull Lond Math Soc 13(1):23–27

    Article  MATH  MathSciNet  Google Scholar 

  • He RJ, Yang ZY (2012) Differential evolution with adaptive mutation and parameter control using lévy probability distribution. J Comput Sci Technol 27(5):1035–1055

    Article  MATH  MathSciNet  Google Scholar 

  • Kukkonen S, Lampinen J (2005) Gde3: The third evolution step of generalized differential evolution. In: The 2005 IEEE congress on evolutionary computation, 2005, vol. 1. p 443–450. IEEE

  • Kukkonen S, Lampinen J (2006) Constrained real-parameter optimization with generalized differential evolution. In: IEEE congress on evolutionary computation, 2006. CEC 2006. p 207–214. IEEE

  • Lampinen J (2002) A constraint handling approach for the differential evolution algorithm. In: Proceedings of the world on congress on computational intelligence, vol. 2. p 1468–1473. IEEE

  • Lee CY, Yao X (2004) Evolutionary programming using mutations based on the lévy probability distribution. IEEE Trans Evolut Comput 8(1):1–13

    Article  Google Scholar 

  • Lévy P (1934) Sur les intégrales dont les éléments sont des variables aléatoires indépendantes. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 3(3–4):337–366

    MATH  MathSciNet  Google Scholar 

  • Noman N, Iba H (2005) Enhancing differential evolution performance with local search for high dimensional function optimization. In: Proceedings of the 2005 conference on Genetic and evolutionary computation, p 967–974. ACM

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

    MATH  Google Scholar 

  • Qin AK, Suganthan PN (2005) Self-adaptive differential evolution algorithm for numerical optimization. In: The 2005 IEEE congress on evolutionary computation, 2005, vol. 2. p 1785–1791. IEEE

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Robič T, Filipič B (2005) Demo: differential evolution for multiobjective optimization. Evolutionary multi-criterion pptimization. Springer, New York, pp 520–533

    Google Scholar 

  • Ronkkonen J, Kukkonen S, Price KV (2005) Real-parameter optimization with differential evolution. Proc IEEE CEC 1:506–513

    Google Scholar 

  • Santana-Quintero LV, Coello CAC (2005) An algorithm based on differential evolution for multi-objective problems. Int J Comput Intell Res 1(1):151–169

    MathSciNet  Google Scholar 

  • Storn R (1999) System design by constraint adaptation and differential evolution. IEEE Trans Evolut Comput 3(1):22–34

    Article  Google Scholar 

  • Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359

    Article  MATH  MathSciNet  Google Scholar 

  • Tanabe R, Fukunaga A (2013) Success-history based parameter adaptation for differential evolution. In: IEEE congress on evolutionary computation (CEC), 2013. p 71–78. IEEE

  • Tang K, Yang P, Yao X (2016) Negatively correlated search. IEEE J Sel Areas Commun 34(3):542–550

    Article  Google Scholar 

  • Tang K, Yáo X, Suganthan PN, MacNish C, Chen YP, Chen CM, Yang Z (2007) Benchmark functions for the cec2008 special session and competition on large scale global optimization. Nature inspired computation and applications laboratory, USTC, China, p 153–177

  • Weron R (1996a) Correction to:“on the chambers-mallows-stuck method for simulating skewed stable random variables”. Hugo Steinhaus Center, Wroclaw University of Technology, Tech. rep

  • Weron R (1996b) On the chambers-mallows-stuck method for simulating skewed stable random variables. Stat Probab Lett 28(2):165–171

    Article  MATH  MathSciNet  Google Scholar 

  • Weron R (2004) Computationally intensive value at risk calculations. Papers/Humboldt-Universität Berlin, Center for Applied Statistics and Economics (CASE), Tech. rep

  • Xue F, Sanderson AC, Graves RJ (2003) Pareto-based multi-objective differential evolution. In: The 2003 congress on evolutionary computation, 2003. CEC’03, vol. 2. p 862–869. IEEE

  • Yang Z, Tang K, Yao X (2008) Large scale evolutionary optimization using cooperative coevolution. Inf Sci 178(15):2985–2999

    Article  MATH  MathSciNet  Google Scholar 

  • Yang Z, Tang K, Yao X (2007) Differential evolution for high-dimensional function optimization. In: IEEE congress on evolutionary computation, 2007. CEC 2007, p 3523–3530. IEEE

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Zhang M, Luo W, Wang X (2008) Differential evolution with dynamic stochastic selection for constrained optimization. Inf Sci 178(15):3043–3074

    Article  Google Scholar 

  • Zielinski K, Laur R (2006) Constrained single-objective optimization using differential evolution. In: IEEE congress on evolutionary computation, 2006. CEC 2006, pp. 223–230. IEEE

Download references

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2015R1D1A1A02062017). Correspondence should be addressed to Dr. Chang Wook Ahn; cwan@skku.edu.

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Authors and Affiliations

  1. Department of Computer Engineering, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea

    Tae Jong Choi & Chang Wook Ahn

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  1. Tae Jong Choi
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  2. Chang Wook Ahn
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Correspondence to Chang Wook Ahn.

Additional information

An erratum to this article is available at https://doi.org/10.1007/s11047-017-9621-6.

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Choi, T.J., Ahn, C.W. Adaptive α-stable differential evolution in numerical optimization. Nat Comput 16, 637–657 (2017). https://doi.org/10.1007/s11047-016-9579-9

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