Journal of Applied Mathematics

Sufficient Conditions for Global Convergence of Differential Evolution Algorithm

Zhongbo Hu, Shengwu Xiong, Qinghua Su, and Xiaowei Zhang

Full-text: Open access

Abstract

The differential evolution algorithm (DE) is one of the most powerful stochastic real-parameter optimization algorithms. The theoretical studies on DE have gradually attracted the attention of more and more researchers. However, few theoretical researches have been done to deal with the convergence conditions for DE. In this paper, a sufficient condition and a corollary for the convergence of DE to the global optima are derived by using the infinite product. A DE algorithm framework satisfying the convergence conditions is then established. It is also proved that the two common mutation operators satisfy the algorithm framework. Numerical experiments are conducted on two parts. One aims to visualize the process that five convergent DE based on the classical DE algorithms escape from a local optimal set on two low dimensional functions. The other tests the performance of a modified DE algorithm inspired of the convergent algorithm framework on the benchmarks of the CEC2005.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 193196, 14 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808166

Digital Object Identifier
doi:10.1155/2013/193196

Mathematical Reviews number (MathSciNet)
MR3122108

Zentralblatt MATH identifier
06950547

Citation

Hu, Zhongbo; Xiong, Shengwu; Su, Qinghua; Zhang, Xiaowei. Sufficient Conditions for Global Convergence of Differential Evolution Algorithm. J. Appl. Math. 2013 (2013), Article ID 193196, 14 pages. doi:10.1155/2013/193196. https://projecteuclid.org/euclid.jam/1394808166


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References

  • R. Storn and K. V. Price, “Differential evolution: a simpleand efficient adaptive scheme for global optimizationover contin-uous spaces,” Tech. Rep., International Computer Science Institute, 1995.
  • S. Das and P. N. Suganthan, “Differential evolution: a survey of the state-of-the-art,” IEEE Transactions on Evolutionary Com-putation, vol. 15, no. 1, pp. 4–31, 2011.
  • S. Das, P. N. Suganthan, and C. A. C. Coello, “Guest editorial special issue on differential evolution,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 1–3, 2011.
  • J. Vesterstrøm and R. Thomsen, “A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems,” in Proceedings of the Congress on Evolutionary Computation (CEC '04), pp. 1980–1987, June 2004.
  • K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evo-lution: A Practical Approach to Global Optimization, Natural Computing Series, Springer, Berlin, Germany, 2005.
  • O. Hrstka and A. Kučerová, “Improvements of real coded gene-tic algorithms based on differential operators preventing premature convergence,” Advances in Engineering Software, vol. 35, no. 3-4, pp. 237–246, 2004.
  • G. Iacca, F. Caraffini, and F. Neri, “Compact differential evolu-tion light: high performance despite limited memory requirement and modest computational overhead,” Journal of Computer Science and Technology, vol. 27, no. 5, pp. 1056–1076, 2012.
  • R. J. He and Z. Y. Yang, “Differential evolution with adaptive mutation and parameter control using Lévy probability distribution,” Journal of Computer Science and Technology, vol. 27, no. 5, pp. 1035–1055, 2012.
  • A. W. Mohamed, H. Z. Sabry, and M. Khorshid, “An alternative differential evolution algorithm for global optimization,” Journal of Advanced Research, vol. 3, no. 2, pp. 149–165, 2012.
  • A. W. Mohamed and H. Z. Sabry, “Constrained optimization based on modified differential evolution algorithm,” Information Sciences, vol. 194, pp. 171–208, 2012.
  • E. Mezura-Montes, M. E. Miranda-Varela, and R. del Carmen Gómez-Ramón, “Differential evolution in constrained numerical optimization: an empirical study,” Information Sciences, vol. 180, no. 22, pp. 4223–4262, 2010.
  • A. M. Gujarathi and B. V. Babu, “Hybrid multi-objective diffe-rential evolution (H-MODE) for optimization of polyethylene terephthalate (PET) reactor,” International Journal of Bio-Inspired Computation, vol. 2, no. 3-4, pp. 213–221, 2010.
  • X. Lu and K. Tang, “Classification- and regressionassisted differential evolution for computationally expensive problems,” Journal of Computer Science and Technology, vol. 27, no. 5, pp. 1024–1034, 2012.
  • Q. K. Pan, L. Wang, and B. Qian, “A novel differential evolution algorithm for bi-criteria no-wait flow shop scheduling problems,” Computers and Operations Research, vol. 36, no. 8, pp. 2498–2511, 2009.
  • K. Zielinski, D. Peters, and R. Laur, “Run time analysis regarding stopping criteria for differential evolution and particle swarm optimization,” in Proceedings of the 1st International Conference on Experiments Process System Modelling, Simulation and Optimization, 2005.
  • D. Zaharie, “On the explorative power of differential evolution,” in Proceedings of the 3rd International Workshop on Symbolic Numerical Algorithms for Scientific Computing, October 2001.
  • D. Zaharie, “Critical values for the control parameters of diffe-rential evolution algorithms,” in Proceedings of the 8th International Mendel Conference on Soft Computing, pp. 62–67, 2002.
  • D. Zaharie, “Parameter adaptation in differential evolution by controlling the population diversity,” in Proceedings of the 4th International Workshop on Symbolic Numeric Algorithms for Scientific Computing, pp. 385–397, 2002.
  • D. Zaharie, “A comparative analysis of crossover variants in dierential evolution,” in Proceedings of the International Multiconference on Computer Science and Information Technology (IMCSIT '07), pp. 171–181, Wisla, Poland, 2007.
  • D. Zaharie, “Statistical properties of differential evolution and related random search algorithms,” in Proceedings of the International Conference on Computational Statistics (COMPSTAT '08), pp. 473–485, 2008.
  • D. Zaharie, “Influence of crossover on the behavior of differential evolution algorithms,” Applied Soft Computing Journal, vol. 9, no. 3, pp. 1126–1138, 2009.
  • S. Dasgupta, S. Das, A. Biswas, and A. Abraham, “The population dynamics of differential evolution: a mathematical model,” in Proceedings of the IEEE Congress on Evolutionary Computation (CEC '08), pp. 1439–1446, Hong Kong, China, June 2008.
  • S. Dasgupta, S. Das, A. Biswas, and A. Abraham, “On stability and convergence of the population-dynamics in differential evolution,” AI Communications, vol. 22, no. 1, pp. 1–20, 2009.
  • L. Wang and F. Z. Huang, “Parameter analysis based on sto-chastic model for differential evolution algorithm,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3263–3273, 2010.
  • A. E. Eiben and G. Rudolph, “Theory of evolutionary algorithms: a bird's eye view,” Theoretical Computer Science, vol. 229, no. 1-2, pp. 3–9, 1999.
  • F. Xue, A. C. Sanderson, and R. J. Graves, “Modeling and convergence analysis of a continuous multi-objective differential evolution algorithm,” in Proceedings of the IEEE Congress on Evolutionary Computation (IEEE CEC '05), pp. 228–235, IEEE Press, Edinburgh, UK, September 2005.
  • F. Xue, A. C. Sanderson, and R. J. Graves, “Multiobjective differential evolution: algorithm, convergence analysis, and appli-cations,” in Proceedings of the IEEE Congress on Evolutionary Computation, vol. 1, pp. 743–745, Edinburgh, UK, September 2005.
  • Y. T. Zhao, J. Wang, and Y. Song, “An improved differential evolution to continuous domains and its convergence,” in Proceedings of the 1st ACM/SIGEVO Summit on Genetic and Evolutionary Computation (GEC '09), pp. 1061–1064, June 2009.
  • C. F. Sun, Differential evolution and its application on the optimal scheduling of electrical power system [Ph.D. thesis], Huazhong University of Science and Technology, 2009 (Chinese).
  • Y. C. He, X. Z. Wang, K. Q. Liu, and Y. Q. Wang, “Convergent analysis and algorithmic improvement of differential evolution,” Journal of Software, vol. 21, no. 5, pp. 875–885, 2010 (Chinese).
  • S. Ghosh, S. Das, A. V. Vasilakos, and K. Suresh, “On convergence of differential evolution over a class of continuous functions with unique global optimum,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 42, no. 1, pp. 107–124, 2012.
  • Z. Zhan and J. Zhang, “Enhance differential evolution with random walk,” in Proceedings of the 14th International Conference on Genetic and Evolutionary Computation Conference Companion (GECCO '12), pp. 1513–1514, 2012.
  • Z. B. Hu, Q. H. Su, S. W. Xiong, and F. G. Hu, “Self-adaptive hybrid differential evolution with simulated annealing algorithm for numerical optimization,” in Proceedings of the IEEE Congress on Evolutionary Computation (CEC '08), pp. 1189–1194, Hong Kong, China, June 2008.
  • A. Ghosh, S. Das, A. Chowdhury, and R. Giri, “An improved differential evolution algorithm with fitness-based adaptation of the control parameters,” Information Sciences, vol. 181, no. 18, pp. 3749–3765, 2011.
  • C. Z. Chen, F. L. Jin, X. Y. Zhu, and G. Z. Ouyan, Mathematics Analysis, Higher Education Press, 2000.
  • C. J. F. T. Braak, “A Markov chain Monte Carlo version of the genetic algorithm differential evolution: easy Bayesian compu-ting for real parameter spaces,” Statistics and Computing, vol. 16, no. 3, pp. 239–249, 2006.
  • J. He and X. H. Yu, “Conditions for the convergence of evo-lutionary algorithms,” Journal of Systems Architecture, vol. 47, no. 6, pp. 601–612, 2001.
  • G. Rudolph, “Convergence of evolutionary algorithms in general search spaces,” in Proceedings of the IEEE International Conference on Evolutionary Computation (ICEC '96), pp. 50–54, May 1996.
  • Z. Michalewicz, “Evolutionary algorithms for constrained para-meter optimization problems,” Evolutionary Computation, vol. 4, no. 1, pp. 1–32, 1996.
  • L. Eshelman and J. Schaffer, “Real-coded genetic algorithms and interval-schemata,” in Foundations of Genetic Algorithms-2, L. Whitley, Ed., pp. 187–202, Morgan Kaufmann, San Mateo, Calif, USA, 1993.
  • T. Bäck, F. Hoffmeister, and H. P. Schwefel, “A survey of evolution stragies,” in Proceedings of the 4th International Conference on Genetic Algorithms, pp. 2–9, 1991.
  • D. B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, IEEE Press, Piscataway, NJ, USA, 1995.
  • X. S. Wang, M. L. Hao, and Y. H. Cheng, “On the use of differential evolution for forward kinematics of parallel manipulators,” Applied Mathematics and Computation, vol. 205, no. 2, pp. 760–769, 2008.
  • C. E. Chiang, W. P. Lee, and J. S. Heh, “A 2-Opt based differential evolution for global optimization,” Applied Soft Computing Journal, vol. 10, no. 4, pp. 1200–1207, 2010.
  • J. Y. Yan, Q. Ling, and D. M. Sun, “A differential evolution with simulated annealing updating method,” in Proceedings of the International Conference on Machine Learning and Cybernetics, pp. 2103–2106, Dalian, China, August 2006.
  • F. Neri and V. Tirronen, “Recent advances in differential evolution: a survey and experimental analysis,” Artificial Intelligence Review, vol. 33, no. 1-2, pp. 61–106, 2010.
  • J. Brest, S. Greiner, B. Bošković, M. Mernik, and V. Zumer, “Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems,” IEEE Transactions on Evolutionary Computation, vol. 10, no. 6, pp. 646–657, 2006.
  • Y. Wang, Z. X. Cai, and Q. F. Zhang, “Differential evolution with composite trial vector generation strategies and control para-meters,” IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 55–66, 2011.
  • J. Derrac, S. García, D. Molina, and F. Herrera, “A practical tutorial on the use of nonparametric statistical tests as a metho-dology for comparing evolutionary and swarm intelligence algorithms,” Swarm and Evolutionary Computation, vol. 1, no. 1, pp. 3–18, 2011.
  • D. E. Goldberg and P. Segrest, “Finite markov chain analysis of genetic algorithm,” in Proceedings of the International Conference on Genetic Algorithms, Hillsdale, NJ, USA, 1987.
  • J. Suzuki, “A Markov chain analysis on simple genetic algorithms,” IEEE Transactions on Systems, Man and Cybernetics, vol. 25, no. 4, pp. 655–659, 1995.
  • L. Ming, Y. P. Wang, and Y. M. Cheng, “On convergencerate of a class of genetic algorithms,” in Proceedings of the World Auto-mation Congress, Budapest, Hungary, 2006.