Adaptive population structure learning in evolutionary multi-objective optimization


Some recent research shows that in multi-objective evolutionary algorithms (MOEAs), mating with similar individuals can improve the quality of new solutions and accelerate the convergence of algorithms. Based on the above finding, some clustering-based mating restriction strategies are proposed. However, those clustering algorithms are not suitable for the population with non-convex structures. Therefore, it may fail to detect population structure in different evolutionary stages. To solve this problem, we propose a normalized hypervolume-based mating transformation strategy (NMTS). In NMTS, the population structure is detected by K-nearest-neighbor graph and spectral clustering before and after the mating transformation condition, respectively. And the parent solutions are chosen according to the founded population structure. The proposed algorithm has been applied to a number of test instances with complex Pareto optimal solution sets or Pareto fronts, and compared with some state-of-the-art MOEAs. The results have demonstrated its advantages over other algorithms.

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  1. Bach FR, Jordan MI (2004) Learning spectral clustering. In: Advances in neural information processing systems, pp 305–312

  2. Bengio Y, Paiement Jf, Vincent P, Delalleau O, Roux NL, Ouimet M (2004) Out-of-sample extensions for lle, isomap, mds, eigenmaps, and spectral clustering. In: Advances in neural information processing systems, pp 177–184

  3. Beume N, Naujoks B, Emmerich M (2007) SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur J Oper Res 181(3):1653–1669

    MATH  Article  Google Scholar 

  4. Cheng R, Jin Y, Narukawa K, Sendhoff B (2015) A multiobjective evolutionary algorithm using gaussian process-based inverse modeling. IEEE Trans Evol Comput 19(6):838–856

    Article  Google Scholar 

  5. Coello CA (2000) An updated survey of GA-based multiobjective optimization techniques. ACM Comput Surv (CSUR) 32(2):109–143

    Article  Google Scholar 

  6. Deb K (2001) Multi objective optimization using evolutionary algorithms. Wiley, New York

    Google Scholar 

  7. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  8. Deb K, Mohan M, Mishra S (2005) Evaluating the \(\varepsilon \)-domination based multi-objective evolutionary algorithm for a quick computation of pareto-optimal solutions. Evol Comput 13(4):501–525

    Article  Google Scholar 

  9. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the em algorithm. J R Stat Soc Ser B (Methodol) 39(1):1–22

    MathSciNet  MATH  Google Scholar 

  10. Domínguez M, Fernández-Cardador A, Cucala AP, Gonsalves T, Fernández A (2014) Multi objective particle swarm optimization algorithm for the design of efficient ato speed profiles in metro lines. Eng Appl of Artif Intell 29:43–53

    Article  Google Scholar 

  11. Dong W, Moses C, Li K (2011) Efficient \(k\)-nearest neighbor graph construction for generic similarity measures. In: Proceedings of the 20th international conference on World Wide Web, ACM, pp 577–586

  12. Frey BJ, Dueck D (2007) Clustering by passing messages between data points. Science 315(5814):972–977

    MathSciNet  MATH  Article  Google Scholar 

  13. Gu F, Liu HL, Tan KC (2012) A multiobjective evolutionary algorithm using dynamic weight design method. Int J Innov Comput Inf Control 8(5 (B)):3677–3688

    Google Scholar 

  14. Hartigan JA, Wong MA (1979) Algorithm as 136: a \(k\)-means clustering algorithm. J R Stat Soc Ser C (Appl Stat) 28(1):100–108

    MATH  Google Scholar 

  15. Hautamaki V, Karkkainen I, Franti P (2004) Outlier detection using \(k\)-nearest neighbour graph. In: Proceedings of the 17th international conference on pattern recognition, 2004. ICPR 2004., IEEE, vol 3, pp 430–433

  16. Hillermeier C (2001) Nonlinear multiobjective optimization: a generalized homotopy approach, vol 135. Springer, Berlin

    Google Scholar 

  17. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506

    MATH  Article  Google Scholar 

  18. Li H, Zhang Q (2009) Multiobjective optimization problems with complicated pareto sets, MOEA/D and NSGA-II. IEEE Trans Evol Comput 13(2):284–302

    Article  Google Scholar 

  19. Li X, Song S, Zhang H (2018) Evolutionary multiobjective optimization with clustering-based self-adaptive mating restriction strategy. Soft Comput 23(10):3303–3325

    Article  Google Scholar 

  20. Maier M, Hein M, von Luxburg U (2009) Optimal construction of k-nearest-neighbor graphs for identifying noisy clusters. Theor Comput Sci 410(19):1749–1764

    MathSciNet  MATH  Article  Google Scholar 

  21. Miettinen K (2012) Nonlinear multiobjective optimization, vol 12. Springer, Berlin

    Google Scholar 

  22. Ng AY, Jordan MI, Weiss Y (2002) On spectral clustering: analysis and an algorithm. In: Advances in neural information processing systems, In NIPS 2001, pp 849–856

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

    Google Scholar 

  24. Schaffer JD (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms and their applications, 1985, Lawrence Erlbaum Associates. Inc., Publishers

  25. Stella XY, Shi J (2003) Multiclass spectral clustering. In: null, IEEE, p 313

  26. Sun J, Zhang H, Zhou A, Zhang Q (2018) Learning from a stream of non-stationary and dependent data in multiobjective evolutionary optimization. IEEE Trans Evol Comput 23(4):541–555

    Article  Google Scholar 

  27. Von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17(4):395–416

    MathSciNet  Article  Google Scholar 

  28. Xu R, Wunsch D (2008) Clustering, vol 10. Wiley, New York

    Google Scholar 

  29. Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 17(5):721–736

    Article  Google Scholar 

  30. Zhang H, Song S, Zhou A, Gao XZ (2014) A clustering based multiobjective evolutionary algorithm. In: 2014 IEEE Congress on evolutionary computation (CEC), IEEE, pp 723–730

  31. Zhang H, Zhang X, Gao XZ, Song S (2016a) Self-organizing multiobjective optimization based on decomposition with neighborhood ensemble. Neurocomputing 173:1868–1884

    Article  Google Scholar 

  32. Zhang H, Zhou A, Song S, Zhang Q, Gao XZ, Zhang J (2016b) A self-organizing multiobjective evolutionary algorithm. IEEE Trans Evol Comput 20(5):792–806

    Article  Google Scholar 

  33. Zhang H, Zhang X, Song S, Gao XZ (2017) An affinity propagation-based multiobjective evolutionary algorithm for selecting optimal aiming points of missiles. Soft Comput 21(11):3013–3031

    Article  Google Scholar 

  34. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731

    Article  Google Scholar 

  35. Zhang Q, Zhou A, Jin Y (2008a) Rm-meda: A regularity model-based multiobjective estimation of distribution algorithm. IEEE Trans Evol Comput 12(1):41–63

    Article  Google Scholar 

  36. Zhang Q, Zhou A, Zhao S, Suganthan PN, Liu W, Tiwari S (2008b) Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms, technical report 264

  37. Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol Comput 1(1):32–49

    Article  Google Scholar 

  38. Zhou A, Zhang Q, Zhang G (2014) Multiobjective evolutionary algorithm based on mixture gaussian models. J Softw 25(5):913–928

    MathSciNet  MATH  Google Scholar 

  39. Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: International conference on parallel problem solving from nature, Springer, pp 832–842

  40. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evol Comput 3(4):257–271

    Article  Google Scholar 

  41. Zitzler E, Laumanns M, Thiele L (2001) Spea2: Improving the strength Pareto evolutionary algorithm. TIK-report 103

  42. Zitzler E, Thiele L, Laumanns M et al (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132

    Article  Google Scholar 

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This study was found by National Natural Science Foundation of China (Grant numbers: 61703382, 51875053, 61673180) and China Ministry of Science and Technology Key Research and Development Program (Grant number: 2018YFC1903101).

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Correspondence to Hu Zhang.

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Wang, S., Zhang, H., Zhang, Y. et al. Adaptive population structure learning in evolutionary multi-objective optimization. Soft Comput 24, 10025–10042 (2020).

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  • Evolutionary algorithm
  • Multi-objective optimization
  • Mating restriction
  • Population structure