Advertisement

Investigating the Normalization Procedure of NSGA-III

  • Julian BlankEmail author
  • Kalyanmoy Deb
  • Proteek Chandan Roy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11411)

Abstract

Most practical optimization problems are multi-objective in nature. Moreover, the objective values are, in general, differently scaled. In order to obtain uniformly distributed set of Pareto-optimal points, the objectives must be normalized so that any distance metric computation in the objective space is meaningful. Thus, normalization becomes a crucial component of an evolutionary multi-objective optimization (EMO) algorithm. In this paper, we investigate and discuss the normalization procedure for NSGA-III, a state-of-the-art multi- and many-objective evolutionary algorithm. First, we show the importance of normalization in higher-dimensional objective spaces. Second, we provide pseudo-codes which presents a clear description of normalization methods proposed in this study. Third, we compare the proposed normalization methods on a variety of test problems up to ten objectives. The results indicate the importance of normalization for the overall algorithm performance and show the effectiveness of the originally proposed NSGA-III’s hyperplane concept in higher-dimensional objective spaces.

Keywords

Many-objective optimization NSGA-III Normalization 

References

  1. 1.
    Moeaframework. http://moeaframework.org. Accessed 26 Sept 2018
  2. 2.
    Bhesdadiya, R.H., Trivedi, I.N., Jangir, P., Jangir, N., Kumar, A.: An NSGA-III algorithm for solving multi-objective economic/environmental dispatch problem. Cogent Eng. 3(1), 1269383 (2016)CrossRefGoogle Scholar
  3. 3.
    Bi, X., Wang, C.: An improved NSGA-III algorithm based on objective space decomposition for many-objective optimization. Soft Comput. 21(15), 4269–4296 (2017)CrossRefGoogle Scholar
  4. 4.
    Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part i: solving problems with box constraints. IEEE Trans. Evol. Comput. 18(4), 577–601 (2014)CrossRefGoogle Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  7. 7.
    Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable test problems for evolutionary multiobjective optimization. In: Abraham, A., Jain, L., Goldberg, R. (eds.) Evolutionary Multiobjective Optimization. AI&KP, pp. 105–145. Springer, London (2005).  https://doi.org/10.1007/1-84628-137-7_6CrossRefzbMATHGoogle Scholar
  8. 8.
    Durillo, J., Nebro, A., Alba, E.: The jmetal framework for multi-objective optimization: design and architecture. In: CEC 2010, Barcelona, Spain, pp. 4138–4325, July 2010Google Scholar
  9. 9.
    Gaur, A., Talukder, A.K.M.K., Deb, K., Tiwari, S., Xu, S., Jones, D.: Finding near-optimum and diverse solutions for a large-scale engineering design problem. In: 2017 IEEE Symposium Series on Computational Intelligence (SSCI), pp. 1–8, November 2017Google Scholar
  10. 10.
    Ibrahim, A., Rahnamayan, S., Martin, M.V., Deb, K.: EliteNSGA-III: an improved evolutionary many-objective optimization algorithm. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 973–982, July 2016Google Scholar
  11. 11.
    Ishibuchi, H., Doi, K., Nojima, Y.: On the effect of normalization in MOEA/D for multi-objective and many-objective optimization. Complex Intell. Syst. 3(4), 279–294 (2017)CrossRefGoogle Scholar
  12. 12.
    Jain, H., Deb, K.: An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part ii: handling constraints and extending to an adaptive approach. IEEE Trans. Evol. Comput. 18(4), 602–622 (2014)CrossRefGoogle Scholar
  13. 13.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  14. 14.
    Seada, H., Deb, K.: A unified evolutionary optimization procedure for single, multiple, and many objectives. IEEE Trans. Evol. Comput. 20(3), 358–369 (2016)CrossRefGoogle Scholar
  15. 15.
    Singh, H.K., Yao, X.: Improvement of reference points for decomposition based multi-objective evolutionary algorithms. In: Shi, Y., et al. (eds.) SEAL 2017. LNCS, vol. 10593, pp. 284–296. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-68759-9_24CrossRefGoogle Scholar
  16. 16.
    Tian, Y., Cheng, R., Zhang, X., Jin, Y.: Platemo: a matlab platform for evolutionary multi-objective optimization [educational forum]. IEEE Comput. Intell. Mag. 12(4), 73–87 (2017)CrossRefGoogle Scholar
  17. 17.
    Wang, R., Xiong, J., Ishibuchi, H., Wu, G., Zhang, T.: On the effect of reference point in MOEA/D for multi-objective optimization. Appl. Soft Comput. 58, 25–34 (2017)CrossRefGoogle Scholar
  18. 18.
    Wierzbicki, A.P.: The use of reference objectives in multiobjective optimization. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making Theory and Applications. LNEMS, vol. 177, pp. 468–486. Springer, Berlin (1980)CrossRefGoogle Scholar
  19. 19.
    Yuan, X., Tian, H., Yuan, Y., Huang, Y., Ikram, R.M.: An extended NSGA-III for solution multi-objective hydro-thermal-wind scheduling considering wind power cost. Energy Convers. Manag. 96, 568–578 (2015)CrossRefGoogle Scholar
  20. 20.
    Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)CrossRefGoogle Scholar
  21. 21.
    Zhou, A., Qu, B.Y., Li, H., Zhao, S.Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol. Comput. 1(1), 32–49 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

Personalised recommendations