Abstract
Multi-objective particle swarm optimization (MOPSO) has been well studied in recent years. However, existing MOPSO methods are not powerful enough when tackling optimization problems with more than three objectives, termed as many-objective optimization problems (MaOPs). In this study, an improved set evolution multi-objective particle swarm optimization (S-MOPSO, for short) is proposed for solving many-objective problems. According to the proposed framework of set evolution MOPSO (S-MOPSO), including quality indicators-based objective transformation, the Pareto dominance on sets, and the particle swarm operators for set evolution, an enhanced S-MOPSO method is developed by updating particles hierarchically, i.e., a set of solutions is first regarded as a particle to be updated and then the solutions in a selected set are further evolved by a modified PSO. In the set evolutionary stage, the strategy for efficiently updating the set particle is proposed. When further evolving a single solution in the initial decision space of the optimized MaOP, the global and local best particles are dynamically determined based on those ideal reference points. The performance of the proposed algorithm is empirically demonstrated by applying it to several scalable benchmark many-objective problems.
Access this article
Rent this article via DeepDyve
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-015-1637-1/MediaObjects/500_2015_1637_Fig6_HTML.gif)
Similar content being viewed by others
![](https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs40747-023-01128-x/MediaObjects/40747_2023_1128_Fig1_HTML.png)
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Andre BD, Aurora P (2012) Measuring the convergence and diversity of CDAS multi-objective particle swarm optimization algorithms: A study of many-objective problems. Neurocomputing 75:43–51
Basseur M, Burke EK (2007) Indicator-based multi-objective local search. Proc IEEE Congress Evolut Comput (CEC), pp 3100–3107
Castro OR, Pozo AA (2014) MOPSO based on hyper-heuristic to optimize many-objective problems. In: Proceedings of IEEE symposium on swarm intelligence (SIS), pp 1–8
Chaman GI, Coello Coello CA, Montano A (2014) MOPSOhv: a new hypervolume-based multi-objective particle swarm optimizer. In Proceedings of IEEE Congress on evolutionary computation (CEC), pp 266–273
Coello Coello CA, Pulido GT, Lechuga MS (2004) Handling multiple objectives with particle swarm optimization. IEEE Trans Evolut Comput 8(3):256–279
Deb K, Prata PA, Agarwal (2002) A fast and elitist multiobjective genetic algorithm: NSGA-2. IEEE Trans Evolut Comput 6(2):182–197
Deb K, Mohan M, Mishra S (2005) Evaluating the \(\epsilon \)-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evolut Comput 13(4):501–525
Gilberto RM, Xavier B, Javier S, Miguel M (2014) Controller tuning using evolutionary multi-objective optimisation: current trends and applications. Control Eng Pract 28:58–73
Goh CK, Tan KC, Liu DS, Chiam SC (2010) A competitive and cooperative co-evolutionary approach to multi-objective particle swarm optimization algorithm design. Europ J Operat Res 202:42–54
Goldberg DE (1988) Genetic algorithms for search, optimization, and machine learning. Addison-Wesley Publishing, Pearson
Gong DW, Ji XF, Sun XY (2014) Solving many-objective optimization problems using set-based evolutionary algorithms. Acta Electronic Sinica 42(1):77–83
Jia SJ, Zhu J, Du B, Yue H (2011) Indicator-based particle swarm optimization with local search. In: Proceedings of International conference on natural computation (ICNC), pp 1180–1184
Jiang S, Zhang J, Ong YS, Zhang AN, Tan PS (2014) A simple and fast hypervolume indicator-based multiobjective evolutionary algorithm. IEEE Trans Cybernet. doi:10.1109/TCYB.2014.2367526
Johannes B, Zitzler E (2008) HypE: an algorithm for fast hypervolume-based many-objective optimization. TIK-Report No. 286, November 26, 2008, pp 1–25
Knowles JD, Corne DW (1999) The pareto archived evolution strategy: a new baseline algorithm for pareto multiobjective optimisation. In: Proceedings of congress on evolutionary computation, Vol 1, Piscataway, NJ, pp 98–105
Li M, Yang S, Liu X (2013) A comparative study on evolutionary algorithms for many-objective optimization. Evolutionary multi-criterion optimization. Springer, Berlin Heidelberg
Li M, Yang S, Liu X (2014) Shift-based density estimation for Pareto-based algorithms in many-objective optimization. IEEE Trans Evolut Comput 18(3):348–365
Li M, Yang S, Liu X (2014) Diversity comparison of Pareto front approximations in many-objecive optimization. IEEE Trans Cybern 44(12):2568–2584
Lobato FS, Sousa MN, Silva MA, Machado AR (2014) Multi-objective optimization and bio-inspired methods applied to machinability of stainless steel. Appl Soft Comput 22:261–271
Margarita RS, Coello Coello CA (2006) Multi-objective particle swarm optimizers: a survey of the state-of -the-art. Int J Comput Intell Res 2(3):287–308
Mario K, Kaori Y (2007) Substitute distance assignments in NSGA-II for handling many-objective optimization problems. Lect Notes Comput Sci 4403:727–741
Mostaghim RS, Schmeck H (2008) Distance based ranking in many-objective particle swarm optimization. In: Proceedings of the international conference on parallel problem solving from bature (PPSN), pp 753–762
Mostaghim S, Teich J (2003) The role of dominance in multi-objective particle swarm optimization methods. In: Proceedings of the 2003 IEEE swarm intelligence symposium, Indianapolis, USA, pp 26–33
Pedro CM, Gonzalo GG, Laureano (2014) MILP-based decomposition algorithm for dimensionality reduction in multi-objective optimization. Comput Chem Eng 67:137–1474
Phan DH, Suzuki J (2013) R2-IBEA: R2 indicator based evolutionary algorithm for multiobjective optimization. In: Proceedings of IEEE congress on evolutionary computation (CEC), pp 1836–1845
Purshouse RC, Fleming PJ (2003) Evolutionary many-objective optimization: An exploratory analysis. In: Proceedings of 2003 IEEE congress on evolutionary computation. Canberra, pp 2066–2073
Reyes-Sierra M, Coello Coello CA (2006) Multi-objective particle swarm optimizers: a survey of the state-of-the-art. Int J Comput Intell Res 2(3):287–308
Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Masters thesis
Singh HK, Isaacs A, Ray TA (2011) Pareto corner search evolutionary algorithm and dimensionality reduction in many-objective optimization problems. IEEE Trans Evolut Comput 15(4):539–556
Sun XY, Chen XZ, Xu RD, Gong DW (2014) Hybrid many-objective particle swarm optimization set-evolution. In: Proceedings of 11th world congress on intelligent control and automation, Shenyang, pp 1324–1329
Sun XY, Xu RD, Zhang Y, Gong DW (2014) Sets evolution-based particle swarm optimization for many-objective problems. In: Proceedings of the 2014 IEEE international conference on information and automation (ICIA), Halaer, pp 1119–1124
Von Lcken C, Barn B, Brizuela C (2014) A survey on multi-objective evolutionary algorithms for many-objective problems. Comput Optim Appl, pp 1–50
Wickramasinghe UK, Li X (2009) Using a distance metric to guide PSO algorithms for many-objective optimization. In: Proceedings of the 11th annual conference on genetic and evolutionary computation conference (GECCO), pp 667–674
Woolard MM, Fieldsend JE (2013) On the effect of selection and archiving operators in many-objective particle swarm optimization. In: Proceedings of 2013 genetic and evolutionary computation conference, Amsterdam, pp 129–136
Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evolut Comput 17(5):721–736
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731
Zhang Q, Zhou A, Zhao S, Suganthan P, Liu W, Tiwari S (2008) Muliobjective optimization test instances for the cec 2009 special session and competition. In: University of Essex, Colchester, UK and Nanyang technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms, technical report, 2008
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evolut Comput 3(3):257–271
Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical Results. Evolut Comput 8(2):173–195
Zitzler E, Knzli S (2004) Indicator-based selection in multiobjective search. Lect Notes Comput Sci 3242:832–842
Zitzler E, Thiele L, Bader J (2010) On set-based multi-objective optimization. IEEE Trans Evolut Comput 14(1):58–79
Zitzler E, Kunzli S (2004) Indicator-based selection in multi-objective search. In: Proceedings of 8th international conference on parallel problem solving from nature, pp 832–842
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61105063 and 61473298, and the Fundamental Research Funds for the Central Universities under Grant 2012QNA58.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Jin.
Rights and permissions
About this article
Cite this article
Sun, X., Chen, Y., Liu, Y. et al. Indicator-based set evolution particle swarm optimization for many-objective problems. Soft Comput 20, 2219–2232 (2016). https://doi.org/10.1007/s00500-015-1637-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-015-1637-1