Skip to main content
SpringerLink
Log in

An improved indicator-based two-archive algorithm for many-objective optimization problems

  • Regular Paper
  • Published:
Computing Aims and scope Submit manuscript

Abstract

The large number of objectives in many-objective optimization problems (MaOPs) has posed significant challenges to the performance of multi-objective evolutionary algorithms (MOEAs) in terms of convergence and diversity. To design a more balanced MOEA, a multiple indicator-based two-archive algorithm named IBTA is proposed to deal with problems with complicated Pareto fronts. Specifically, a two-archive framework is introduced to focus on convergence and diversity separately. In IBTA, we assign different selection principles to the two archives. In the convergence archive, the inverted generational distance with noncontributing solution detection (IGD-NS) indicator is applied to choose the solutions with favorable convergence in each generation. In the diversity archive, we use crowdedness and fitness to select solutions with favorable diversity. To evaluate the performance of IBTA on MaOPs, we compare it with several state-of-the-art MOEAs on various benchmark problems with different Pareto fronts. The experimental results demonstrate that IBTA can deal with multi-objective optimization problems (MOPs)/MaOPs with satisfactory convergence and diversity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Algorithm 1
Algorithm 2
Algorithm 3
Algorithm 4
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

  1. For \( x_1, x_2 \in X \), \(x_1\) Pareto dominates \(x_2\)(denoted as \(x_1 \prec x_2\)) means: 1) for all objectives, \(f_i(x_1) \le f_i(x_2)\) \(i = 1,..., m\), and 2) at least one objective satisfies that \(f_j(x_1) < f_j(x_2)\). Sequentially, A solution \(x^* \in X\) is Pareto optimal if there is no solution \(x \in X \) satisfies \(x \succ x^* \).

References

  1. Farina M, Amato P (2002) On the optimal solution definition for many-criteria optimization problems. In: 2002 annual meeting of the North American fuzzy information processing society proceedings. NAFIPS-FLINT 2002 (Cat. No. 02TH8622), pp 233–238

  2. Li B, Li J, Tang K, Yao X (2015) Many-objective evolutionary algorithms: a survey. ACM Comput Surv 48(1):1–35

    Article  ADS  Google Scholar 

  3. Coello CDA, Van Veldhuizen CA, Lamont GB (2007) Evolutionary algorithms for solving multi-objective problems. Springer, New York

    Google Scholar 

  4. Ishibuchi H, Yu S, Masuda H, Nojima Y (2017) Performance of decomposition-based many-objective algorithms strongly depends on pareto front shapes. IEEE Trans Evol Comput 21(2):169–190

    Article  Google Scholar 

  5. Falcón-Cardona JG, Emmerich MTM, Coello CCA (2019a) On the cooperation of multiple indicator-based multi-objective evolutionary algorithms. In: 2019 IEEE congress on evolutionary computation (CEC), pp 2050–2057

  6. Falcón-Cardona JG, Ishibuchi H, Coello CC (2020) A Exploiting the trade-off between convergence and diversity indicators. In: 2020 IEEE symposium series on computational intelligence (SSCI), pp 141–148

  7. Gómez RH, Coello CA (2017) A hyper-heuristic of scalarizing functions. Association for Computing Machinery

  8. Falcón-Cardona JG, Coello Coello CA, Emmerich M (2019b) CRI-EMOA: a pareto-front shape invariant evolutionary multi-objective algorithm. In: Evolutionary multi-criterion optimization, Springer International Publishing, pp 307–318

  9. Márquez-Vega LA, Falcón-Cardona JG, COE (2021) Towards a pareto front shape invariant multi-objective evolutionary algorithm using pair-potential functions. In: Advances in computational intelligence, Springer International Publishing, pp 369–382

  10. Wang Z, Zhang Q, Li H, Ishibuchi H, Jiao L (2017) On the use of two reference points in decomposition based multiobjective evolutionary algorithms. Swarm Evol Comput 34:89–102

    Article  CAS  Google Scholar 

  11. Saborido R, Ruiz AB, Luque M (2017) Global WASF-GA: an evolutionary algorithm in multiobjective optimization to approximate the whole pareto optimal front. Evolut Comput 25(2):309–349

    Article  Google Scholar 

  12. Hoai NB, Bing X, Peter A, Hisao I, Mengjie Z (2020) Multiple reference points-based decomposition for multiobjective feature selection in classification: Static and dynamic mechanisms. IEEE Trans Evol Comput 24(1):170–184

    Article  Google Scholar 

  13. Zhou Aimin Q, Bo-Yang LH, Shi-Zheng Z, Nagaratnam SP, Qingfu Z (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol Comput 1(1):32–49

    Article  Google Scholar 

  14. Trivedi A, Srinivasan D, Sanyal K, Ghosh A (2017) A survey of multiobjective evolutionary algorithms based on decomposition. Trans Evol Comp 21(3):440–462

    Google Scholar 

  15. 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 

  16. Li K, Deb K, Zhang Q, Kwong S (2014) An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans Evol Comput 19(5):694–716

    Article  Google Scholar 

  17. Deb K, Jain H (2014) 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

    Article  Google Scholar 

  18. Cheng R, Jin Y, Olhofer M, Sendhoff B (2016) A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 20(5):773–791

    Article  Google Scholar 

  19. Asafuddoula M, Ray T, Sarker R (2015) A decomposition-based evolutionary algorithm for many objective optimization. IEEE Trans Evol Comput 19(3):445–460

    Article  Google Scholar 

  20. Ishibuchi H, Hitotsuyanagi Y, Ohyanagi H, Nojima Y (2011) Effects of the existence of highly correlated objectives on the behavior of MOEA/D. In: Evolutionary multi-criterion optimization–6th international conference, EMO 2011, Ouro Preto, Brazil, April 5-8, 2011. Proceedings, pp 166–181

  21. Giagkiozis I, Purshouse RC, Fleming PJ (2012) Generalized decomposition and cross entropy methods for many-objective optimization. Inf Sci 21:169–190

    Google Scholar 

  22. Sato H (2014) Inverted PBI in MOEA/D and its impact on the search performance on multi and many-objective optimization. In: Genetic and evolutionary computation conference (GECCO), Vancouver, pp 645–652

  23. 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 

  24. Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. In: Evolutionary methods for design, optimization and control with applications to industrial problems. Proceedings of the EUROGEN’2001. Athens. Greece, September 19-21, pp 95–100

  25. Corne DW, Jerram NR, Knowles JD, Oates MJ (2002) PESA-II: region-based selection in evolutionary multiobjective optimization. Morgan Kaufmann Publishers Inc., UK, pp 283–290

    Google Scholar 

  26. Purshouse RC, Fleming PJ (2007) On the evolutionary optimization of many conflicting objectives. IEEE Trans Evol Comput 11(6):770–784

    Article  Google Scholar 

  27. Ikeda K, Kita H, Kobayashi S (2001) Failure of pareto-based MOEAs: does non-dominated really mean near to optimal? In: Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No.01TH8546), vol. 2, pp 957–962

  28. Laumanns M, Thiele L, Deb K, Zitzler E (2002) Combining convergence and diversity in evolutionary multiobjective optimization. Evol Comput 10(3):263–282

    Article  PubMed  Google Scholar 

  29. Yuan Y, Xu H, Wang B, Yao X (2016) A new dominance relation-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 20(1):16–37

    Article  Google Scholar 

  30. 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 

  31. Li M, Yang S, Liu X (2014) Shift-based density estimation for pareto-based algorithms in many-objective optimization. IEEE Trans Evol Comput 18(3):348–365

    Article  Google Scholar 

  32. Falcón-Cardona JG, Coello CA (2020) Indicator-based multi-objective evolutionary algorithms: a comprehensive survey. ACM Comput Surv 53(2):1–35

    Article  Google Scholar 

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

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

    Article  Google Scholar 

  35. Sun Y, Yen GG, Yi Z (2019) IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Trans Evol Comput 23(2):173–187

    Article  Google Scholar 

  36. Gómez RH, Coello CA (2015) Improved metaheuristic based on the R2 indicator for many-objective optimization. In: Genetic and evolutionary computation conference, pp 679–686

  37. While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evol Comput 10(1):29–38

    Article  Google Scholar 

  38. Coello Coello CA, Reyes Sierra M (2004) A study of the parallelization of a coevolutionary multi-objective evolutionary algorithm. In: Mexican international conference on artificial intelligence (MICAI), pp 688–697

  39. Brockhoff D, Wagner T, Trautmann H (2012) On the properties of the R2 indicator. In: Genetic and evolutionary computation conference, pp 465–472

  40. Wang H, Jiao L, Yao X (2015) Two_Arch2: an improved two-archive algorithm for many-objective optimization. IEEE Trans Evol Comput 19(4):524–541. https://doi.org/10.1109/TEVC.2014.2350987

    Article  Google Scholar 

  41. Guillermo F-CJ, Hisao I, Coello Coello Carlos A, Michael E (2021) On the effect of the cooperation of indicator-based multiobjective evolutionary algorithms. IEEE Trans Evol Comput 25(4):681–695

    Article  Google Scholar 

  42. Li B, Tang K, Li J, Yao X (2016) Stochastic ranking algorithm for many-objective optimization based on multiple indicators. IEEE Trans Evol Comput 20(6):924–938

    Article  Google Scholar 

  43. Phan DH, Suzuki J (2011) Boosting indicator-based selection operators for evolutionary multiobjective optimization algorithms. In: 2011 IEEE 23rd international conference on tools with artificial intelligence, pp 276–281

  44. Phan DH, Suzuki J, Hayashi I (2012) Leveraging indicator-based ensemble selection in evolutionary multiobjective optimization algorithms. In: Proceedings of the 14th annual conference on genetic and evolutionary computation, pp 497–504

  45. Tian Y, Zhang X, Cheng R, Jin Y (2016) A multi-objective evolutionary algorithm based on an enhanced inverted generational distance metric. In: 2016 IEEE congress on evolutionary computation (CEC), pp 5222–5229

  46. Tian Y, Cheng R, Zhang X, Cheng F, Jin Y (2018) An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Trans Evol Comput 22(4):609–622

    Article  Google Scholar 

  47. Glamsch J, Rosnitschek T, Rieg F (2021) Initial population influence on hypervolume convergence of NSGA-III. Int J Simul Modell 20(1):123–133

    Article  Google Scholar 

  48. Zhou Y, Chen Z, Huang Z, Xiang Y (2022) A multiobjective evolutionary algorithm based on objective-space localization selection. IEEE Trans Cybern 52(5):3888–3901. https://doi.org/10.1109/TCYB.2020.3016426

    Article  PubMed  Google Scholar 

  49. He C, Cheng R, Yazdani D (2022) Adaptive offspring generation for evolutionary large-scale multiobjective optimization. IEEE Trans Syst Man Cyber Syst 52(2):786–798

    Article  Google Scholar 

  50. He C, Cheng R, Li L, Tan KC, Jin Y (2022) Large-scale multiobjective optimization via reformulated decision variable analysis. IEEE Trans Evol Comput 28:1–1

    CAS  Google Scholar 

  51. Tian Y, Cheng R, Zhang X, Jin Y (2017) PlatEMO: a MATLAB platform for evolutionary multi-objective optimization [educational forum]. IEEE Comput Intell Mag 12(4):73–87

    Article  Google Scholar 

  52. Indraneel D, Dennis JE (1998) Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657

    Article  MathSciNet  Google Scholar 

  53. Agrawal RB, Deb K, Agrawal RB (1994) Simulated binary crossover for continuous search space. Complex Syst 9(3):115–148

    MathSciNet  Google Scholar 

  54. Deb K, Goyal M (1996) A combined genetic adaptive search (geneas) for engineering design. Comput Sci Inf 26(4):30–45

    Google Scholar 

  55. Jain H, Deb K (2014) 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

    Article  Google Scholar 

  56. Tian Y, Cheng R, Zhang X, Li M, Jin Y (2019) Diversity assessment of multi-objective evolutionary algorithms: performance metric and benchmark problems [research frontier]. IEEE Comput Intell Mag 14(3):61–74

    Article  Google Scholar 

  57. Ishibuchi H, Masuda H, Tanigaki Y, Nojima Y (2015) Modified distance calculation in generational distance and inverted generational distance. In: Evolutionary multi-criterion optimization, Springer International Publishing, pp 110–125

  58. Wang H, Jin Y, Yao X (2017) Diversity assessment in many-objective optimization. IEEE Trans Cybern 47(6):1510–1522

    Article  PubMed  Google Scholar 

  59. Tian Y, Xiang X, Zhang X, Cheng R, Jin Y (2018b) Sampling reference points on the pareto fronts of benchmark multi-objective optimization problems. In: 2018 IEEE congress on evolutionary computation (CEC), pp 1–6

Download references

Funding

This work is supported by the National Natural Science Foundation of China under Grant 61802153.

Author information

Authors and Affiliations

  1. School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, Jiangsu, China

    Weida Song, Shanxin Zhang, Wenlong Ge & Wei Wang

Authors
  1. Weida Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shanxin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenlong Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Weida Song: Methodology, Investigation, Writing—original draft. Shanxin Zhang: Writing—review & editing. Wenlong Ge: Methodology. Wei Wang: Methodology.

Corresponding author

Correspondence to Shanxin Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, W., Zhang, S., Ge, W. et al. An improved indicator-based two-archive algorithm for many-objective optimization problems. Computing (2024). https://doi.org/10.1007/s00607-024-01272-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00607-024-01272-3

Keywords

Mathematics Subject Classification

Search

Navigation