Skip to main content
SpringerLink
Log in

Multimodal multi-objective optimization via determinantal point process-assisted evolutionary algorithm

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

Multimodal multi-objective optimization problems (MMOPs) are widely present in real life. Due to the need of balancing between convergence and diversity in multi-objective optimization, as well as the need of balancing between diversities in objective and decision spaces, exploring the Pareto optimal front and Pareto optimal solution set becomes rather difficult in solving MMOPs. Recently, some multimodal multi-objective optimization algorithms have emerged. However, most of them are convergence-first, which may result in poor diversity of the solution set in decision space. To remedy this defect, in this paper, a determinantal point process (DPP)-assisted evolutionary algorithm is proposed to effectively solve MMOPs. In the proposed method, i) the DPPs are used to select subsets to consider convergence and diversity in both objective and decision spaces; ii) a kernel matrix is designed to retain solutions with poor convergence but good diversity in the decision space to explore the equivalent Pareto optimal solution sets; and iii) we propose a framework that combines the population and archive to better solve MMOPs. The results show that the proposed algorithm achieves the best performance in 18 of the 28 benchmark problems compared to six state-of-the-art algorithms.

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
Fig. 2
Algorithm 1
Algorithm 2
Algorithm 3
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

Data availability

Comparison algorithms and benchmarks are available on Platemo [35] (https://github.com/BIMK/PlatEMO).

References

  1. Zheng T, Liu J, Liu Y, Tan S (2022) Hybridizing multi-objective, clustering and particle swarm optimization for multimodal optimization. Neural Comput Appl 34:2247–2274

    Article  Google Scholar 

  2. Babaee Tirkolaee E, Goli A, Weber G-W (2020) Fuzzy mathematical programming and self-adaptive artificial fish swarm algorithm for just-in-time energy-aware flow shop scheduling problem with outsourcing option. IEEE Trans Fuzzy Syst 28(11):2772–2783

    Article  Google Scholar 

  3. Yue C, Qu B, Liang J (2018) A multiobjective particle swarm optimizer using ring topology for solving multimodal multiobjective problems. IEEE Trans Evolut Computat 22(5):805–817

    Article  Google Scholar 

  4. Sebag M, Tarrisson N, Teytaud O, Lefevre J, Baillet S (2005) A multi-objective multi-modal optimization approach for mining stable spatio-temporal patterns. In: IJCAI-05, proceedings of the nineteenth international joint conference on artificial intelligence, Edinburgh, Scotland, UK, July 30-August 5, pp. 859–864

  5. Jaszkiewicz A (2002) On the performance of multiple-objective genetic local search on the 0/1 knapsack problem-a comparative experiment. IEEE Trans Evolut Computat 6(4):402–412

    Article  Google Scholar 

  6. Ishibuchi H, Yamane M, Akedo N, Nojima Y (2013) Many-objective and many-variable test problems for visual examination of multiobjective search. In: Evolutionary Computation

  7. Han S, Zhu K, Zhou M, Cai X (2021) Information-utilization-method-assisted multimodal multiobjective optimization and application to credit card fraud detection. IEEE Trans Computat Soc Syst 8(4):856–869

    Article  Google Scholar 

  8. Peng Y, Ishibuchi H (2021) A diversity-enhanced subset selection framework for multi-modal multi-objective optimization. IEEE Trans Evolut Computat 26(5):886–900

    Article  Google Scholar 

  9. Rizk-Allah AE, Rizk M, Hassanien Slowik A (2020) Multi-objective orthogonal opposition-based crow search algorithm for large-scale multi-objective optimization. Neural Comput Appl 32:13715–13746

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Zitzler E, Laumanns M, Thiele L (2001) Spea 2: improving the strength pareto evolutionary algorithm. TIK-Report 103:201

    Google Scholar 

  12. Menchaca-Mendez A, Coello CAC (2015) Gde-moea: A new moea based on the generational distance indicator and \(\varepsilon \)-dominance. In: 2015 IEEE congress on evolutionary computation (CEC), pp 947–955

  13. Wang H, Jiao L, Yao X (2015) Two_arch2: an improved two-archive algorithm for many-objective optimization. IEEE Trans Evolut Computat 19(4):524–541

    Article  Google Scholar 

  14. Zhang Q, Li H (2007) Moea/d: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Liang JJ, Yue CT, Qu BY (2016) Multimodal multi-objective optimization: A preliminary study. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 2454–2461

  17. Lin Q, Lin W, Zhu Z, Gong M, Li J, Coello CAC (2021) Multimodal multiobjective evolutionary optimization with dual clustering in decision and objective spaces. IEEE Trans Evolut Computat 25(1):130–144

    Article  Google Scholar 

  18. Zhang K, Shen C, Yen GG, Xu Z, He J (2021) Two-stage double niched evolution strategy for multimodal multiobjective optimization. IEEE Trans Evolut Computat 25(4):754–768

    Article  Google Scholar 

  19. Li W, Zhang T, Wang R, Ishibuchi H (2021) Weighted indicator-based evolutionary algorithm for multimodal multiobjective optimization. IEEE Trans Evolut Computat 25(6):1064–1078

    Article  Google Scholar 

  20. Li W, Yao X, Zhang T, Wang R, Wang L (2022) Hierarchy ranking method for multimodal multi-objective optimization with local pareto fronts. IEEE Trans Evolut Computat 27(1):98–110

    Article  Google Scholar 

  21. Liu Y, Ishibuchi H, Yen GG, Nojima Y, Masuyama N (2020) Handling imbalance between convergence and diversity in the decision space in evolutionary multimodal multiobjective optimization. IEEE Trans Evolut Computat 24(3):551–565

    Google Scholar 

  22. Kulesza A, Taskar B (2011) k-dpps: fixed-size determinantal point processes. In: proceedings of the 28th international conference on machine learning, ICML 2011, Bellevue, Washington, USA, June 28 - July 2, 2011

  23. Gartrell M, Paquet U, Koenigstein N (2016) The bayesian low-rank determinantal point process mixture model. In: proceedings of the 10th ACM conference on recommender systems, pp. 349–356

  24. Tremblay N, Amblard P-O, Barthelmé S (2017) Graph sampling with determinantal processes. In: 2017 25th European signal processing conference (EUSIPCO), pp. 1674–1678

  25. Liu Y, Yen GG, Gong D (2019) A multimodal multiobjective evolutionary algorithm using two-archive and recombination strategies. IEEE Trans Evolut Computat 23(4):660–674

    Article  Google Scholar 

  26. Li Z, Zou J, Yang S, Zheng J (2021) A two-archive algorithm with decomposition and fitness allocation for multi-modal multi-objective optimization. Inf Sci 574:413–430

    Article  MathSciNet  Google Scholar 

  27. Tanabe R, Ishibuchi H (2020) A framework to handle multimodal multiobjective optimization in decomposition-based evolutionary algorithms. IEEE Trans Evolut Computat 24(4):720–734

    Article  Google Scholar 

  28. Wang W, Li G, Wang Y, Wu F, Zhang W, Li L (2022) Clearing-based multimodal multi-objective evolutionary optimization with layer-to-layer strategy. Swarm Evolut Computat 68:100976

    Article  Google Scholar 

  29. Li G, Wang W, Zhang W, You W, Wu F, Tu H (2021) Handling multimodal multi-objective problems through self-organizing quantum-inspired particle swarm optimization. Inf Sci 577:510–540

    Article  MathSciNet  Google Scholar 

  30. Fan Q, Yan X (2021) Solving multimodal multiobjective problems through zoning search. IEEE Trans Syst Man Cybern Syst 51(8):4836–4847

    Article  Google Scholar 

  31. Li G, Wang W, Zhang W, Wang Z, Tu H, You W (2021) Grid search based multi-population particle swarm optimization algorithm for multimodal multi-objective optimization. Swarm Evolut Computat 62:100843

    Article  Google Scholar 

  32. Maity G, Roy SK, Verdegay JL (2020) Analyzing multimodal transportation problem and its application to artificial intelligence. Neural Comput Appl 32:2243–2256

    Article  Google Scholar 

  33. Zhang P, Li J, Li T, Chen H (2021) A new many-objective evolutionary algorithm based on determinantal point processes. IEEE Trans Evolut Computat 25(2):334–345

    Article  Google Scholar 

  34. Liu Y, Ishibuchi H, Yen GG, Nojima Y, Masuyama N, Han Y (2020) On the normalization in evolutionary multi-modal multi-objective optimization. In: 2020 IEEE congress on evolutionary computation (CEC), pp. 1–8

  35. Tian Y, Cheng R, Zhang X, Jin Y (2017) PlatEMO: a MATLAB platform for evolutionary multi-objective optimization. IEEE Computat Intell Magaz 12:73–87

    Article  Google Scholar 

  36. Agrawal R, Deb K, Agrawal R (2000) Simulated binary crossover for continuous search space. Complex Syst 9:115–148

    MathSciNet  Google Scholar 

  37. Deb K, Tiwari S (2008) Omni-optimizer: a generic evolutionary algorithm for single and multi-objective optimization. Europ J Operat Res 185(3):1062–1087

    Article  MathSciNet  Google Scholar 

  38. Zhou A, Zhang Q, Jin Y (2009) Approximating the set of pareto-optimal solutions in both the decision and objective spaces by an estimation of distribution algorithm. IEEE Trans Evolut Computat 13(5):1167–1189

    Article  Google Scholar 

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

    Article  Google Scholar 

  40. Alcalá-Fdez J, Sánchez L, García S, del Jesus MJ, Ventura S, Garrell JM, Otero J, Romero C, Bacardit J, Rivas VM, Fernández JC, Herrera F (2009) KEEL: a software tool to assess evolutionary algorithms for data mining problems. Soft Comput 13(3):307–318

    Article  Google Scholar 

  41. Zhou Y, Xiang Y, He X (2021) Constrained multiobjective optimization: test problem construction and performance evaluations. IEEE Trans Evolut Computat 25(1):172–186

    Article  Google Scholar 

  42. Ishibuchi H, Pang LM, Shang K (2022) Difficulties in fair performance comparison of multi-objective evolutionary algorithms [research frontier]. IEEE Computat Intell Magaz 17(1):86–101

    Article  Google Scholar 

  43. Liang J, Lin H, Yue C, Yu K, Guo Y, Qiao K (2023) Multiobjective differential evolution with speciation for constrained multimodal multiobjective optimization. IEEE Trans Evolut Computat 27(4):1115–1129

    Article  Google Scholar 

Download references

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China under Grant No. 62076225.

Author information

Authors and Affiliations

  1. School of Computer Science, China University of Geosciences, Wuhan, 430074, Hubei, China

    Xinyu Cheng, Wenyin Gong & Fei Ming

  2. School of Computer Science and Artificial Intelligence, Wuhan University of Technology, Wuhan, 430071, Hubei, China

    Xiaofang Zhu

Authors
  1. Xinyu Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenyin Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fei Ming
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaofang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenyin Gong.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts of interest to this work. The article contains no violation of any existing copyright or other third party right.

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

Cheng, X., Gong, W., Ming, F. et al. Multimodal multi-objective optimization via determinantal point process-assisted evolutionary algorithm. Neural Comput & Applic 36, 1381–1411 (2024). https://doi.org/10.1007/s00521-023-09110-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-023-09110-x

Keywords

Search

Navigation