Abstract
In the past few decades, to solve the multi-objective optimization problems, many multi-objective evolutionary algorithms (MOEAs) have been proposed. However, MOEAs have a common difficulty: because the diversity and convergence of solutions are often two conflicting conditions, the balance between the diversity and convergence directly determines the quality of the solutions obtained by the algorithms. Meanwhile, the nondominated sorting method is a costly operation in part Pareto-based MOEAs and needs to be optimized. In this article, we propose a multi-objective evolutionary algorithm framework with convergence and diversity adjusted adaptively. Our contribution is mainly reflected in the following aspects: firstly, we propose a nondominated sorting-based MOEA framework with convergence and diversity adjusted adaptively; secondly, we propose a novel fast nondominated sorting algorithm; thirdly, we propose a convergence improvement strategy and a diversity improvement strategy. In the experiments, we compare our method with several popular MOEAs based on two widely used performance indicators in several multi-objective problem test instances, and the empirical results manifest the proposed method performs the best on most test instances, which further demonstrates that it outperforms all the comparison algorithms.
Similar content being viewed by others
Data availability
All data and materials are available on request from the authors of this paper.
References
Xue Y, Zhu H, Neri F (2022) A self-adaptive multi-objective feature selection approach for classification problems. Integr Comput Aided Eng 29(1):3–21
Xue Y, Tang Y, Xu X, Liang J, Neri F (2021) Multi-objective feature selection with missing data in classification. IEEE Trans Emerg Top Comput Intell 6:1–10
Gupta S, Garg H, Chaudhary S (2020) Parameter estimation and optimization of multi-objective capacitated stochastic transportation problem for gamma distribution. Complex Intell Syst 6(3):651–667
Garg H, Rizk-Allah RM (2021) A novel approach for solving rough multi-objective transportation problem: development and prospects. Comput Appl Math 40(4):1–24
Liang Y, He F, Zeng X, Luo J (2022) An improved loop subdivision to coordinate the smoothness and the number of faces via multi-objective optimization. Integr Comput Aided Eng 29(1):23–41
Gholizadeh H, Javadian N, Fazlollahtabar H (2020) An integrated fuzzy-genetic failure mode and effect analysis for aircraft wing reliability. Soft Comput 24(17):13401–13412
Zhang J, He F, Duan Y, Yang S (2023) Aidednet: anti-interference and detail enhancement dehazing network for real-world scenes. Front Comput Sci 17(2):172703
Wang F, Liao F, Li Y, Wang H (2021) A new prediction strategy for dynamic multi-objective optimization using Gaussian Mixture Model. Inf Sci 580:331–351
Wang G-G, Gao D, Pedrycz W (2022) Solving multi-objective fuzzy job-shop scheduling problem by a hybrid adaptive differential evolution algorithm. IEEE Trans Ind Inform 18:8519
Wu H, He F, Duan Y, Yan X (2022) Perceptual metric-guided human image generation. Integr Comput Aided Eng 29(2):141–151
Gholizadeh H, Goh M, Fazlollahtabar H, Mamashli Z (2022) Modelling uncertainty in sustainable-green integrated reverse logistics network using metaheuristics optimization. Comput Ind Eng 163:107828
Li H, He F, Chen Y, Luo J (2020) Multi-objective self-organizing optimization for constrained sparse array synthesis. Swarm Evolut Comput 58:100743
Osyczka A (1978) An approach to multicriterion optimization problems for engineering design. Comput Methods Appl Mech Eng 15(3):309–333
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evolut Comput 11(6):712–731
Zhou A, Qu B-Y, Li H, Zhao S-Z, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut Comput 1(1):32–49
Ghane-Kanafi A, Khorram E (2015) A new scalarization method for finding the efficient frontier in non-convex multi-objective problems. Appl Math Model 39(23–24):7483–7498
Li H, He F, Liang Y, Quan Q (2020) A dividing-based many-objective evolutionary algorithm for large-scale feature selection. Soft Comput 24(9):6851–6870
Fakhfakh F, Cheikhrouhou S, Dammak B, Hamdi M, Rekik M (2022) Multi-objective approach for scheduling time-aware business processes in cloud-fog environment. J Supercomput 79:1–25
Luo J, He F, Gao X (2023) An enhanced grey wolf optimizer with fusion strategies for identifying the parameters of photovoltaic models. Integr Comput Aided Eng 30(1):89–104
Zhou J, Zhang Y, Zheng J, Li M (2022) Domination-based selection and shift-based density estimation for constrained multiobjective optimization. IEEE Trans Evolut Comput. https://doi.org/10.1109/TEVC.2022.3190401
De Moraes MB, Coelho GP (2022) A random forest-assisted decomposition-based evolutionary algorithm for multi-objective combinatorial optimization problems. In: 2022 IEEE Congress on Evolutionary Computation (CEC). IEEE, pp 1–8
Zhao C, Zhou Y, Hao Y (2022) Decomposition-based evolutionary algorithm with dual adjustments for many-objective optimization problems. Swarm Evolut Comput 75:101168
Rostami S, Neri F (2017) A fast hypervolume driven selection mechanism for many-objective optimisation problems. Swarm Evolut Comput 34:50–67
Liu Z, Wang H, Jin Y (2022) Performance indicator-based adaptive model selection for offline data-driven multiobjective evolutionary optimization. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2022.3170344
Mefgouda B, Idoudi H (2022) New network interface selection based on MADM and multi-objective whale optimization algorithm in heterogeneous wireless networks. J Supercomput 79:1–36
Cheng R, Jin Y, Olhofer M, Sendhoff B (2016) A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Trans Evolut Comput 20(5):773–791
Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evolut Comput 2(3):221–248
Fang H, Wang Q, Tu Y-C, Horstemeyer MF (2008) An efficient non-dominated sorting method for evolutionary algorithms. Evolut Comput 16(3):355–384
Zhang X, Tian Y, Cheng R, Jin Y (2015) An efficient approach to nondominated sorting for evolutionary multiobjective optimization. IEEE Trans Evolut Comput 19(2):201–213
Roy PC, Deb K, Islam MM (2019) An efficient nondominated sorting algorithm for large number of fronts. IEEE Trans Cybern 49(3):859–869
Chen B, Zeng W, Lin Y, Zhang D (2014) A new local search-based multiobjective optimization algorithm. IEEE Trans Evolut Comput 19(1):50–73
Tian Y, Cheng R, Zhang X, Su Y, Jin Y (2018) A strengthened dominance relation considering convergence and diversity for evolutionary many-objective optimization. IEEE Trans Evolut Comput 23(2):331–345
Li H, Zhang Q (2008) Multiobjective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II. IEEE Trans Evolut Comput 13(2):284–302
Yuan Y, Xu H, Wang B, Zhang B, Yao X (2015) Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans Evolut Comput 20(2):180–198
Guo X, Wang X, Wei Z (2015) MOEA/D with adaptive weight vector design. In: 2015 11th International Conference on Computational Intelligence and Security (CIS). IEEE, pp 291–294
Li W, Yuan J, Wang L (2023) An enhanced decomposition-based multiobjective evolutionary algorithm with adaptive neighborhood operator and extended distance-based environmental selection. J Supercomput 79:1–53
Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evolut Comput 19(1):45–76
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). IEEE, pp 947–955
Sun Y, Yen GG, Yi Z (2018) IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Trans Evolut Comput 23(2):173–187
Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength Pareto evolutionary algorithm. TIK-Report 103
Zhang X, Zheng X, Cheng R, Qiu J, Jin Y (2018) A competitive mechanism based multi-objective particle swarm optimizer with fast convergence. Inf Sci 427:63–76
Tian Y, He C, Cheng R, Zhang X (2019) A multistage evolutionary algorithm for better diversity preservation in multiobjective optimization. IEEE Trans Syst Man Cybern Syst 51(9):5880–5894
Moreno J, Rodriguez D, Nebro AJ, Lozano JA (2021) Merge nondominated sorting algorithm for many-objective optimization. IEEE Trans Cybern 51(12):6154–6164
Spurlock K, Elgazzar H (2022) A genetic mixed-integer optimization of neural network hyper-parameters. J Supercomput 78(12):14680–14702
Dhal KG, Das A, Ray S, Rai R, Ghosh TK (2022) Archimedes optimizer-based fast and robust fuzzy clustering for noisy image segmentation. J Supercomput 79:1–40
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
Farias LR, Araújo AF (2021) IM-MOEA/D: an inverse modeling multi-objective evolutionary algorithm based on decomposition. In: 2021 IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEE, pp 462–467
Dai C, Wang Y, Ye M (2015) A new multi-objective particle swarm optimization algorithm based on decomposition. Inf Sci 325:541–557
Lin Q, Li J, Du Z, Chen J, Ming Z (2015) A novel multi-objective particle swarm optimization with multiple search strategies. Eur J Oper Res 247(3):732–744
Pan L, Xu W, Li L, He C, Cheng R (2021) Adaptive simulated binary crossover for rotated multi-objective optimization. Swarm Evolut Comput 60:100759
Li L, He C, Cheng R, Li H, Pan L, Jin Y (2022) A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization. Swarm Evolut Comput 75:101181
He C, Cheng R, Li L, Tan KC, Jin Y (2022) Large-scale multiobjective optimization via reformulated decision variable analysis. IEEE Trans Evolut Comput. https://doi.org/10.1109/TEVC.2022.3213006
Wang G-G, Gandomi AH, Alavi AH, Deb S (2016) A hybrid method based on krill herd and quantum-behaved particle swarm optimization. Neural Computing and Applications 27(4):989–1006
Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evolut Comput 8(2):173–195
Deb K, Thiele L, Laumanns M, Zitzler E (2002) Scalable multi-objective optimization test problems. In: Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600), vol 1. IEEE, pp 825–830
Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evolut Comput 10(5):477–506
Wang Z, Ong Y-S, Ishibuchi H (2019) On scalable multiobjective test problems with hardly dominated boundaries. IEEE Trans Evolut Comput 23(2):217–231
Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evolut Comput 7(2):117–132
While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evolut Comput 10(1):29–38
Gonzalez-Sanchez B, Vega-Rodríguez MA, Santander-Jiménez S (2022) Parallel multi-objective optimization approaches for protein encoding. J Supercomput 78:1–31
McClymont K, Keedwell E (2012) Deductive sort and climbing sort: new methods for non-dominated sorting. Evolut Comput 20(1):1–26
Wang H, Yao X (2013) Corner sort for Pareto-based many-objective optimization. IEEE Trans Cybern 44(1):92–102
Buzdalov M, Shalyto A (2014) A provably asymptotically fast version of the generalized Jensen algorithm for non-dominated sorting. In: Parallel Problem Solving from Nature—PPSN XIII: 13th International Conference, Ljubljana, Slovenia, September 13–17, 2014. Proceedings, vol 13. Springer, pp 528–537
Li L, He F, Fan R, Fan B, Yan X (2023) 3D Reconstruction based on hierarchical reinforcement learning with transferability. Integr Comput Aided Eng. https://doi.org/10.3233/ICA-230710
Li P, He F, Fan B, Song Y (2023) TPNet: a novel mesh analysis method via topology preservation and perception enhancement. Comput Aided Geom Des. https://doi.org/10.1016/j.cagd.2023.102219
Funding
This work was supported by the National Natural Science Foundation of China under Grant 62072348 and China Yunnan province major science and technology special plan project No. 202202AF080004. The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.
Author information
Authors and Affiliations
Contributions
XG wrote the main manuscript text and designed the method. FH supervised and revised the manuscript. JL participated in the experiment. SZ and BF revised and proofread the manuscript. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human or animal subjects performed by any of the authors.
Informed consent
Informed consent was obtained from all individual participants included in the study.
Consent to participate
Not applicable.
Consent for publication
We declare that we consented for the publication of this research work.
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.
About this article
Cite this article
Gao, X., He, F., Zhang, S. et al. A fast nondominated sorting-based MOEA with convergence and diversity adjusted adaptively. J Supercomput 80, 1426–1463 (2024). https://doi.org/10.1007/s11227-023-05516-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-023-05516-5