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A general framework for enhancing relaxed Pareto dominance methods in evolutionary many-objective optimization

Abstract

In the last decade, it is widely known that the Pareto dominance-based evolutionary algorithms (EAs) are unable to deal with many-objective optimization problems (MaOPs) well, as it is hard to maintain a good balance between convergence and diversity. Instead, most researchers in this domain tend to develop EAs that do not rely on Pareto dominance (e.g., decomposition-based and indicator-based techniques) to solve MaOPs. However, it is still hard for these non-Pareto-dominance-based methods to solve MaOPs with unknown irregular PF shapes. In this paper, we develop a general framework for enhancing relaxed Pareto dominance methods to solve MaOPs, which can promote both convergence and diversity. During the environmental selection step, we use M different cases of relaxed Pareto dominance simultaneously, where each expands the dominance area of solutions for \(M\,-\) 1 objectives to improve the selection pressure, while the remaining one objective keeps unchanged. We conduct the experiments on a variety of test problems, the result shows that our proposed framework can obviously improve the performance of relaxed Pareto dominance in solving MaOPs, and is very competitive against or outperform some state-of-the-art many-objective EAs.

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References

  • Batista LS, Campelo F, Guimarães FG, Ramírez JA (2011) Pareto cone \(\varepsilon\)-dominance: improving convergence and diversity in multiobjective evolutionary algorithms. In: Proceedings of international conference evolutionary multi-criterion optimization, pp 76–90

  • Breiman L (1996) Bagging predictors. Mach Learn 24(2):123–140

    MATH  Google Scholar 

  • Chen L, Liu HL, Tan KC, Cheung YM, Wang Y (2019) Evolutionary many-objective algorithm using decomposition-based dominance relationship. IEEE Trans Cybern 49(12):4129–4139

    Article  Google Scholar 

  • Chen L, Deb K, Liu HL, Zhang Q (2021) Effect of objective normalization and penalty parameter on penalty boundary intersection decomposition-based evolutionary many-objective optimization algorithms. Evol Comput 29(1):157–186

    Article  Google Scholar 

  • 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 

  • Cheng R, Li M, Tian Y, Zhang X, Yang S, Jin Y, Yao X (2017) A benchmark test suite for evolutionary many-objective optimization. Complex Intell Syst 3(1):67–81

    Article  Google Scholar 

  • Cui M, Li L, Zhou M, Abusorrah A (2021) Surrogate-assisted autoencoder-embedded evolutionary optimization algorithm to solve high-dimensional expensive problems. IEEE Trans Evol Comput. https://doi.org/10.1109/TEVC.2021.3113923

    Article  Google Scholar 

  • Cui M, Li L, Zhou M, Li J, Abusorrah A, Sedraoui K (2022) A bi-population cooperative optimization algorithm assisted by an autoencoder for medium-scale expensive problems. IEEE/CAA J Automatica Sinica. https://doi.org/10.1109/JAS.2022.105425

    Article  Google Scholar 

  • Dai C, Wang Y, Hu L (2016) An improved \(\alpha\)-dominance strategy for many-objective optimization problems. Soft Comput 20(3):1105–1111

    Article  Google Scholar 

  • 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 

  • 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 

  • Deb K, Mohan M, Mishra S (2005) Evaluating the \(\varepsilon\)-domination based multi-objective evolutionary algorithm for a quick computation of pareto-optimal solutions. Evol Comput 13(4):501–525

    Article  Google Scholar 

  • Deb K, Thiele L, Laumanns M, Zitzler E (2005b) Scalable test problems for evolutionary multiobjective optimization. In: Evolutionary multiobjective optimization, Springer, pp 105–145

  • Elarbi M, Bechikh S, Coello CAC, Makhlouf M, Said LB (2020) Approximating complex pareto fronts with pre-defined normal-boundary intersection directions. IEEE Trans Evol Comput 24(5):809–823

    Article  Google Scholar 

  • Falcón-Cardona JG, Ishibuchi H, Coello CAC, Emmerich M (2021) On the effect of the cooperation of indicator-based multi-objective evolutionary algorithms. IEEE Trans Evol Comput 25(4):9681–8695

    Article  Google Scholar 

  • Ishibuchi H, Setoguchi Y, 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 

  • Ishibuchi H, Matsumoto T, Masuyama N, Nojima Y (2020a) Effects of dominance resistant solutions on the performance of evolutionary multi-objective and many-objective algorithms. In: Proceedings of annual conference on genetic and evolutionary computation (GECCO), pp 507–515

  • Ishibuchi H, Matsumoto T, Masuyama N, Nojima Y (2020b) Many-objective problems are not always difficult for pareto dominance-based evolutionary algorithms. In: Proceedings of 24th European conference on artificial intelligence (ECAI), pp 291–298

  • Ikeda K, Kobayashi S, Kita H(2001) Failure of pareto-based MOEAs: Does non-dominated really mean near to optimal? In: Proceedings of the 2001 congress on evolutionary computation, vol 2, pp 957–962

  • 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 

  • Li K, Deb K, Zhang Q, Kwong S (2015) An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans Evol Comput 19(5):694–716

    Article  Google Scholar 

  • Liu HL, Chen L, Zhang Q, Deb K (2018) Adaptively allocating search effort in challenging many-objective optimization problems. IEEE Trans Evol Comput 22(3):433–448

    Article  Google Scholar 

  • Liu J, Wang Y, Wang X, Guo S, Sui X (2019) A new dominance method based on expanding dominated area for many-objective optimization. Int J Pattern Recognit Artif Intell 33(03):1959008

    Article  Google Scholar 

  • Lu Z, Deb K, Goodman E, Banzhaf W, Boddeti VN (2020a) NSGANetV2: Evolutionary multi-objective surrogate-assisted neural architecture search. In: European conference on computer vision. Springer, pp 35–51

  • Lu Z, Whalen I, Dhebar Y, Deb K, Goodman E, Banzhaf W, Boddeti VN (2020b) Multi-objective evolutionary design of deep convolutional neural networks for image classification. IEEE Trans Evol Comput 25(2):277–291

  • Lu Z, Sreekumar G, Goodman E, Banzhaf W, Deb K, Boddeti VN (2021) Neural architecture transfer. IEEE Trans Pattern Anal Mach Intell 43(9):2971–2989

    Article  Google Scholar 

  • Peng C, Liu HL, Goodman ED (2021) A cooperative evolutionary framework based on an improved version of directed weight vectors for constrained multiobjective optimization with deceptive constraints. IEEE Trans Cybern 51(11):5546–5558

    Article  Google Scholar 

  • Peng C, Liu HL, Goodman ED, Tan KC (2022) A two-phase framework of locating the reference point for decomposition-based constrained multi-objective evolutionary algorithms. Knowl Based Syst 239:107933

    Article  Google Scholar 

  • Santos T, Takahashi RH (2018) On the performance degradation of dominance-based evolutionary algorithms in many-objective optimization. IEEE Trans Evol Comput 22(1):19–31

    Article  Google Scholar 

  • Sato H, Aguirre HE, Tanaka K (2007) Controlling dominance area of solutions and its impact on the performance of MOEAs. In: Proceedings of international conference on evolutionary multi-criterion optimization (EMO 2007), ACM, pp 5–20

  • Sato H, Aguirre H, Tanaka K (2010) Self-controlling dominance area of solutions in evolutionary many-objective optimization. Simul Evol Learn 455–465

  • Shang K, Ishibuchi H (2020) A new hypervolume-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 24(5):839–852

    Article  Google Scholar 

  • Singh HK, Bhattacharjee KS, Ray T (2019) Distance-based subset selection for benchmarking in evolutionary multi/many-objective optimization. IEEE Trans Evol Comput 23(5):904–912

    Article  Google Scholar 

  • 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 

  • Tian Y, Cheng R, Zhang X, Su Y, Jin Y (2019) A strengthened dominance relation considering convergence and diversity for evolutionary many-objective optimization. IEEE Trans Evol Comput 23(2):331–345

    Article  Google Scholar 

  • Xiang Y, Zhou Y, Li M, Chen Z (2016) A vector angle-based evolutionary algorithm for unconstrained many-objective optimization. IEEE Trans Evol Comput 21(1):131–152

    Article  Google Scholar 

  • Xiang Y, Zhou Y, Yang X, Huang H (2020) A many-objective evolutionary algorithm with pareto-adaptive reference points. IEEE Trans Evol Comput 24(1):99–113

    Article  Google Scholar 

  • 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 

  • Zhao Z, Liu S, Zhou M, Abusorrah A (2020) Dual-objective mixed integer linear program and memetic algorithm for an industrial group scheduling problem. IEEE/CAA J Automatica Sinica 8(6):1199–1209

    Article  Google Scholar 

  • Zhao Z, Zhou M, Liu S (2021) Iterated greedy algorithms for flow-shop scheduling problems: a tutorial. IEEE Trans Autom Sci Eng. https://doi.org/10.1109/TASE.2021.3062994

    Article  Google Scholar 

  • Zhou Z, Zhu S (2018) Kernel-based multiobjective clustering algorithm with automatic attribute weighting. Soft Comput 22(11):3685–3709

    Article  Google Scholar 

  • Zhu C, Xu L, Goodman ED (2016) Generalization of pareto-optimality for many-objective evolutionary optimization. IEEE Trans Evol Comput 20(2):299–315

    Article  Google Scholar 

  • Zhu S, Xu L, Goodman ED (2020) Evolutionary multi-objective automatic clustering enhanced with quality metrics and ensemble strategy. Knowl Based Syst 188(105018):1–21

    Google Scholar 

  • Zhu S, Xu L, Goodman ED (2021) Hierarchical topology-based cluster representation for scalable evolutionary multiobjective clustering. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2021.3081988

    Article  Google Scholar 

  • Zhu S, Xu L, Goodman ED, Lu Z (2022) A new many-objective evolutionary algorithm based on generalized pareto dominance. IEEE Trans Cybern. 52(8):7776–7790  https://doi.org/10.1109/TCYB.2021.3051078

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Acknowledgements

This work was supported by the Natural Science Foundation of China under Grant 61973337, 62073155, 62002137, 62106088 and the Guangdong Provincial Key Laboratory under Grant 2020B121201001

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Correspondence to Lihong Xu.

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Zhu, S., Xu, L., Goodman, E. et al. A general framework for enhancing relaxed Pareto dominance methods in evolutionary many-objective optimization. Nat Comput (2022). https://doi.org/10.1007/s11047-022-09889-z

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Keywords

  • Evolutionary algorithms
  • Many-objective optimization
  • Pareto dominance
  • Enhance selection pressure