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Volume 11 Issue 6
Jun.  2024

IEEE/CAA Journal of Automatica Sinica

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Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2024  >  11(6): 1458-1475
J. Liang, H. Lin, C. Yue, P. Suganthan, and  Y. Wang,  “Multiobjective differential evolution for higher-dimensional multimodal multiobjective optimization,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 6, pp. 1458–1475, Jun. 2024. doi: 10.1109/JAS.2024.124377
Citation: J. Liang, H. Lin, C. Yue, P. Suganthan, and  Y. Wang,  “Multiobjective differential evolution for higher-dimensional multimodal multiobjective optimization,” IEEE/CAA J. Autom. Sinica, vol. 11, no. 6, pp. 1458–1475, Jun. 2024. doi: 10.1109/JAS.2024.124377
shu

Multiobjective Differential Evolution for Higher-Dimensional Multimodal Multiobjective Optimization

doi: 10.1109/JAS.2024.124377
Funds:  This work was supported in part by National Natural Science Foundation of China (62106230, U23A20340, 62376253, 62176238), China Postdoctoral Science Foundation (2023M743185), and Key Laboratory of Big Data Intelligent Computing, Chongqing University of Posts and Telecommunications Open Fundation (BDIC-2023-A-007)
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    Abstract

  • In multimodal multiobjective optimization problems (MMOPs), there are several Pareto optimal solutions corresponding to the identical objective vector. This paper proposes a new differential evolution algorithm to solve MMOPs with higher-dimensional decision variables. Due to the increase in the dimensions of decision variables in real-world MMOPs, it is difficult for current multimodal multiobjective optimization evolutionary algorithms (MMOEAs) to find multiple Pareto optimal solutions. The proposed algorithm adopts a dual-population framework and an improved environmental selection method. It utilizes a convergence archive to help the first population improve the quality of solutions. The improved environmental selection method enables the other population to search the remaining decision space and reserve more Pareto optimal solutions through the information of the first population. The combination of these two strategies helps to effectively balance and enhance convergence and diversity performance. In addition, to study the performance of the proposed algorithm, a novel set of multimodal multiobjective optimization test functions with extensible decision variables is designed. The proposed MMOEA is certified to be effective through comparison with six state-of-the-art MMOEAs on the test functions.

     

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  • [1]
    K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, Apr. 2002. doi: 10.1109/4235.996017
    [2]
    F. Kudo, T. Yoshikawa, and T. Furuhashi, “A study on analysis of design variables in Pareto solutions for conceptual design optimization problem of hybrid rocket engine,” in Proc. IEEE Congr. Evolutionary Computation, New Orleans, USA, 2011, pp. 2558–2562.
    [3]
    D.-H. Cho, H.-K. Jung, and C.-G. Lee, “Induction motor design for electric vehicle using a niching genetic algorithm,” IEEE Trans. Ind. Appl., vol. 37, no. 4, pp. 994–999, Jul.–Aug. 2001. doi: 10.1109/28.936389
    [4]
    X. Yao, W. Li, X. Pan, and R. Wang, “Multimodal multi-objective evolutionary algorithm for multiple path planning,” Comput. Ind. Eng., vol. 169, p. 108145, Jul. 2022. doi: 10.1016/j.cie.2022.108145
    [5]
    J. Liang, C. Yue, G. Li, B. Qu, P. N. Suganthan, and K. Yu, “Problem definitions and evaluation criteria for the CEC 2021 on multimodal multiobjective path planning optimization,” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China, Computational Intelligence Laboratory, Nanyang Technological University, Singapore, Tech. Rep. 202012, 2020.
    [6]
    C. T. Yue, J. J. Liang, B. Y. Qu, K. J. Yu, and H. Song, “Multimodal multiobjective optimization in feature selection,” in Proc. IEEE Congr. Evolutionary Computation, Wellington, New Zealand, 2019, pp. 302–309.
    [7]
    K. Jha and S. Saha, “Incorporation of multimodal multiobjective optimization in designing a filter based feature selection technique,” Appl. Soft Comput., vol. 98, p. 106823, Jan. 2021. doi: 10.1016/j.asoc.2020.106823
    [8]
    M. Wang, X. Li, and L. Chen, “An enhance multimodal multiobjective optimization genetic algorithm with special crowding distance for pulmonary hypertension feature selection,” Comput. Biol. Med., vol. 146, p. 105536, Jul. 2022. doi: 10.1016/j.compbiomed.2022.105536
    [9]
    E. Pérez, M. Posada, and A. Lorenzana, “Taking advantage of solving the resource constrained multi-project scheduling problems using multi-modal genetic algorithms,” Soft Comput., vol. 20, no. 5, pp. 1879–1896, May 2016. doi: 10.1007/s00500-015-1610-z
    [10]
    T. Yang, S.-Y. Lin, Y.-H. Hung, and C.-C. Hong, “A study on the optimization of in-process inspection procedure for active pharmaceutical ingredients manufacturing process,” Sustainability, vol. 14, no. 6, p. 3706, Mar. 2022. doi: 10.3390/su14063706
    [11]
    K.-L. Zhao, J. Liu, X.-G. Zhou, J.-Z. Su, Y. Zhang, and G.-J. Zhang, “MMpred: A distance-assisted multimodal conformation sampling for de novo protein structure prediction,” Bioinformatics, vol. 37, no. 23, pp. 4350–4356, Dec. 2021. doi: 10.1093/bioinformatics/btab484
    [12]
    D. Pršić, N. Nedić, and V. Stojanović, “A nature inspired optimal control of pneumatic-driven parallel robot platform,” Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci., vol. 231, no. 1, pp. 59–71, Jan. 2017. doi: 10.1177/0954406216662367
    [13]
    V. Stojanovic and N. Nedic, “A nature inspired parameter tuning approach to cascade control for hydraulically driven parallel robot platform,” J. Optim. Theory Appl., vol. 168, no. 1, pp. 332–347, Jan. 2016. doi: 10.1007/s10957-015-0706-z
    [14]
    N. Nedic, D. Prsic, L. Dubonjic, V. Stojanovic, and V. Djordjevic, “Optimal cascade hydraulic control for a parallel robot platform by PSO,” Int. J. Adv. Manuf. Technol., vol. 72, pp. 5–8, May 1085.
    [15]
    J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. Int. Conf. Neural Networks, Perth, Australia, 1995, pp. 1942–1948.
    [16]
    K. Deb and S. Tiwari, “Omni-optimizer: A procedure for single and multi-objective optimization,” in Proc. 3rd Int. Conf. Evolutionary Multi-Criterion Optimization, Guanajuato, Mexico, 2005, pp. 47–61.
    [17]
    K. Qiao, K. Yu, B. Qu, J. Liang, H. Song, and C. Yue, “An evolutionary multitasking optimization framework for constrained multiobjective optimization problems,” IEEE Trans. Evol. Comput., vol. 26, no. 2, pp. 263–277, Apr. 2022. doi: 10.1109/TEVC.2022.3145582
    [18]
    C. Yue, P. N. Suganthan, J. Liang, B. Qu, K. Yu, Y. Zhu, and L. Yan, “Differential evolution using improved crowding distance for multimodal multiobjective optimization,” Swarm Evol. Comput., vol. 62, p. 100849, Apr. 2021. doi: 10.1016/j.swevo.2021.100849
    [19]
    T. Zhou, Z. Hu, Q. Su, and W. Xiong, “A clustering differential evolution algorithm with neighborhood-based dual mutation operator for multimodal multiobjective optimization,” Expert Syst. Appl., vol. 216, p. 119438, Apr. 2023. doi: 10.1016/j.eswa.2022.119438
    [20]
    Y. Liu, H. Ishibuchi, G. G. Yen, Y. Nojima, and N. Masuyama, “Handling imbalance between convergence and diversity in the decision space in evolutionary multimodal multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 24, no. 3, pp. 551–565, Jun. 2020.
    [21]
    W. Li, T. Zhang, R. Wang, and H. Ishibuchi, “Weighted indicator-based evolutionary algorithm for multimodal multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 25, no. 6, pp. 1064–1078, Dec. 2021. doi: 10.1109/TEVC.2021.3078441
    [22]
    R. Tanabe and H. Ishibuchi, “A decomposition-based evolutionary algorithm for multi-modal multi-objective optimization,” in Proc. 15th Int. Conf. Parallel Problem Solving from Nature, Coimbra, Portugal, 2018, pp. 249–261.
    [23]
    K. Qiao, J. Liang, K. Yu, C. Yue, H. Lin, D. Zhang, and B. Qu, “Evolutionary constrained multiobjective optimization: Scalable high-dimensional constraint benchmarks and algorithm,” IEEE Trans. Evol. Comput., 2023. doi: 10.1109/TEVC.2023.3281666
    [24]
    B. Cao, J. Zhao, Y. Gu, Y. Ling, and X. Ma, “Applying graph-based differential grouping for multiobjective large-scale optimization,” Swarm Evol. Comput., vol. 53, p. 100626, Mar. 2020. doi: 10.1016/j.swevo.2019.100626
    [25]
    L. M. Antonio, C. A. C. Coello, S. G. Brambila, J. F. González, and G. C. Tapia, “Operational decomposition for large scale multi-objective optimization problems,” in Proc. Genetic and Evolutionary Computation Conf. Companion, Prague, Czech Republic, 2019, pp. 225–226.
    [26]
    X. Zhang, Y. Tian, R. Cheng, and Y. Jin, “A decision variable clustering-based evolutionary algorithm for large-scale many-objective optimization,” IEEE Trans. Evol. Comput., vol. 22, no. 1, pp. 97–112, Feb. 2018. doi: 10.1109/TEVC.2016.2600642
    [27]
    S. Qin, C. Sun, Y. Jin, Y. Tan, and J. Fieldsend, “Large-scale evolutionary multiobjective optimization assisted by directed sampling,” IEEE Trans. Evol. Comput., vol. 25, no. 4, pp. 724–738, Aug. 2021. doi: 10.1109/TEVC.2021.3063606
    [28]
    Y. Tian, C. Lu, X. Zhang, F. Cheng, and Y. Jin, “A pattern mining-based evolutionary algorithm for large-scale sparse multiobjective optimization problems,” IEEE Trans. Cybern., vol. 52, no. 7, pp. 6784–6797, Jul. 2022. doi: 10.1109/TCYB.2020.3041325
    [29]
    Y. Tian, C. Lu, X. Zhang, K. C. Tan, and Y. Jin, “Solving large-scale multiobjective optimization problems with sparse optimal solutions via unsupervised neural networks,” IEEE Trans. Cybern., vol. 51, no. 6, pp. 3115–3128, Jun. 2021. doi: 10.1109/TCYB.2020.2979930
    [30]
    Y. Tian, X. Zheng, X. Zhang, and Y. Jin, “Efficient large-scale multiobjective optimization based on a competitive swarm optimizer,” IEEE Trans. Cybern., vol. 50, no. 8, pp. 3696–3708, Aug. 2020. doi: 10.1109/TCYB.2019.2906383
    [31]
    Y. Zhang, G.-G. Wang, K. Li, W.-C. Yeh, M. Jian, and J. Dong, “Enhancing MOEA/D with information feedback models for large-scale many-objective optimization,” Inf. Sci., vol. 522, pp. 1–16, Jun. 2020. doi: 10.1016/j.ins.2020.02.066
    [32]
    H. Chen, R. Cheng, J. Wen, H. Li, and J. Weng, “Solving large-scale many-objective optimization problems by covariance matrix adaptation evolution strategy with scalable small subpopulations,” Inf. Sci., vol. 509, pp. 457–469, Jan. 2020. doi: 10.1016/j.ins.2018.10.007
    [33]
    M. Preuss, B. Naujoks, and G. Rudolph, “Pareto set and EMOA behavior for simple multimodal multiobjective functions,” in Proc. 9th Int. Conf. Parallel Problem Solving from Nature, Reykjavik, Iceland, 2006, pp. 513–522.
    [34]
    G. Rudolph, B. Naujoks, and M. Preuss, “Capabilities of EMOA to detect and preserve equivalent Pareto subsets,” in Proc. 4th Int. Conf. Evolutionary Multi-Criterion Optimization, Matsushima, Japan, 2007, pp. 36–50.
    [35]
    K. Deb and S. Tiwari, “Omni-optimizer: A generic evolutionary algorithm for single and multi-objective optimization,” Eur. J. Oper. Res., vol. 185, no. 3, pp. 1062–1087, Mar. 2008. doi: 10.1016/j.ejor.2006.06.042
    [36]
    H. Ishibuchi, N. Akedo, and Y. Nojima, “A many-objective test problem for visually examining diversity maintenance behavior in a decision space,” in Proc. 13th Annu. Conf. Genetic and Evolutionary Computation, Dublin, Ireland, 2011, pp. 649–656.
    [37]
    C. Yue, B. Qu, and J. Liang, “A multiobjective particle swarm optimizer using ring topology for solving multimodal multiobjective problems,” IEEE Trans. Evol. Comput., vol. 22, no. 5, pp. 805–817, Oct. 2018. doi: 10.1109/TEVC.2017.2754271
    [38]
    C. Yue, B. Qu, K. Yu, J. Liang, and X. Li, “A novel scalable test problem suite for multimodal multiobjective optimization,” Swarm Evol. Comput., vol. 48, pp. 62–71, Aug. 2019. doi: 10.1016/j.swevo.2019.03.011
    [39]
    Y. Liu, G. G. Yen, and D. Gong, “A multimodal multiobjective evolutionary algorithm using two-archive and recombination strategies,” IEEE Trans. Evol. Comput., vol. 23, no. 4, pp. 660–674, Aug. 2019. doi: 10.1109/TEVC.2018.2879406
    [40]
    W. Li, X. Yao, T. Zhang, R. Wang, and L. Wang, “Hierarchy ranking method for multimodal multiobjective optimization with local Pareto fronts,” IEEE Trans. Evol. Comput., vol. 27, no. 1, pp. 98–110, Feb. 2023. doi: 10.1109/TEVC.2022.3155757
    [41]
    Y. Tian, R. Liu, X. Zhang, H. Ma, K. C. Tan, and Y. Jin, “A multipopulation evolutionary algorithm for solving large-scale multimodal multiobjective optimization problems,” IEEE Trans. Evol. Comput., vol. 25, no. 3, pp. 405–418, Jun. 2021. doi: 10.1109/TEVC.2020.3044711
    [42]
    Y. Liu, H. Ishibuchi, Y. Nojima, N. Masuyama, and K. Shang, “A double-niched evolutionary algorithm and its behavior on polygon-based problems,” in Proc. 15th Int. Conf. Parallel Problem Solving from Nature, Coimbra, Portugal, 2018, pp. 262–273.
    [43]
    X. Li, A. Engelbrecht, and M. G. Epitropakis, “Benchmark functions for CEC’2013 special session and competition on niching methods for multimodal function optimization,” Evolutionary Computation and Machine Learning Group, RMIT University, Melbourne, Australia, Tech. Rep., 2013.
    [44]
    B. Y. Qu, J. J. Liang, Z. Y. Wang, Q. Chen, and P. N. Suganthan, “Novel benchmark functions for continuous multimodal optimization with comparative results,” Swarm Evol. Comput., vol. 26, pp. 23–34, Feb. 2016. doi: 10.1016/j.swevo.2015.07.003
    [45]
    Q. Zhang, A. Zhou, S. Zhao, P. N. Suganthan, W. Liu, and S. Tiwari, “Multiobjective optimization test instances for the CEC 2009 special session and competition,” University of Essex, Colchester, UK, Nanyang Technological University, Singapore, Tech. Rep. CES–487, 2009.
    [46]
    K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable test problems for evolutionary multiobjective optimization,” in Evolutionary Multiobjective Optimization: Theoretical Advances and Applications, A. Abraham, L. Jain, and R. Goldberg, Eds. London, UK: Springer, 2005, pp. 105–145.
    [47]
    J. J. Liang, C. Yue, and B. Y. Qu, “Multimodal multi-objective optimization: A preliminary study,” in Proc. IEEE Congr. Evolutionary Computation, Vancouver, Canada, 2016, pp. 2454–2461.
    [48]
    B. Qu, C. Li, J. Liang, L. Yan, K. Yu, and Y. Zhu, “A self-organized speciation based multi-objective particle swarm optimizer for multimodal multi-objective problems,” Appl. Soft Comput., vol. 86, p. 105886, Jan. 2020. doi: 10.1016/j.asoc.2019.105886
    [49]
    Q. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposition,” IEEE Trans. Evol. Comput., vol. 11, no. 6, pp. 712–731, Dec. 2007. doi: 10.1109/TEVC.2007.892759
    [50]
    Q. Lin, W. Lin, Z. Zhu, M. Gong J. Li, and C. A. C. Coello, “Multimodal multiobjective evolutionary optimization with dual clustering in decision and objective spaces,” IEEE Trans. Evol. Comput., vol. 25, no. 1, pp. 130–144, Feb. 2021. doi: 10.1109/TEVC.2020.3008822
    [51]
    J. Liang, K. Qiao, C. Yue, K Yu, B. Qu, R. Xu, Z. Li, and Y. Hu, “A clustering-based differential evolution algorithm for solving multimodal multi-objective optimization problems,” Swarm Evol. Comput., vol. 60, p. 100788, Feb. 2021. doi: 10.1016/j.swevo.2020.100788
    [52]
    Z. Ding, L. Cao, L. Chen, D. Sun, X. Zhang, and Z. Tao, “Large-scale multimodal multiobjective evolutionary optimization based on hybrid hierarchical clustering,” Knowl.-Based Syst., vol. 266, p. 110398, Apr. 2023. doi: 10.1016/j.knosys.2023.110398
    [53]
    Y. Peng and H. Ishibuchi, “A decomposition-based large-scale multi-modal multi-objective optimization algorithm,” in Proc. IEEE Congr. Evolutionary Computation, Glasgow, UK, 2020, pp. 1–8.
    [54]
    J. Liang, H. Lin, C. Yue, K. Yu, Y. Guo, and K. Qiao, “Multiobjective differential evolution with speciation for constrained multimodal multiobjective optimization,” IEEE Trans. Evol. Comput., vol. 27, no. 4, pp. 1115–1129, Aug. 2023. doi: 10.1109/TEVC.2022.3194253
    [55]
    K. Deb, “Multi-objective genetic algorithms: Problem difficulties and construction of test problems,” Evol. Comput., vol. 7, no. 3, pp. 205–230, Sep. 1999. doi: 10.1162/evco.1999.7.3.205
    [56]
    Y. Zhou, Y. Xiang, and X. He, “Constrained multiobjective optimization: Test problem construction and performance evaluations,” IEEE Trans. Evol. Comput., vol. 25, no. 1, pp. 172–186, Feb. 2021. doi: 10.1109/TEVC.2020.3011829
    [57]
    J. J. Liang, B. Y. Qu, and P. N. Suganthan, “Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization,” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China, Computational Intelligence Laboratory, Nanyang Technological University, Singapore, Tech. Rep. 201311, 2013.
    [58]
    B. Y. Qu, J. J. Liang, P. N. Suganthan, and Q. Chen, “Problem definitions and evaluation criteria for the CEC 2015 competition on single objective multi-niche optimization,” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou, China, Computational Intelligence Laboratory, Nanyang Technological University, Singapore, Tech. Rep. 201411B, 2014.
    [59]
    J. J. Liang, T. P. Runarsson, E. Mezura-Montes, M. Clerc, P. N. Suganthan, C. A. C. Coello, and K. Deb, “Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization,” Nanyang Technological University, Singapore, Tech. Rep., 2006.
    [60]
    R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, “Test problems for large-scale multiobjective and many-objective optimization,” IEEE Trans. Cybern., vol. 47, no. 12, pp. 4108–4121, Dec. 2017. doi: 10.1109/TCYB.2016.2600577
    [61]
    R. Storn and K. Price, “Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim., vol. 11, no. 4, pp. 341–359, Dec. 1997. doi: 10.1023/A:1008202821328
    [62]
    H. Ma, H. Wei, Y. Tian, R. Cheng, and X. Zhang, “A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints,” Inf. Sci., vol. 560, pp. 68–91, Jun. 2021. doi: 10.1016/j.ins.2021.01.029
    [63]
    V. L. Huang, P. N Suganthan, A. K. Qin, and S. Baskar, “Multiobjective differential evolution with external archive and harmonic distance-based diversity measure,” Nanyang Technological University, Singapore, Tech. Rep., 2005.
    [64]
    A. Zhou, J. Sun, and Q. Zhang, “An estimation of distribution algorithm with cheap and expensive local search methods,” IEEE Trans. Evol. Comput., vol. 19, no. 6, pp. 807–822, Dec. 2015. doi: 10.1109/TEVC.2014.2387433
    [65]
    Z.-J. Wang, Y.-R. Zhou, and J. Zhang, “Adaptive estimation distribution distributed differential evolution for multimodal optimization problems,” IEEE Trans. Cybern., vol. 52, no. 7, pp. 6059–6070, Jul. 2022. doi: 10.1109/TCYB.2020.3038694
    [66]
    C. A. C. Coello, G. B. Lamont, and D. A. Van Veldhuizen, Evolutionary Algorithms for Solving Multi-Objective Problems. 2nd ed. New York, USA: Springer, 2007.
    [67]
    A. Zhou, Q. Zhang, and Y. Jin, “Approximating the set of Pareto-optimal solutions in both the decision and objective spaces by an estimation of distribution algorithm,” IEEE Trans. Evol. Comput., vol. 13, no. 5, pp. 1167–1189, Oct. 2009. doi: 10.1109/TEVC.2009.2021467
    [68]
    E. Zitzler and L. Thiele, “Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach,” IEEE Trans. Evol. Comput., vol. 3, no. 4, pp. 257–271, Nov. 1999. doi: 10.1109/4235.797969
    [69]
    W. Li, T. Zhang, R. Wang, S. Huang, and J. Liang, “Multimodal multi-objective optimization: Comparative study of the state-of-the-art,” Swarm Evol. Comput., vol. 77, p. 101253, Mar. 2023. doi: 10.1016/j.swevo.2023.101253
    [70]
    M. Mitchell, An Introduction to Genetic Algorithms. Cambridge, UK: MIT Press, 1998.
    [71]
    L. van der Maaten and G. Hinton, “Visualizing data using t-SNE,” J. Mach. Learn. Res., vol. 9, no. 86, pp. 2579–2605, Nov. 2008.
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    Highlights

    • A new framework of MMO test functions is proposed, incorporating various components with different functions to control convergence difficulty, multimodal characteristics, and the extensibility of decision variables
    • Fifteen test instances are generated, allowing for a more comprehensive evaluation of the performance of multimodal multiobjective optimization evolutionary algorithms (MMOEAs)
    • A new MMOEA is developed, called HDMMODE
    • A dual-population coevolution method is designed to significantly enhance search capability in the decision space
    • An improved environmental selection method is developed to preserve multiple Pareto optimal solutions

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