An Efficient Evolutionary Algorithm for Few-for-Many Optimization

Ke Shang; Hisao Ishibuchi; Zexuan Zhu; Qingfu Zhang

doi:10.1109/JAS.2026.125852

Volume 13 Issue 6

Jun. 2026

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2026 > 13(6): 1362-1377

K. Shang, H. Ishibuchi, Z. Zhu, and Q. Zhang, “An efficient evolutionary algorithm for few-for-many optimization,” IEEE/CAA J. Autom. Sinica, vol. 13, no. 6, pp. 1362–1377, Jun. 2026. doi: 10.1109/JAS.2026.125852

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K. Shang, H. Ishibuchi, Z. Zhu, and Q. Zhang, “An efficient evolutionary algorithm for few-for-many optimization,” IEEE/CAA J. Autom. Sinica, vol. 13, no. 6, pp. 1362–1377, Jun. 2026. doi: 10.1109/JAS.2026.125852

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An Efficient Evolutionary Algorithm for Few-for-Many Optimization

doi: 10.1109/JAS.2026.125852

Funds: This work was supported by the National Natural Science Foundation of China (62472292, 62471310, 62376115), Guangdong Basic and Applied Basic Research Foundation (2025A1515011638), and the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF Project No. CityU11215622)

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Author Bio:
Ke Shang (Senior Member, IEEE) received the B.S. and Ph.D. degrees from Xi’an Jiaotong University in 2009 and 2016, respectively. He is currently an Assistant Professor with the School of Artificial Intelligence, Shenzhen University. His research interests include multiobjective optimization and learning. Dr. Shang received the GECCO 2018, 2021, and 2024 Best Paper Awards

Hisao Ishibuchi (Fellow, IEEE) received the B.S. and M.S. degrees from Kyoto University, Japan, and the Ph.D. degree from Osaka Prefecture University, Japan. Since 2017, he has been a Chair Professor at Southern University of Science and Technology. His research interests include fuzzy rule-based classifiers, evolutionary multiobjective optimization, many-objective optimization, and memetic algorithms. He was the IEEE Computational Intelligence Society (CIS) Vice-President for Technical Activities in 2010−2013, and the Editor-in-Chief of IEEE Computational Intelligence Magazine in 2014−2019. Currently, he is an IEEE CIS Administrative Committee Member (2021−2026), and an Associate Editor of several journals such as IEEE Transactions on Evolutionary Computation, Evolutionary Computation Journal, and ACM Computing Surveys. He is also General Chair of EMO 2027, and Program Chair of FUZZ-IEEE 2027. He received a Fuzzy Systems Pioneer Award from IEEE CIS in 2019, an Outstanding Paper Award from IEEE Transactions on Evolutionary Computation in 2020, an Enrique Ruspini Award for Meritorious Service from IEEE CIS in 2023, and Best Paper Awards from FUZZ-IEEE 2009, 2011, EMO 2019, 2025 and GECCO 2004, 2017, 2018, 2020, 2021, 2024. He also received a JSPS prize in 2007

Zexuan Zhu (Senior Member, IEEE) received the B.S. degree in computer science and technology from Fudan University in 2003, and the Ph.D. degree in computer engineering from Nanyang Technological University, Singapore, in 2008. He is currently a Professor with the School of Artificial Intelligence and the National Engineering Laboratory for Big Data System Computing Technology, Shenzhen University. His research interests include computational intelligence, machine learning, and bioinformatics. Dr. Zhu is an Associate Editor for IEEE Transactions on Evolutionary Computation

Qingfu Zhang (Fellow, IEEE) received the B.Sc. degree in mathematics from Shanxi University in 1984, and the M.Sc. degree in applied mathematics and the Ph.D. degree in information engineering from Xidian University in 1991 and 1994, respectively. He is currently a Chair Professor at the Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China. His research interests include evolutionary computation, optimization, machine learning, and their applications. He has been a Highly Cited Researcher in Computer Science since 2016
Corresponding author: Ke Shang, e-mail: shangk@szu.edu.cn
Received Date: 2025-05-27
Revised Date: 2025-11-15
Accepted Date: 2025-12-24

Available Online: 2026-06-22

Abstract

Abstract

Few-for-many (F4M) optimization, recently introduced as a novel paradigm in multi-objective optimization, aims to find a small set of solutions that effectively handle a large number of conflicting objectives. Unlike traditional many-objective optimization methods, which typically attempt comprehensive coverage of the Pareto front, F4M optimization emphasizes finding a small representative solution set to efficiently address high-dimensional objective spaces. Motivated by the computational complexity and practical relevance of F4M optimization, this paper proposes a new evolutionary algorithm explicitly tailored for efficiently solving F4M optimization problems. Inspired by S-metric selection evolutionary multi-objective optimization algorithm (SMS-EMOA), our proposed approach employs a $ (\mu+1) $-evolution strategy guided by the objective of F4M optimization. Furthermore, to facilitate rigorous performance assessment, we propose a novel benchmark test suite specifically designed for F4M optimization by leveraging the similarity between the R2 indicator and F4M formulations. Our test suite is highly flexible, allowing any existing multi-objective optimization problem to be transformed into a corresponding F4M instance via scalarization using the weighted Tchebycheff function. Comprehensive experimental evaluations on benchmarks demonstrate the superior performance of our algorithm compared to existing state-of-the-art algorithms, especially on instances involving a large number of objectives. The source code of the proposed algorithm will be released publicly. Source code is available at https://github.com/MOL-SZU/SoM-EMOA.
- Evolutionary algorithm,
- few-for-many optimization,
- many-objective optimization (MOO),
- multi-objective optimization

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¹ https://github.com/nataliemaus/mocobo

References(54)

References

[1]	K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ,” IEEE Trans. Evol. Computat., vol. 6, no. 2, pp. 182–197, Apr. 2002. doi: 10.1109/4235.996017
[2]	Q. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposition,” IEEE Trans. Evol. Computat., vol. 11, no. 6, pp. 712–731, Dec. 2007. doi: 10.1109/TEVC.2007.892759
[3]	K. Deb and H. Jain, “An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints,” IEEE Trans. Evol. Computat., vol. 18, no. 4, pp. 577–601, Aug. 2014. doi: 10.1109/TEVC.2013.2281535
[4]	R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, “A reference vector guided evolutionary algorithm for many-objective optimization,” IEEE Trans. Evol. Comput., vol. 20, no. 5, pp. 773–791, Oct. 2016. doi: 10.1109/TEVC.2016.2519378
[5]	Y. Tian, X. Li, H. Ma, X. Zhang, K. C. Tan, and Y. Jin, “Deep reinforcement learning based adaptive operator selection for evolutionary multi-objective optimization,” IEEE Trans. Emerg. Top. Comput. Intell., vol. 7, no. 4, pp. 1051–1064, Aug. 2023. doi: 10.1109/TETCI.2022.3146882
[6]	T. Huang, S. Wang, and K. Li, “Direct preference-based evolutionary multi-objective optimization with dueling bandits,” in Proc. 38th Int. Conf. Neural Information Processing Systems, Vancouver, Canada, 2024.
[7]	Y. Hua, Q. Liu, K. Hao, and Y. Jin, “A survey of evolutionary algorithms for multi-objective optimization problems with irregular Pareto fronts,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 303–318, Feb. 2021. doi: 10.1109/JAS.2021.1003817
[8]	S. Yang, M. Li, X. Liu, and J. Zheng, “A grid-based evolutionary algorithm for many-objective optimization,” IEEE Trans. Evol. Computat., vol. 17, no. 5, pp. 721–736, Oct. 2013. doi: 10.1109/TEVC.2012.2227145
[9]	H. Wang, Y. Jin, and X. Yao, “Diversity assessment in many-objective optimization,” IEEE Trans. Cybern., vol. 47, no. 6, pp. 1510–1522, 2017. doi: 10.1109/TCYB.2016.2550502
[10]	H. Ishibuchi, N. Tsukamoto, and Y. Nojima, “Evolutionary many-objective optimization: A short review,” in Proc. IEEE Congr. Evolutionary Computation, Hong Kong, China, 2008, pp. 2419−2426.
[11]	A. López Jaimes and C. A. Coello Coello, “Many-objective problems: Challenges and methods,” in Springer Handbook of Computational Intelligence, J. Kacprzyk and W. Pedrycz, Eds. Berlin, Germany: Springer, pp. 1033−1046, 2015.
[12]	H. Sato and H. Ishibuchi, “Evolutionary many-objective optimization: Difficulties, approaches, and discussions,” IEEJ Trans. Electr. Electr. Eng., vol. 18, no. 7, pp. 1048–1058, Jul. 2023. doi: 10.1002/tee.23796
[13]	Y. Liu, C. Lu, X. Lin, and Q. Zhang, “Many-objective cover problem: Discovering few solutions to cover many objectives,” in Proc. 18th Int. Conf. Parallel Problem Solving From Nature, Hagenberg, Austria, 2024, pp. 68−82.
[14]	X. Lin, Y. Liu, X. Zhang, F. Liu, Z. Wang, and Q. Zhang, “Few for many: Tchebycheff set scalarization for many-objective optimization,” in Proc. 13th Int. Conf. Learning Representations, Singapore, Singapore, 2024.
[15]	H. Ishibuchi, N. Akedo, and Y. Nojima, “Behavior of multiobjective evolutionary algorithms on many-objective knapsack problems,” IEEE Trans. Evol. Computat., vol. 19, no. 2, pp. 264–283, Apr. 2015. doi: 10.1109/TEVC.2014.2315442
[16]	Y. Yuan, Y.-S. Ong, A. Gupta, and H. Xu, “Objective reduction in many-objective optimization: Evolutionary multiobjective approaches and comprehensive analysis,” IEEE Trans. Evol. Computat., vol. 22, no. 2, pp. 189–210, Apr. 2018. doi: 10.1109/TEVC.2017.2672668
[17]	Z. Li, T. Li, V. Smith, J. Bilmes, and T. Zhou, “Many-objective multi-solution transport,” Proc. 13th Int. Conf. Learning Representations, Singapore, Singapore, 2025.
[18]	J. S. Angelo, I. A. Guedes, H. J. C. Barbosa, and L. E. Dardenne, “Multi-and many-objective optimization: Present and future in de novo drug design,” Front. Chem., vol. 11, Art. no. 1288626, Dec. 2023.
[19]	S. C. Matz, M. Kosinski, G. Nave, and D. J. Stillwell, “Psychological targeting as an effective approach to digital mass persuasion,” in Proc. Natl. Acad. Sci. USA, vol. 114, no. 48, pp. 12714–12719, Nov. 2017. doi: 10.1073/pnas.1710966114
[20]	D. Eckles, R. F. Kizilcec, and E. Bakshy, “Estimating peer effects in networks with peer encouragement designs,” in Proc. Natl. Acad. Sci. USA, vol. 113, no. 27, pp. 7316–7322, Jul. 2016. doi: 10.1073/pnas.1511201113
[21]	T. Standley, A. R. Zamir, D. Chen, L. J. Guibas, J. Malik, and S. Savarese, “Which tasks should be learned together in multi-task learning?” in Proc. 37th Int. Conf. Machine Learning, 2020, pp. 9120−9132.
[22]	S. Wu, T. Li, Z. Charles, Y. Xiao, Z. Liu, Z. Xu, and V. Smith, “Motley: Benchmarking heterogeneity and personalization in federated learning,” arXiv preprint arXiv: 2206.09262, 2022.
[23]	N. Maus, K. Kim, Y. Zeng, H. T. Jones, F. Wan, M. D. T. Torres, C. De La Fuente-Nunez, and J. R. Gardner, “Covering multiple objectives with a small set of solutions using Bayesian optimization,” arXiv preprint arXiv: 2501.19342, 2025.
[24]	K. Miettinen, Nonlinear Multiobjective Optimization. Boston, America: Kluwer Academic Publishers, 1999.
[25]	K. Deb, K. Sindhya, and J. Hakanen, “Multi-objective optimization,” in Decision Sciences, R. N. Sengupta, A. Gupta, and J. Dutta, Eds. Boca Raton, America: CRC Press, 2016, pp. 161−200.
[26]	K. S. Bhattacharjee, H. K. Singh, M. Ryan, and T. Ray, “Bridging the gap: Many-objective optimization and informed decision-making,” IEEE Trans. Evol. Computat., vol. 21, no. 5, pp. 813–820, Oct. 2017. doi: 10.1109/TEVC.2017.2687320
[27]	B. Dy, N. Ibrahim, A. Poorthuis, and S. Joyce, “Improving visualization design for effective multi-objective decision making,” IEEE Trans. Visual. Comput. Graph., vol. 28, no. 10, pp. 3405–3416, Oct. 2022. doi: 10.1109/TVCG.2021.3065126
[28]	C. Contardo and A. Hertz, “An exact algorithm for a class of geometric set-cover problems,” Discrete Appl. Math., vol. 300, pp. 25–35, Sep. 2021. doi: 10.1016/j.dam.2021.05.005
[29]	Y. J. Zhang, X. D. Mu, X. W. Liu, X. Y. Wang, X. Zhang, K. Li, T. Y. Wu, D. Zhao, and C. Dong, “Applying the quantum approximate optimization algorithm to the minimum vertex cover problem,” Appl. Soft Comput., vol. 118, Art. no. 108554, Mar. 2022. doi: 10.1016/j.asoc.2022.108554
[30]	L. Ding, Z. Chen, X. Wang, and W. Yin, “Efficient algorithms for sum-of-minimum optimization,” in Proc. 41st Int. Conf. Machine Learning, Vienna, Austria, 2024, pp. 10927−10959.
[31]	X. Lin, H.-L. Zhen, Z. Li, Q.-F. Zhang, and S. Kwong, “Pareto multi-task learning,” in Proc. 33rd Int. Conf. Neural Information Processing Systems, Vancouver, Canada, 2019.
[32]	G. Qin and J. Eisner, “Learning how to ask: Querying LMs with mixtures of soft prompts,” in Proc. Conf. North American Chapter Association Computational Linguistics: Human Language Technologies, 2021, pp. 5203−5212.
[33]	I. Bajaj, A. Arora, and M. M. F. Hasan, “Black-box optimization: Methods and applications,” in Black Box Optimization, Machine Learning, and No-free Lunch Theorems, P. M. Pardalos, V. Rasskazova, and M. N. Vrahatis, Eds. Cham, Germany: Springer, 2021, pp. 35−65.
[34]	B. H. Korte, J. Vygen, B. Korte, and J. Vygen, Combinatorial Optimization. Springer, 2011.
[35]	N. Beume, B. Naujoks, and M. Emmerich, “SMS-EMOA: Multiobjective selection based on dominated hypervolume,” Eur. J. Oper. Res., vol. 181, no. 3, pp. 1653–1669, Sep. 2007. doi: 10.1016/j.ejor.2006.08.008
[36]	J. Knowles, “ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems,” IEEE Trans. Evol. Computat., vol. 10, no. 1, pp. 50–66, Feb. 2006. doi: 10.1109/TEVC.2005.851274
[37]	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.
[38]	S. Huband, P. Hingston, L. Barone, and L. While, “A review of multiobjective test problems and a scalable test problem toolkit,” IEEE Trans. Evol. Computat., vol. 10, no. 5, pp. 477–506, Oct. 2006. doi: 10.1109/TEVC.2005.861417
[39]	H. Ishibuchi, Y. Setoguchi, H. Masuda, and Y. Nojima, “Performance of decomposition-based many-objective algorithms strongly depends on Pareto front shapes,” IEEE Trans. Evol. Computat., vol. 21, no. 2, pp. 169–190, Apr. 2017. doi: 10.1109/TEVC.2016.2587749
[40]	A. Jaszkiewicz and P. Zielniewicz, “Exact calculation and properties of the R2 multiobjective quality indicator,” IEEE Trans. Evol. Computati., vol. 29, no. 4, pp. 1227–1238, Aug. 2025. doi: 10.1109/TEVC.2024.3440571
[41]	L. Schäpermeier and P. Kerschke, “Reinvestigating the R2 indicator: Achieving Pareto compliance by integration,” in Proc. 18th Int. Conf. Parallel Problem Solving From Nature, Hagenberg, Austria, 2024, pp. 202−216.
[42]	I. Das and J. E. Dennis, “Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems,” SIAM J. Optim., vol. 8, no. 3, pp. 631–657, 1998. doi: 10.1137/S1052623496307510
[43]	B. S. Saini, H. K. Singh, B. Shavazipour, and K. Miettinen, “An efficient iterative approach for uniformly representing Pareto fronts,” in Proc. 13th Int. Conf. Evolutionary Multi-Criterion Optimization, Canberra, Australia, 2025, pp. 241−256.
[44]	Y. Tian, R. Cheng, X. Zhang, and Y. Jin, “PlatEMO: A MATLAB platform for evolutionary multi-objective optimization [educational forum],” IEEE Comput. Intell. Mag., vol. 12, no. 4, pp. 73–87, Nov. 2017. doi: 10.1109/MCI.2017.2742868
[45]	C. He, Y. Tian, H. Wang, and Y. Jin, “A repository of real-world datasets for data-driven evolutionary multiobjective optimization,” Complex Intell. Syst., vol. 6, no. 1, pp. 189–197, Apr. 2020. doi: 10.1007/s40747-019-00126-2
[46]	J. Yuan, H.-L. Liu, F. Gu, Q. Zhang, and Z. He, “Investigating the properties of indicators and an evolutionary many-objective algorithm using promising regions,” IEEE Trans. Evol. Computat., vol. 25, no. 1, pp. 75–86, Feb. 2021. doi: 10.1109/TEVC.2020.2999100
[47]	Y. Hua, Q. Liu, and K. Hao, “Adaptive normal vector guided evolutionary multi- and many-objective optimization,” Complex Intell. Syst., vol. 10, no. 3, pp. 3709–3726, Jun. 2024. doi: 10.1007/s40747-024-01353-y
[48]	L. R. De Farias and A. F. R. Araújo, “A decomposition-based many-objective evolutionary algorithm updating weights when required,” Swarm Evol. Comput., vol. 68, Art. no. 100980, Feb. 2022. doi: 10.1016/j.swevo.2021.100980
[49]	Z. Liu, F. Han, Q. Ling, H. Han, and J. Jiang, “A many-objective optimization evolutionary algorithm based on hyper-dominance degree,” Swarm Evol. Comput., vol. 83, Art. no. 101411, Dec. 2023. doi: 10.1016/j.swevo.2023.101411
[50]	L. Pan, W. Xu, L. Li, C. He, and R. Cheng, “Adaptive simulated binary crossover for rotated multi-objective optimization,” Swarm Evol. Comput., vol. 60, Art. no. 100759, Feb. 2021. doi: 10.1016/j.swevo.2020.100759
[51]	J. L. Carles-Bou and S. F. Galán, “Self-adaptive polynomial mutation in NSGA-II,” Soft Comput., vol. 27, no. 23, pp. 17711–17727, Dec. 2023. doi: 10.1007/s00500-023-09049-0
[52]	K. Zhang, L. Xu, X. Yi, Z. Ding, K. H. Johansson, T. Chai, and T. Yang, “Predefined-time distributed multiobjective optimization for network resource allocation,” Sci. China Inform. Sci., vol. 66, no. 7, Art. no. 170204, Jul. 2023. doi: 10.1007/s11432-022-3791-8
[53]	K. Zhang, X. Yi, G. Wen, M. Cao, K. H. Johansson, T. Chai, and T. Yang, “Distributed event-triggered bandit convex optimization with time-varying constraints,” IEEE Trans. Control Network Syst., vol. 12, no. 3, pp. 2242–2253, Sep. 2025. doi: 10.1109/TCNS.2025.3558791
[54]	K. Zhang, L. Xu, X. Yi, G. Wen, L. Xie, T. Chai, and T. Yang, “One-point sampling for distributed bandit convex optimization with time-varying constraints,” IEEE Trans. Control Network Syst., vol. 12, no. 4, pp. 3062–3072, Dec. 2025. doi: 10.1109/TCNS.2025.3608004