{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T12:59:13Z","timestamp":1782737953474,"version":"3.54.5"},"reference-count":38,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,29]],"date-time":"2024-12-29T00:00:00Z","timestamp":1735430400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11872190"],"award-info":[{"award-number":["11872190"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Surrogate-assisted evolutionary algorithms (SAEAs) are widely used in the field of high-dimensional expensive optimization. However, real-world problems are usually complex and characterized by a variety of features. Therefore, it is very challenging to choose the most appropriate surrogate. It has been shown that multiple surrogates can characterize the fitness landscape more accurately than a single surrogate. In this work, a multi-surrogate-assisted multi-tasking optimization algorithm (MSAMT) is proposed that solves high-dimensional problems by simultaneously optimizing multiple surrogates as related tasks using the generalized multi-factorial evolutionary algorithm. In the MSAMT, all exactly evaluated samples are initially grouped to form a collection of clusters. Subsequently, the search space can be divided into several areas based on the clusters, and surrogates are constructed in each region that are capable of completely describing the entire fitness landscape as a way to improve the exploration capability of the algorithm. Near the current optimal solution, a novel ensemble surrogate is adopted to achieve local search in speeding up the convergence process. In the framework of a multi-tasking optimization algorithm, several surrogates are optimized simultaneously as related tasks. As a result, several optimal solutions spread throughout disjoint regions can be found for real function evaluation. Fourteen 10- to 100-dimensional test functions and a spatial truss design problem were used to compare the proposed approach with several recently proposed SAEAs. The results show that the proposed MSAMT performs better than the comparison algorithms in most test functions and real engineering problems.<\/jats:p>","DOI":"10.3390\/a18010004","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T13:26:25Z","timestamp":1735651585000},"page":"4","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Multi-Surrogate Assisted Multi-Tasking Optimization Algorithm for High-Dimensional Expensive Problems"],"prefix":"10.3390","volume":"18","author":[{"given":"Hongyu","family":"Li","sequence":"first","affiliation":[{"name":"Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lei","family":"Chen","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7497-0646","authenticated-orcid":false,"given":"Jian","family":"Zhang","sequence":"additional","affiliation":[{"name":"Ocean Institute, Northwestern Polytechnical University, Taicang 215400, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muxi","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Tianjin University, Tianjin 300354, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8091","DOI":"10.1007\/s11042-020-10139-6","article-title":"A review on genetic algorithm: Past, present, and future","volume":"80","author":"Katoch","year":"2020","journal-title":"Multimedia Tools Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TEVC.2010.2059031","article-title":"Differential evolution: A survey of the state-of-the-art","volume":"15","author":"Das","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_3","unstructured":"Yang, G.Y. (2007, January 16\u201318). A modified particle swarm optimizer algorithm. Proceedings of the International Conference on Electronic Measurement and Instruments, Xi\u2019an, China."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1109\/TEVC.2003.819944","article-title":"Meta-Lamarckian learning in memetic algorithms","volume":"8","author":"Ong","year":"2004","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/j.robot.2008.09.009","article-title":"Fitness functions in evolutionary robotics: A survey and analysis","volume":"57","author":"Nelson","year":"2009","journal-title":"Robot. Auton. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2008","DOI":"10.1109\/TVT.2008.2004588","article-title":"Dynamics of network selection in heterogeneous wireless networks: An evolutionary game approach","volume":"58","author":"Niyato","year":"2009","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1109\/TCYB.2018.2881190","article-title":"Solving multiobjective constrained trajectory optimization problem by an extended evolutionary algorithm","volume":"50","author":"Chai","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"10809","DOI":"10.1109\/TIE.2019.2962482","article-title":"Multiobjective optimal parking maneuver planning of autonomous wheeled vehicles","volume":"67","author":"Chai","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1109\/TEVC.2018.2869001","article-title":"Data-driven evolutionary optimization: An overview and case studies","volume":"23","author":"Jin","year":"2019","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Shi, L., and Rasheed, K. (2010). A survey of fitness approximation methods applied in evolutionary algorithms. Computational Intelligence in Expensive Optimization Problems, Springer.","DOI":"10.1007\/978-3-642-10701-6_1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s00500-003-0329-4","article-title":"Faster convergence by means of fitness estimation","volume":"9","author":"Branke","year":"2003","journal-title":"Soft Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1109\/TEVC.2019.2890818","article-title":"A novel evolutionary sampling assisted optimization method for high-dimensional expensive problems","volume":"23","author":"Wang","year":"2019","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1109\/TCYB.2018.2794503","article-title":"Heterogeneous ensemble-based infill criterion for evolutionary multiobjective optimization of expensive problems","volume":"49","author":"Guo","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107747","DOI":"10.1016\/j.knosys.2021.107747","article-title":"A twofold infill criterion-driven heterogeneous ensemble surrogate-assisted evolutionary algorithm for computationally expensive problems","volume":"236","author":"Yu","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"106303","DOI":"10.1016\/j.asoc.2020.106303","article-title":"A fast surrogate-assisted particle swarm optimization algorithm for computationally expensive problems","volume":"92","author":"Li","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1109\/TCYB.2020.2967553","article-title":"A surrogate-assisted multiswarm optimization algorithm for high-dimensional computationally expensive problems","volume":"51","author":"Li","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Nourian, N., El-Badry, M., and Jamshidi, M. (2023). Design optimization of truss structures using a graph neural network-based surrogate model. Algorithms, 16.","DOI":"10.3390\/a16080380"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1109\/TEVC.2019.2919762","article-title":"Efficient generalized surrogate-assisted evolutionary algorithm for high-dimensional expensive problems","volume":"24","author":"Cai","year":"2020","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.ins.2020.11.056","article-title":"An efficient surrogate-assisted hybrid optimization algorithm for expensive optimization problems","volume":"561","author":"Pan","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1007\/s00500-014-1283-z","article-title":"A two-layer surrogate-assisted particle swarm optimization algorithm","volume":"19","author":"Sun","year":"2014","journal-title":"Soft Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2685","DOI":"10.1109\/TCYB.2022.3175533","article-title":"A surrogate-assisted differential evolution algorithm for high-dimensional expensive optimization problems","volume":"53","author":"Wang","year":"2023","journal-title":"IEEE Trans. Cybern."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.ins.2022.11.045","article-title":"Combining Lipschitz and RBF surrogate models for high-dimensional computationally expensive problems","volume":"619","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rumpfkeil, M.P., Bryson, D., and Beran, P. (2022). Multi-fidelity sparse polynomial chaos and kriging surrogate models applied to analytical benchmark problems. Algorithms, 15.","DOI":"10.3390\/a15030101"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"12448","DOI":"10.1007\/s10489-022-04080-4","article-title":"A surrogate-assisted bi-swarm evolutionary algorithm for expensive optimization","volume":"53","author":"Liu","year":"2022","journal-title":"Appl. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s40747-021-00277-1","article-title":"A bi-stage surrogate-assisted hybrid algorithm for expensive optimization problems","volume":"7","author":"Ren","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"15051","DOI":"10.1007\/s00500-021-06348-2","article-title":"An adaptive surrogate-assisted particle swarm optimization for expensive problems","volume":"25","author":"Li","year":"2021","journal-title":"Soft Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"119075","DOI":"10.1016\/j.eswa.2022.119075","article-title":"Decision space partition based surrogate-assisted evolutionary algorithm for expensive optimization","volume":"214","author":"Liu","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"111212","DOI":"10.1016\/j.asoc.2023.111212","article-title":"A hierarchical surrogate assisted optimization algorithm using teaching-learning-based optimization and differential evolution for high-dimensional expensive problems","volume":"152","author":"Zhang","year":"2024","journal-title":"Appl. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1016\/j.swevo.2018.08.015","article-title":"Ensemble strategies for population-based optimization algorithms\u2013A survey","volume":"44","author":"Wu","year":"2019","journal-title":"Swarm Evol. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"106262","DOI":"10.1016\/j.knosys.2020.106262","article-title":"Multi-surrogate multi-tasking optimization of expensive problems","volume":"205","author":"Liao","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"109775","DOI":"10.1016\/j.asoc.2022.109775","article-title":"A surrogate assisted evolutionary multitasking optimization algorithm","volume":"132","author":"Yang","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Akopov, A.S., and Hevencev, M.A. (2013, January 13\u201316). A multi-agent genetic algorithm for multi-objective optimization. Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Manchester, UK.","DOI":"10.1109\/SMC.2013.240"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.1109\/TSMCB.2003.821456","article-title":"A multiagent genetic algorithm for global numerical optimization","volume":"34","author":"Zhong","year":"2004","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1109\/TEVC.2015.2458037","article-title":"Multifactorial evolution: Toward evolutionary multitasking","volume":"20","author":"Gupta","year":"2016","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1109\/TEVC.2017.2785351","article-title":"Generalized multitasking for evolutionary optimization of expensive problems","volume":"23","author":"Ding","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Box, G.E.P., and Wilson, K.B. (1992). On the experimental attainment of optimum conditions. Breakthroughs in Statistics: Methodology and Distribution, Springer.","DOI":"10.1007\/978-1-4612-4380-9_23"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1029\/JB076i008p01905","article-title":"Multiquadric equations of topography and other irregular surfaces","volume":"76","author":"Hardy","year":"1971","journal-title":"J. Geophys. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/0305215X.2020.1739280","article-title":"A unified ensemble of surrogates with global and local measures for global metamodelling","volume":"53","author":"Zhang","year":"2021","journal-title":"Eng. Optim."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:56:02Z","timestamp":1760115362000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,29]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["a18010004"],"URL":"https:\/\/doi.org\/10.3390\/a18010004","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,29]]}}}