{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T06:03:28Z","timestamp":1770357808761,"version":"3.49.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Solving nonlinear equation systems (NESs) requires locating different roots in one run. To effectively deal with NESs, a multi-population cooperative teaching\u2013learning-based optimization, named MCTLBO, is presented. The innovations of MCTLBO are as follows: (i) two niching technique (crowding and improved speciation) are integrated into the algorithm to enhance population diversity; (ii) an adaptive selection scheme is proposed to select the learning rules in the teaching phase; (iii) the new learning rules based on experience learning are developed to promote the search efficiency in the teaching and learning phases. MCTLBO was tested on 30 classical problems and the experimental results show that MCTLBO has better root finding performance than other algorithms. In addition, MCTLBO achieves competitive results in eighteen new test sets.<\/jats:p>","DOI":"10.1007\/s40747-023-01074-8","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T01:02:21Z","timestamp":1684976541000},"page":"6593-6609","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Multi-population cooperative teaching\u2013learning-based optimization for nonlinear equation systems"],"prefix":"10.1007","volume":"9","author":[{"given":"Liao","family":"Zuowen","sequence":"first","affiliation":[]},{"given":"Li","family":"Shuijia","sequence":"additional","affiliation":[]},{"given":"Gong","family":"Wenyin","sequence":"additional","affiliation":[]},{"given":"Gu","family":"Qiong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"issue":"11","key":"1074_CR1","doi-asserted-by":"crossref","first-page":"2210","DOI":"10.1109\/TSMC.2018.2836968","volume":"49","author":"L Xiao","year":"2019","unstructured":"Xiao L, Zhang Z, Li S (2019) Solving time-varying system of nonlinear equations by finite-time recurrent neural networks with application to motion tracking of robot manipulators. IEEE Trans Syst Man Cybern Syst 49(11):2210\u20132220","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"issue":"8","key":"1074_CR2","doi-asserted-by":"crossref","first-page":"3781","DOI":"10.1109\/TAC.2020.3028566","volume":"66","author":"Z Bartosiewicz","year":"2021","unstructured":"Bartosiewicz Z, Kaldm\u00e4e A, Kawano Y, Kotta U, Pawluszewicz E, Simha A, Wyrwas M (2021) Accessibility and system reduction of nonlinear time-delay control systems. IEEE Trans Autom Control 66(8):3781\u20133788","journal-title":"IEEE Trans Autom Control"},{"issue":"5","key":"1074_CR3","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1109\/TAC.2017.2736961","volume":"63","author":"AI Doban","year":"2018","unstructured":"Doban AI, Lazar M (2018) Computation of lyapunov functions for nonlinear differential equations via a massera-type construction. IEEE Trans Autom Control 63(5):1259\u20131272","journal-title":"IEEE Trans Autom Control"},{"issue":"7","key":"1074_CR4","first-page":"1230","volume":"28","author":"R Jafari","year":"2020","unstructured":"Jafari R, Razvarz S, Gegov A (2020) Neural network approach to solving fuzzy nonlinear equations using z-numbers. IEEE Trans Fuzzy Syst 28(7):1230\u20131241","journal-title":"IEEE Trans Fuzzy Syst"},{"issue":"1","key":"1074_CR5","doi-asserted-by":"crossref","first-page":"15","DOI":"10.23919\/CSMS.2021.0002","volume":"1","author":"W Gong","year":"2021","unstructured":"Gong W, Liao Z, Mi X, Wang L, Guo Y (2021) Nonlinear equations solving with intelligent optimization algorithms: a survey. Compl Syst Model Simul 1(1):15\u201332","journal-title":"Compl Syst Model Simul"},{"issue":"2","key":"1074_CR6","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/j.cam.2005.07.042","volume":"199","author":"H Schwandt","year":"2007","unstructured":"Schwandt H (2007) Parallel interval Newton-like Schwarz methods for almost linear parabolic problems. J Comput Appl Math 199(2):437\u2013444","journal-title":"J Comput Appl Math"},{"issue":"7\u20138","key":"1074_CR7","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1016\/S0098-1354(01)00675-5","volume":"25","author":"KS Gritton","year":"2001","unstructured":"Gritton KS, Seader J, Lin W-J (2001) Global homotopy continuation procedures for seeking all roots of a nonlinear equation. Comput Chem Eng 25(7\u20138):1003\u20131019","journal-title":"Comput Chem Eng"},{"issue":"1","key":"1074_CR8","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/4235.585888","volume":"1","author":"T Back","year":"1997","unstructured":"Back T, Hammel U, Schwefel H-P (1997) Evolutionary computation: comments on the history and current state. IEEE Trans Evol Comput 1(1):3\u201317","journal-title":"IEEE Trans Evol Comput"},{"key":"1074_CR9","volume":"141","author":"S Li","year":"2021","unstructured":"Li S, Gong W, Gu Q (2021) A comprehensive survey on meta-heuristic algorithms for parameter extraction of photovoltaic models. Renew Sustain Energy Rev 141:110828","journal-title":"Renew Sustain Energy Rev"},{"key":"1074_CR10","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.enconman.2019.02.048","volume":"186","author":"S Li","year":"2019","unstructured":"Li S, Gong W, Yan X, Hu C, Bai D, Wang L, Gao L (2019) Parameter extraction of photovoltaic models using an improved teaching-learning-based optimization. Energy Convers Manag 186:293\u2013305","journal-title":"Energy Convers Manag"},{"issue":"6","key":"1074_CR11","doi-asserted-by":"crossref","first-page":"3171","DOI":"10.1109\/TCYB.2019.2955599","volume":"51","author":"L Feng","year":"2021","unstructured":"Feng L, Zhou L, Gupta A, Zhong J, Zhu Z, Tan K-C, Qin K (2021) Solving generalized vehicle routing problem with occasional drivers via evolutionary multitasking. IEEE Trans Cybern 51(6):3171\u20133184","journal-title":"IEEE Trans Cybern"},{"issue":"9","key":"1074_CR12","doi-asserted-by":"crossref","first-page":"4553","DOI":"10.1109\/TCYB.2019.2960302","volume":"51","author":"Z Liang","year":"2021","unstructured":"Liang Z, Luo T, Hu K, Ma X, Zhu Z (2021) An indicator-based many-objective evolutionary algorithm with boundary protection. IEEE Trans Cybern 51(9):4553\u20134566","journal-title":"IEEE Trans Cybern"},{"issue":"5","key":"1074_CR13","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1109\/TEVC.2011.2161873","volume":"16","author":"B Qu","year":"2012","unstructured":"Qu B, Suganthan P, Liang J (2012) Differential evolution with neighborhood mutation for multimodal optimization. IEEE Trans Evol Comput 16(5):601\u2013614","journal-title":"IEEE Trans Evol Comput"},{"key":"1074_CR14","volume":"198","author":"S Li","year":"2020","unstructured":"Li S, Gong W, Wang L, Yan X, Hu C (2020) Optimal power flow by means of improved adaptive differential evolution. Energy 198:117314","journal-title":"Energy"},{"key":"1074_CR15","volume":"114","author":"S Li","year":"2022","unstructured":"Li S, Gong W, Wang L, Gu Q (2022) Multi-objective optimal power flow with stochastic wind and solar power. Appl Soft Comput 114:108045","journal-title":"Appl Soft Comput"},{"issue":"4","key":"1074_CR16","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1109\/TSMC.2018.2828018","volume":"50","author":"W Gong","year":"2020","unstructured":"Gong W, Wang Y, Cai Z, Wang L (2020) Finding multiple roots of nonlinear equation systems via a repulsion-based adaptive differential evolution. IEEE Trans Syst Man Cybern Syst 50(4):1499\u20131513","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"issue":"4","key":"1074_CR17","doi-asserted-by":"crossref","first-page":"1590","DOI":"10.1109\/TSMC.2018.2852798","volume":"50","author":"Z Liao","year":"2020","unstructured":"Liao Z, Gong W, Yan X, Wang L, Hu C (2020) Solving nonlinear equations system with dynamic repulsion-based evolutionary algorithms. IEEE Trans Syst Man Cybern Syst 50(4):1590\u20131601","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"1074_CR18","volume":"182","author":"W He","year":"2019","unstructured":"He W, Gong W, Wang L, Yan X, Hu C (2019) Fuzzy neighborhood-based differential evolution with orientation for nonlinear equation systems. Knowl-Based Syst 182:104796","journal-title":"Knowl-Based Syst"},{"key":"1074_CR19","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2020.113261","volume":"149","author":"Z Liao","year":"2020","unstructured":"Liao Z, Gong W, Wang L (2020) Memetic niching-based evolutionary algorithms for solving nonlinear equation system. Expert Syst Appl 149:113261","journal-title":"Expert Syst Appl"},{"key":"1074_CR20","doi-asserted-by":"crossref","DOI":"10.1016\/j.knosys.2019.105312","volume":"191","author":"Z Liao","year":"2020","unstructured":"Liao Z, Gong W, Wang L, Yan X, Hu C (2020) A decomposition-based differential evolution with reinitialization for nonlinear equations systems. Knowl-Based Syst 191:105312","journal-title":"Knowl-Based Syst"},{"key":"1074_CR21","volume":"98","author":"J Wu","year":"2021","unstructured":"Wu J, Gong W, Wang L (2021) A clustering-based differential evolution with different crowding factors for nonlinear equations system. Appl Soft Comput 98:106733","journal-title":"Appl Soft Comput"},{"issue":"12","key":"1074_CR22","doi-asserted-by":"crossref","first-page":"7469","DOI":"10.1109\/TSMC.2022.3157816","volume":"52","author":"K Wang","year":"2022","unstructured":"Wang K, Gong W, Liao Z, Wang L (2022) Hybrid niching-based differential evolution with two archives for nonlinear equation system. IEEE Trans Syst Man Cybern Syst 52(12):7469\u20137481","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"1074_CR23","doi-asserted-by":"crossref","DOI":"10.1016\/j.asoc.2022.108818","volume":"122","author":"Z Liao","year":"2022","unstructured":"Liao Z, Zhu F, Gong W, Li S, Mi X (2022) AGSDE: archive guided speciation-based differential evolution for nonlinear equations. Appl Soft Comput 122:108818","journal-title":"Appl Soft Comput"},{"issue":"3","key":"1074_CR24","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1109\/TEVC.2014.2336865","volume":"19","author":"W Song","year":"2015","unstructured":"Song W, Wang Y, Li H-X, Cai Z (2015) Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization. IEEE Trans Evol Comput 19(3):414\u2013431","journal-title":"IEEE Trans Evol Comput"},{"issue":"5","key":"1074_CR25","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1109\/TEVC.2017.2670779","volume":"21","author":"W Gong","year":"2017","unstructured":"Gong W, Wang Y, Cai Z, Yang S (2017) A weighted biobjective transformation technique for locating multiple optimal solutions of nonlinear equation systems. IEEE Trans Evol Comput 21(5):697\u2013713","journal-title":"IEEE Trans Evol Comput"},{"key":"1074_CR26","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/j.ins.2020.06.042","volume":"541","author":"W Gao","year":"2020","unstructured":"Gao W, Luo Y, Xu J, Zhu S (2020) Evolutionary algorithm with multiobjective optimization technique for solving nonlinear equation systems. Inf Sci 541:345\u2013361","journal-title":"Inf Sci"},{"issue":"9","key":"1074_CR27","doi-asserted-by":"crossref","first-page":"5652","DOI":"10.1109\/TSMC.2019.2957324","volume":"51","author":"W Gao","year":"2021","unstructured":"Gao W, Li G, Zhang Q, Luo Y, Wang Z (2021) Solving nonlinear equation systems by a two-phase evolutionary algorithm. IEEE Trans Syst Man Cybern Syst 51(9):5652\u20135663","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"key":"1074_CR28","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ins.2021.06.070","volume":"576","author":"J-Y Ji","year":"2021","unstructured":"Ji J-Y, Wong ML (2021) An improved dynamic multi-objective optimization approach for nonlinear equation systems. Inf Sci 576:204\u2013227","journal-title":"Inf Sci"},{"issue":"3","key":"1074_CR29","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.cad.2010.12.015","volume":"43","author":"R Rao","year":"2023","unstructured":"Rao R, Savsani V, Vakharia D (2023) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303\u2013315","journal-title":"Comput Aided Des"},{"key":"1074_CR30","doi-asserted-by":"crossref","first-page":"4114","DOI":"10.1016\/j.egyr.2021.06.097","volume":"7","author":"X Mi","year":"2021","unstructured":"Mi X, Liao Z, Li Sh, Gu Q (2021) Adaptive teaching-learning-based optimization with experience learning to identify photovoltaic cell parameters. Energy Rep 7:4114\u20134125","journal-title":"Energy Rep"},{"key":"1074_CR31","doi-asserted-by":"crossref","DOI":"10.1016\/j.knosys.2019.07.007","volume":"187","author":"Y Zhang","year":"2020","unstructured":"Zhang Y, Jin Zh, Chen Y (2020) Hybrid teaching-learning-based optimization and neural network algorithm for engineering design optimization problems. Knowl Based Syst 187:104836","journal-title":"Knowl Based Syst"},{"key":"1074_CR32","volume":"112","author":"J Alneamy","year":"2019","unstructured":"Alneamy J, Alnaish Z, Hashim S, Alnaish R (2019) Utilizing hybrid functional fuzzy wavelet neural networks with a teaching learning-based optimization algorithm for medical disease diagnosis. Comput Biol Med 112:103348","journal-title":"Comput Biol Med"},{"key":"1074_CR33","doi-asserted-by":"crossref","first-page":"3299","DOI":"10.1007\/s40747-022-00670-4","volume":"8","author":"N Yang","year":"2022","unstructured":"Yang N, Tang Z, Cai X (2022) Cooperative multi-population Harris Hawks optimization for many-objective optimization. Complex Intell Syst 8:3299\u20133332","journal-title":"Complex Intell Syst"},{"key":"1074_CR34","doi-asserted-by":"crossref","unstructured":"Zheng M, Fukuyama Y, El-Abd M, Iizaka T, Matsui T (2020) Optimization Overall, of Smart City by Multi-population Global-best Brain Storm Optimization using Cooperative Coevolution, IEEE Congress on Evolutionary Computation (CEC). Glasgow 2020:1\u20137","DOI":"10.1109\/CEC48606.2020.9185789"},{"key":"1074_CR35","doi-asserted-by":"crossref","unstructured":"Storn R, Price K (1997) Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341\u2013359","DOI":"10.1023\/A:1008202821328"},{"key":"1074_CR36","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s40747-020-00178-9","volume":"7","author":"J Jin","year":"2021","unstructured":"Jin J (2021) A robust zeroing neural network for solving dynamic nonlinear equations and its application to kinematic control of mobile manipulator. Complex Intell Syst 7:87\u201399","journal-title":"Complex Intell Syst"},{"key":"1074_CR37","doi-asserted-by":"crossref","unstructured":"Gao W, Li Y (2021) Solving a new test set of nonlinear equation systems by evolutionary algorithm. IEEE Trans Cybern 53(1):406\u2013415","DOI":"10.1109\/TCYB.2021.3108563"},{"issue":"15","key":"1074_CR38","doi-asserted-by":"crossref","first-page":"6676","DOI":"10.1016\/j.eswa.2014.05.009","volume":"41","author":"G Manizheh","year":"2014","unstructured":"Manizheh G, Mohammad R (2014) Forest optimization algorithm. Expert Syst Appl 41(15):6676\u20136687","journal-title":"Expert Syst Appl"},{"key":"1074_CR39","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","volume":"69","author":"M Seyedali","year":"2014","unstructured":"Seyedali M, Seyed M, Mirjalilib A (2014) Grey wolf optimizer. Adv Eng Softw 69:46\u201361","journal-title":"Adv Eng Softw"},{"issue":"15","key":"1074_CR40","first-page":"293","volume":"786","author":"Sh Li","year":"2019","unstructured":"Li Sh, Gong W, Yan X (2019) Parameter extraction of photovoltaic models using an improved teaching-learning-based optimization. Energy Convers Manag 786(15):293\u2013305","journal-title":"Energy Convers Manag"},{"issue":"11","key":"1074_CR41","doi-asserted-by":"crossref","first-page":"4063","DOI":"10.1109\/TIE.2011.2174540","volume":"59","author":"TD Do","year":"2012","unstructured":"Do TD, Choi HH, Jung J (2012) SDRE-based near optimal control system design for PM synchronous motor. IEEE Trans Ind Electron 59(11):4063\u20134074","journal-title":"IEEE Trans Ind Electron"},{"issue":"12","key":"1074_CR42","doi-asserted-by":"crossref","first-page":"2188","DOI":"10.1109\/TSMC.2017.2705160","volume":"48","author":"D Guo","year":"2017","unstructured":"Guo D, Nie Z, Yan L (2017) The application of noise-tolerant zd design formula to robots kinematic control via time-varying nonlinear equations solving. IEEE Trans Syst Man Cybern Syst 48(12):2188\u20132197","journal-title":"IEEE Trans Syst Man Cybern Syst"},{"issue":"3","key":"1074_CR43","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1109\/TEVC.2015.2458037","volume":"20","author":"A Gupta","year":"2016","unstructured":"Gupta A, Ong Y-S, Feng L (2016) Multifactorial evolution: toward evolutionary multitasking. IEEE Trans Evol Comput 20(3):343\u2013357","journal-title":"IEEE Trans Evol Comput"},{"key":"1074_CR44","first-page":"1","volume":"2","author":"T Wei","year":"2021","unstructured":"Wei T, Wang S, Zhong J, Liu D, Zhang J (2021) A review on evolutionary multi-task optimization: Trends and challenges. IEEE Trans Evol Comput 2:1","journal-title":"IEEE Trans Evol Comput"}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-023-01074-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-023-01074-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-023-01074-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T19:19:52Z","timestamp":1698434392000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-023-01074-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":44,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["1074"],"URL":"https:\/\/doi.org\/10.1007\/s40747-023-01074-8","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,25]]},"assertion":[{"value":"12 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest, and the data generated during and\/or analysed during the current study are available from the corresponding author on reasonable request.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}