{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T03:55:55Z","timestamp":1760586955852,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T00:00:00Z","timestamp":1551744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Doctoral Foundation of Xi\u2019an University of Technology","award":["112-451116017"],"award-info":[{"award-number":["112-451116017"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The standard covariance matrix adaptation evolution strategy (CMA-ES) is highly effective at locating a single global optimum. However, it shows unsatisfactory performance for solving multimodal optimization problems (MMOPs). In this paper, an improved algorithm based on the MA-ES, which is called the matrix adaptation evolution strategy with multi-objective optimization algorithm, is proposed to solve multimodal optimization problems (MA-ESN-MO). Taking advantage of the multi-objective optimization in maintaining population diversity, MA-ESN-MO transforms an MMOP into a bi-objective optimization problem. The archive is employed to save better solutions for improving the convergence of the algorithm. Moreover, the peaks found by the algorithm can be maintained until the end of the run. Multiple subpopulations are used to explore and exploit in parallel to find multiple optimal solutions for the given problem. Experimental results on CEC2013 test problems show that the covariance matrix adaptation with Niching and the multi-objective optimization algorithm (CMA-NMO), CMA Niching with the Mahalanobis Metric and the multi-objective optimization algorithm (CMA-NMM-MO), and matrix adaptation evolution strategy Niching with the multi-objective optimization algorithm (MA-ESN-MO) have overall better performance compared with the covariance matrix adaptation evolution strategy (CMA-ES), matrix adaptation evolution strategy (MA-ES), CMA Niching (CMA-N), CMA-ES Niching with Mahalanobis Metric (CMA-NMM), and MA-ES-Niching (MA-ESN).<\/jats:p>","DOI":"10.3390\/a12030056","type":"journal-article","created":{"date-parts":[[2019,3,5]],"date-time":"2019-03-05T11:19:50Z","timestamp":1551784790000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Matrix Adaptation Evolution Strategy with Multi-Objective Optimization for Multimodal Optimization"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4336-5582","authenticated-orcid":false,"given":"Wei","family":"Li","sequence":"first","affiliation":[{"name":"School of Computer Science and Engineering, Xi\u2019an University of Technology, Xi\u2019an 710048, China"},{"name":"Shaanxi Key Laboratory for Network Computing and Security Technology, Xi\u2019an 710048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Singh, G., and Deb, K. (2006, January 8\u201312). Comparison of multimodal optimization algorithms based on evolutionary algorithms. Proceedings of the Genetic and Evolutionary Computation Conference, Seattle, WA, USA.","DOI":"10.1145\/1143997.1144200"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, X. (2005, January 25\u201329). Efficient differential evolution using speciation for multimodal function optimization. Proceedings of the Genetic and Evolutionary Computation Conference, Washington DC, USA.","DOI":"10.1145\/1068009.1068156"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1109\/TEVC.2010.2050024","article-title":"Niching without niching parameters: Particle swarm optimization using a ring topology","volume":"14","author":"Li","year":"2010","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1162\/evco.2010.18.1.18104","article-title":"Adaptive niche radii and niche shapes approaches for niching with the CMA-ES","volume":"18","author":"Shir","year":"2010","journal-title":"Evol. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1162\/106365602760234081","article-title":"A species conserving genetic algorithm for multimodal function optimization","volume":"10","author":"Li","year":"2002","journal-title":"Evol. Comput."},{"key":"ref_6","unstructured":"Thomsen, R. (2004, January 19\u201323). Multimodal optimization using crowding-based differential evolution. Proceedings of the IEEE Congress on Evolutionary Computation, Portland, OR, USA."},{"key":"ref_7","unstructured":"Harik, G.R. (1995, January 15\u201319). Finding multimodal solutions using restricted tournament selection. Proceedings of the International Conference on Genetic Algorithms, Pittsburgh, PA, USA."},{"key":"ref_8","unstructured":"Goldberg, D.E., and Richardson, J. (1987, January 6\u20138). Genetic algorithms with sharing for multimodal function optimization. Proceedings of the International Conference on Genetic Algorithms, Cambridge, MA, USA."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Petrowski, A. (1996, January 20\u201322). A clearing procedure as a niching method for genetic algorithms. Proceedings of the IEEE Congress on Evolutionary Computation, Nagoya, Japan.","DOI":"10.1109\/ICEC.1996.542703"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1162\/106365603321828970","article-title":"Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES)","volume":"11","author":"Hansen","year":"2003","journal-title":"Evol. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1109\/TEVC.2017.2680320","article-title":"Simplify Your Covariance Matrix Adaptation Evolution Strategy","volume":"21","author":"Beyer","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hansen, N., and Kern, S. (2004). Evaluating the CMA Evolution Strategy on Multimodal Test Functions. Parallel Problem Solving from Nature\u2014PPSN VIII, Springer.","DOI":"10.1007\/978-3-540-30217-9_29"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shir, O.M., and B\u00e4ck, T. (2006). Niche Radius Adaptation in the CMA-ES Niching Algorithm. Parallel Problem Solving from Nature\u2014PPSN IX, Springer.","DOI":"10.1007\/11844297_15"},{"key":"ref_14","unstructured":"Shir, O.M., and Back, T. (2005, January 2\u20135). Dynamic niching in evolution strategies with covariance matrix adaptation. Proceedings of the IEEE Congress on Evolutionary Computation, Edinburgh, UK."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1109\/TEVC.2012.2231685","article-title":"Multimodal Optimization Using a Biobjective Differential Evolution Algorithm Enhanced with Mean Distance-Based Selection","volume":"17","author":"Basak","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1162\/EVCO_a_00042","article-title":"Multimodal optimization using a bi-objective evolutionary algorithm","volume":"20","author":"Deb","year":"2012","journal-title":"Evol. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1109\/TCYB.2014.2337117","article-title":"MOMMOP: Multiobjective Optimization for Locating Multiple Optimal Solutions of Multimodal Optimization Problems","volume":"45","author":"Wang","year":"2015","journal-title":"IEEE Trans. Cybern."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1109\/TEVC.2009.2017517","article-title":"Bi-Objective Multipopulation Genetic Algorithm for Multimodal Function Optimization","volume":"14","author":"Yao","year":"2010","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2017.09.044","article-title":"A Tri-Objective Differential Evolution Approach for Multimodal Optimization","volume":"423","author":"Yu","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shir, O.M., Emmerich, M., and Back, T. (2007, January 25\u201328). Self-Adaptive Niching CMA-ES with Mahalanobis Metric. Proceedings of the IEEE Congress on Evolutionary Computation, Singapore.","DOI":"10.1109\/CEC.2007.4424555"},{"key":"ref_21","unstructured":"Li, X., Engelbrecht, A., and Epitropakis, M. (2013). Benchmark Functions for CEC 2013 Special Session and Competition on Niching Methods for Multimodal Function Optimization, Royal Melbourne Institute of Technology. Technical Report."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1109\/TEVC.2017.2744328","article-title":"Evolutionary Multiobjective Optimization Based Multimodal Optimization: Fitness Landscape Approximation and Peak Detection","volume":"22","author":"Cheng","year":"2018","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1109\/TEVC.2011.2161873","article-title":"Differential Evolution with neighborhood mutation for multimodal optimization","volume":"16","author":"Qu","year":"2012","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_24","unstructured":"SMahfoud, W. (1995). Niching Methods for Genetic Algorithms. [Ph.D. Thesis, University of Illinois at Urbana-Champaign]."},{"key":"ref_25","unstructured":"Mengsheol, O., and Goldberg, D. (1999, January 13\u201317). Probabilistic crowding: Deterministic crowding with probabilistic replacement. Proceedings of the Genetic and Evolutionary Computation Conference, Orlando, FL, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.jare.2010.09.001","article-title":"Context based clearing procedure: A niching method for genetic algorithms","volume":"1","author":"Fayek","year":"2010","journal-title":"J. Adv. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1109\/TCYB.2015.2394466","article-title":"Ensemble and Arithmetic Recombination-Based Speciation Differential Evolution for Multimodal Optimization","volume":"46","author":"Hui","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1109\/TEVC.2016.2591064","article-title":"Adaptive Multimodal Continuous Ant Colony Optimization","volume":"21","author":"Yang","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.swevo.2017.03.002","article-title":"A niche GSA method with nearest neighbor scheme for multimodal optimization","volume":"35","author":"Haghbayan","year":"2017","journal-title":"Swarm Evol. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1109\/TEVC.2012.2203138","article-title":"A Distance-Based Locally Informed Particle Swarm Model for Multimodal;Optimization","volume":"17","author":"Qu","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.asoc.2017.11.050","article-title":"Enhancing the performance of differential evolution with covariance matrix self-adaptation","volume":"64","author":"He","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.asoc.2014.01.038","article-title":"Differential evolution based on covariance matrix learning and bimodal distribution parameter setting","volume":"18","author":"Wang","year":"2014","journal-title":"Appl. Soft Comput. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.asoc.2016.04.022","article-title":"Biogeography-based optimization with covariance matrix based migration","volume":"45","author":"Chen","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.ins.2011.08.014","article-title":"A Differential Covariance Matrix Adaptation Evolutionary Algorithm for real parameter optimization","volume":"182","author":"Ghosh","year":"2012","journal-title":"Inf. Sci."},{"key":"ref_35","unstructured":"He, X.Y., Zhou, Y.R., and Chen, Z.F. (2018). Evolutionary Bilevel Optimization based on Covariance Matrix Adaptation. IEEE Trans. Evol. Comput., 1\u201321."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Awad, N., Ali, M.Z., Suganthan, P.N., and Reynolds, R.G. (2017, January 5\u20138). A novel differential crossover strategy based on covariance matrix learning with Euclidean neighborhood for solving real-world problems. Proceedings of the IEEE Congress on Evolutionary Computation, San Sebastian, Spain.","DOI":"10.1109\/CEC.2017.7969337"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.knosys.2017.01.020","article-title":"Multi-objective differential evolution with dynamic covariance matrix learning for multi-objective optimization problems with variable linkages","volume":"121","author":"Jiang","year":"2017","journal-title":"Knowl. Based Syst."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wang, T.C., Liaw, R.T., and Ting, C.K. (2016, January 24\u201329). MOEA\/D using covariance matrix adaptation evolution strategy for complex multi-objective optimization problems. Proceedings of the IEEE Congress on Evolutionary Computation, Vancouver, BC, Canada.","DOI":"10.1109\/CEC.2016.7743896"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2792","DOI":"10.1109\/TCYB.2014.2316552","article-title":"Differential Evolution Enhanced with Multiobjective Sorting-Based Mutation Operators","volume":"44","author":"Wang","year":"2014","journal-title":"IEEE Trans. Cybern."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/TEVC.2010.2087271","article-title":"Differential evolution with composite trial vector generation strategies and control parameters","volume":"15","author":"Wang","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s00500-008-0323-y","article-title":"KEEL: A software tool to assess evolutionary algorithms to data mining problems","volume":"13","author":"Ventura","year":"2009","journal-title":"Soft Comput."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/3\/56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:36:26Z","timestamp":1760186186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/3\/56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,5]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["a12030056"],"URL":"https:\/\/doi.org\/10.3390\/a12030056","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2019,3,5]]}}}