{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:43:52Z","timestamp":1780361032894,"version":"3.54.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T00:00:00Z","timestamp":1717113600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T00:00:00Z","timestamp":1717113600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["05M2020"],"award-info":[{"award-number":["05M2020"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["90685813"],"award-info":[{"award-number":["90685813"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["90685689"],"award-info":[{"award-number":["90685689"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Deutsches Elektronen-Synchrotron (DESY)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper we propose polarized consensus-based dynamics in order to make consensus-based optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions with many modes, respectively. For this, we \u201cpolarize\u201d the dynamics with a localizing kernel and the resulting model can be viewed as a bounded confidence model for opinion formation in the presence of common objective. Instead of being attracted to a common weighted mean as in the original consensus-based methods, which prevents the detection of more than one minimum or mode, in our method every particle is attracted to a weighted mean which gives more weight to nearby particles. We prove that in the mean-field regime the polarized CBS dynamics are unbiased for Gaussian targets. We also prove that in the zero temperature limit and for sufficiently well-behaved strongly convex objectives the solution of the Fokker\u2013Planck equation converges in the Wasserstein-2 distance to a Dirac measure at the minimizer. Finally, we propose a computationally more efficient generalization which works with a predefined number of clusters and improves upon our polarized baseline method for high-dimensional optimization.\n<\/jats:p>","DOI":"10.1007\/s10107-024-02095-y","type":"journal-article","created":{"date-parts":[[2024,5,31]],"date-time":"2024-05-31T06:01:53Z","timestamp":1717135313000},"page":"125-155","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Polarized consensus-based dynamics for optimization and sampling"],"prefix":"10.1007","volume":"211","author":[{"given":"Leon","family":"Bungert","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8440-2928","authenticated-orcid":false,"given":"Tim","family":"Roith","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philipp","family":"Wacker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,5,31]]},"reference":[{"key":"2095_CR1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718768","volume-title":"Introduction to Derivative-free Optimization","author":"AR Conn","year":"2009","unstructured":"Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-free Optimization. SIAM, Philadelphia, PA (2009)"},{"key":"2095_CR2","doi-asserted-by":"crossref","unstructured":"Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN\u201995 - International Conference on Neural Networks, vol. 4, pp. 1942\u20131948 (1995)","DOI":"10.1109\/ICNN.1995.488968"},{"key":"2095_CR3","doi-asserted-by":"crossref","unstructured":"Reynolds, C.W.: Flocks, herds and schools: A distributed behavioral model. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, pp. 25\u201334 (1987)","DOI":"10.1145\/37401.37406"},{"issue":"01","key":"2095_CR4","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1142\/S0218202517400061","volume":"27","author":"R Pinnau","year":"2017","unstructured":"Pinnau, R., Totzeck, C., Tse, O., Martin, S.: A consensus-based model for global optimization and its mean-field limit. Math. Models Methods Appl. Sci. 27(01), 183\u2013204 (2017)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"06","key":"2095_CR5","doi-asserted-by":"publisher","first-page":"1037","DOI":"10.1142\/S0218202518500276","volume":"28","author":"JA Carrillo","year":"2018","unstructured":"Carrillo, J.A., Choi, Y.-P., Totzeck, C., Tse, O.: An analytical framework for consensus-based global optimization method. Math. Models Methods Appl. Sci. 28(06), 1037\u20131066 (2018)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"12","key":"2095_CR6","doi-asserted-by":"publisher","first-page":"2417","DOI":"10.1142\/S0218202520500463","volume":"30","author":"S-Y Ha","year":"2020","unstructured":"Ha, S.-Y., Jin, S., Kim, D.: Convergence of a first-order consensus-based global optimization algorithm. Math. Models Methods Appl. Sci. 30(12), 2417\u20132444 (2020)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"2095_CR7","unstructured":"Fornasier, M., Klock, T., Riedl, K.: Consensus-based optimization methods converge globally in mean-field law (2021). arXiv:2103.15130"},{"issue":"5","key":"2095_CR8","doi-asserted-by":"publisher","first-page":"6026","DOI":"10.3934\/mbe.2020320","volume":"17","author":"C Totzeck","year":"2020","unstructured":"Totzeck, C., Wolfram, M.-T.: Consensus-based global optimization with personal best. Math. Biosci. Eng. 17(5), 6026\u20136044 (2020). https:\/\/doi.org\/10.3934\/mbe.2020320","journal-title":"Math. Biosci. Eng."},{"key":"2095_CR9","unstructured":"Carrillo, J.A., Totzeck, C., Vaes, U.: Consensus-based optimization and ensemble kalman inversion for global optimization problems with constraints (2021). arXiv:2111.02970"},{"key":"2095_CR10","first-page":"5","volume":"27","author":"JA Carrillo","year":"2021","unstructured":"Carrillo, J.A., Jin, S., Li, L., Zhu, Y.: A consensus-based global optimization method for high dimensional machine learning problems. ESAIM: Control Optim. Calculus of Varia. 27, 5 (2021)","journal-title":"ESAIM: Control Optim. Calculus of Varia."},{"issue":"14","key":"2095_CR11","doi-asserted-by":"publisher","first-page":"2725","DOI":"10.1142\/S0218202520500530","volume":"30","author":"M Fornasier","year":"2020","unstructured":"Fornasier, M., Huang, H., Pareschi, L., S\u00fcnnen, P.: Consensus-based optimization on hypersurfaces: well-posedness and mean-field limit. Math. Models Methods Appl. Sci. 30(14), 2725\u20132751 (2020)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"237","key":"2095_CR12","first-page":"1","volume":"22","author":"M Fornasier","year":"2021","unstructured":"Fornasier, M., Pareschi, L., Huang, H., S\u00fcnnen, P.: Consensus-based optimization on the sphere: convergence to global minimizers and machine learning. J. Mach. Learn. Res. 22(237), 1\u201355 (2021)","journal-title":"J. Mach. Learn. Res."},{"key":"2095_CR13","doi-asserted-by":"crossref","unstructured":"Schillings, C., Totzeck, C., Wacker, P.: Ensemble-based gradient inference for particle methods in optimization and sampling (2022). arXiv:2209.15420","DOI":"10.1137\/22M1533281"},{"key":"2095_CR14","doi-asserted-by":"crossref","unstructured":"Totzeck, C.: Trends in consensus-based optimization. In: Active Particles, volume 3, pp. 201\u2013226. Springer, Cham (2022)","DOI":"10.1007\/978-3-030-93302-9_6"},{"issue":"3","key":"2095_CR15","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1111\/sapm.12470","volume":"148","author":"JA Carrillo","year":"2022","unstructured":"Carrillo, J.A., Hoffmann, F., Stuart, A.M., Vaes, U.: Consensus-based sampling. Stud. Appl. Math. 148(3), 1069\u20131140 (2022)","journal-title":"Stud. Appl. Math."},{"issue":"01\u201304","key":"2095_CR16","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1142\/S0219525900000078","volume":"3","author":"G Deffuant","year":"2000","unstructured":"Deffuant, G., Neau, D., Amblard, F., Weisbuch, G.: Mixing beliefs among interacting agents. Adv. Complex Syst. 3(01\u201304), 87\u201398 (2000)","journal-title":"Adv. Complex Syst."},{"key":"2095_CR17","unstructured":"Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5(3) (2002)"},{"issue":"02","key":"2095_CR18","doi-asserted-by":"publisher","first-page":"1150007","DOI":"10.1142\/S0218202511500072","volume":"22","author":"J G\u00f3mez-Serrano","year":"2012","unstructured":"G\u00f3mez-Serrano, J., Graham, C., Le Boudec, J.-Y.: The bounded confidence model of opinion dynamics. Math. Models Methods Appl. Sci. 22(02), 1150007 (2012)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"10","key":"2095_CR19","doi-asserted-by":"publisher","first-page":"1535","DOI":"10.1142\/S0129183105008126","volume":"16","author":"S Fortunato","year":"2005","unstructured":"Fortunato, S., Latora, V., Pluchino, A., Rapisarda, A.: Vector opinion dynamics in a bounded confidence consensus model. Int. J. Mod. Phys. C 16(10), 1535\u20131551 (2005)","journal-title":"Int. J. Mod. Phys. C"},{"issue":"2","key":"2095_CR20","doi-asserted-by":"publisher","first-page":"187","DOI":"10.3934\/krm.2021051","volume":"15","author":"M Burger","year":"2022","unstructured":"Burger, M.: Kinetic equations for processes on co-evolving networks. Kinetic Relat. Models 15(2), 187\u2013212 (2022). https:\/\/doi.org\/10.3934\/krm.2021051","journal-title":"Kinetic Relat. Models"},{"issue":"3","key":"2095_CR21","doi-asserted-by":"publisher","first-page":"937","DOI":"10.1007\/s10013-021-00505-8","volume":"49","author":"M Burger","year":"2021","unstructured":"Burger, M.: Network structured kinetic models of social interactions. Vietnam J. Math. 49(3), 937\u2013956 (2021)","journal-title":"Vietnam J. Math."},{"issue":"1","key":"2095_CR22","first-page":"47","volume":"6","author":"P Schnell","year":"1964","unstructured":"Schnell, P.: Eine methode zur auffindung von gruppen. Biom. Z. 6(1), 47\u201348 (1964)","journal-title":"Biom. Z."},{"issue":"1","key":"2095_CR23","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1109\/TIT.1975.1055330","volume":"21","author":"K Fukunaga","year":"1975","unstructured":"Fukunaga, K., Hostetler, L.: The estimation of the gradient of a density function, with applications in pattern recognition. IEEE Trans. Inf. Theory 21(1), 32\u201340 (1975)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"15","key":"2095_CR24","first-page":"510","volume":"7","author":"KB Petersen","year":"2008","unstructured":"Petersen, K.B., Pedersen, M.S., et al.: The matrix cookbook. Tech. Univ. Denmark 7(15), 510 (2008)","journal-title":"Tech. Univ. Denmark"},{"key":"2095_CR25","doi-asserted-by":"crossref","unstructured":"Bailo, R., Barbaro, A., Gomes, S. N., Riedl, K., Roith, T., Totzeck, C., Vaes, U. (2024). CBX: Python and Julia packages for consensus-based interacting particle methods. arXiv preprint arXiv:2403.14470","DOI":"10.21105\/joss.06611"},{"key":"2095_CR26","volume-title":"A Connectionist Machine for Genetic Hillclimbing","author":"D Ackley","year":"2012","unstructured":"Ackley, D.: A Connectionist Machine for Genetic Hillclimbing. The Springer International Series in Engineering and Computer Science. Springer, New York, NY (2012)"},{"key":"2095_CR27","unstructured":"Rastrigin, L.A.: Systems of extremal control. Nauka (1974)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02095-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-024-02095-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02095-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,28]],"date-time":"2025-04-28T13:56:14Z","timestamp":1745848574000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-024-02095-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,31]]},"references-count":27,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["2095"],"URL":"https:\/\/doi.org\/10.1007\/s10107-024-02095-y","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,31]]},"assertion":[{"value":"25 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author declares that there are no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}