{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T10:07:27Z","timestamp":1780567647106,"version":"3.54.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Projekt DEAL"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.<\/jats:p>","DOI":"10.1007\/s10898-020-00984-y","type":"journal-article","created":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T06:03:05Z","timestamp":1611036185000},"page":"195-227","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["A general branch-and-bound framework for continuous global multiobjective optimization"],"prefix":"10.1007","volume":"80","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1938-6316","authenticated-orcid":false,"given":"Gabriele","family":"Eichfelder","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Kirst","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Laura","family":"Meng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9514-6317","authenticated-orcid":false,"given":"Oliver","family":"Stein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,1,19]]},"reference":[{"key":"984_CR1","doi-asserted-by":"publisher","first-page":"1137","DOI":"10.1016\/S0098-1354(98)00027-1","volume":"22","author":"CS Adjiman","year":"1998","unstructured":"Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, aBB, for general twice-differentiable constrained NLPs: I. Theoretical advances. Comput. Chem. Eng. 22, 1137\u20131158 (1998)","journal-title":"Comput. Chem. Eng."},{"key":"984_CR2","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1287\/moor.8.2.273","volume":"8","author":"FA Al-Khayyal","year":"1983","unstructured":"Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273\u2013286 (1983)","journal-title":"Math. Oper. Res."},{"key":"984_CR3","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF01099647","volume":"7","author":"IP Androulakis","year":"1995","unstructured":"Androulakis, I.P., Maranas, C.D., Floudas, C.A.: $$\\alpha $$BB: a global optimization method for general constrainted nonconvex problems. J. Glob. Optim. 7, 337\u2013363 (1995)","journal-title":"J. Glob. Optim."},{"key":"984_CR4","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/978-1-4614-1927-3_5","volume-title":"Mixed Integer Nonlinear Programming","author":"P Belotti","year":"2012","unstructured":"Belotti, P.: Disjunctive cuts for nonconvex MINLP. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 117\u2013144. Springer, Berlin (2012)"},{"key":"984_CR5","unstructured":"Binh, T.: A multiobjective evolutionary algorithm: the study cases. In: Proceedings of the 1999 Genetic and Evolutionary Computation Conference, pp 127\u2013128 (1999)"},{"key":"984_CR6","doi-asserted-by":"crossref","unstructured":"Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. In: Parallel Problem Solving from Nature: PPSN VIII, Springer, pp 722\u2013731 (2004)","DOI":"10.1007\/978-3-540-30217-9_73"},{"key":"984_CR7","volume-title":"Vector Optimization","author":"G Chen","year":"2005","unstructured":"Chen, G., Huang, X., Yang, X.: Vector Optimization. Springer, Berlin (2005)"},{"key":"984_CR8","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1016\/j.ejor.2016.05.029","volume":"260","author":"K D\u00e4chert","year":"2017","unstructured":"D\u00e4chert, K., Klamroth, K., Lacour, R., Vanderpooten, D.: Efficient computation of the search region in multi-objective optimization. Eur. J. Oper. Res. 260, 841\u2013855 (2017)","journal-title":"Eur. J. Oper. Res."},{"key":"984_CR9","volume-title":"Multi-objective optimization using evolutionary algorithms","author":"K Deb","year":"2001","unstructured":"Deb, K.: Multi-objective optimization using evolutionary algorithms. Wiley, New York (2001)"},{"key":"984_CR10","volume-title":"Multicriteria Optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)"},{"key":"984_CR11","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1134\/S0005117914060046","volume":"75","author":"YuG Evtushenko","year":"2014","unstructured":"Evtushenko, YuG, Posypkin, M.A.: Method of non-uniform coverages to solve the multicriteria optimization problems with guaranteed accuracy. Autom. Remote Control 75, 1025\u20131040 (2014)","journal-title":"Autom. Remote Control"},{"key":"984_CR12","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/s10589-007-9135-8","volume":"42","author":"J Fern\u00e1ndez","year":"2009","unstructured":"Fern\u00e1ndez, J., T\u00f3th, B.: Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods. Comput. Optim. Appl. 42, 393\u2013419 (2009)","journal-title":"Comput. Optim. Appl."},{"key":"984_CR13","doi-asserted-by":"crossref","unstructured":"Fonseca, C.A., Fleming, P.J.: Multiobjective genetic algorithms made easy: selection sharing and mating restriction. In: Proceedings of the 1st International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications, pp 45\u201352. IEEE Press, Piscataway, NJ (1995)","DOI":"10.1049\/cp:19951023"},{"key":"984_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-020-01493-2","author":"C F\u00fcllner","year":"2020","unstructured":"F\u00fcllner, C., Kirst, P., Stein, O.: Convergent upper bounds in global minimization with nonlinear equality constraints. Math. Program. (2020). https:\/\/doi.org\/10.1007\/s10107-020-01493-2","journal-title":"Math. Program."},{"key":"984_CR15","doi-asserted-by":"crossref","unstructured":"Gerth (Tammer), C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 97\u2013320 (1990)","DOI":"10.1007\/BF00940478"},{"key":"984_CR16","series-title":"CMS Books","volume-title":"Variational Methods in Partially Ordered Spaces","author":"A G\u00f6pfert","year":"2003","unstructured":"G\u00f6pfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books. Springer, New York (2003)"},{"key":"984_CR17","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1080\/02331934.2018.1474469","volume":"67","author":"C G\u00fcnther","year":"2018","unstructured":"G\u00fcnther, C., Popovici, N.: New algorithms for discrete vector optimization based on the Graef\u2013Younes method and cone-monotone sorting functions. Optimization 67, 975\u20131003 (2018)","journal-title":"Optimization"},{"key":"984_CR18","volume-title":"Global Optimization Using Interval Analysis","author":"E Hansen","year":"2004","unstructured":"Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis. Marcel Dekker Inc., New York (2004)"},{"key":"984_CR19","series-title":"Methods and Applications","first-page":"75","volume-title":"Multicriteria Decision Making and Fuzzy Systems-Theory","author":"J Jahn","year":"2006","unstructured":"Jahn, J., Rathje, U.: Graef\u2013Younes method with backward iteration. In: K\u00fcfer, K.-H., et al. (eds.) Multicriteria Decision Making and Fuzzy Systems-Theory. Methods and Applications, pp. 75\u201381. Shaker, Aachen (2006)"},{"key":"984_CR20","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1007\/s11750-015-0387-7","volume":"23","author":"P Kirst","year":"2015","unstructured":"Kirst, P., Stein, O., Steuermann, P.: Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints. TOP 23, 591\u2013616 (2015)","journal-title":"TOP"},{"key":"984_CR21","unstructured":"Klamroth, K.: Personal communication (2017)"},{"key":"984_CR22","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1016\/j.ejor.2015.03.031","volume":"245","author":"K Klamroth","year":"2015","unstructured":"Klamroth, K., Lacour, R., Vanderpooten, D.: On the representation of the search region in multi-objective optimization. Eur. J. Oper. Res. 245, 767\u2013778 (2015)","journal-title":"Eur. J. Oper. Res."},{"key":"984_CR23","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/BF00936165","volume":"43","author":"P Loridan","year":"1984","unstructured":"Loridan, P.: $$\\epsilon $$-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265\u2013276 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"984_CR24","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF01580665","volume":"10","author":"GP McCormick","year":"1976","unstructured":"McCormick, G.P.: Computability of gobal solutions to factorable nonconvex programs: part I\u2014convex underestimating problems. Math. Program. 10, 147\u2013175 (1976)","journal-title":"Math. Program."},{"key":"984_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-5563-6","volume-title":"Nonlinear Multiobjective Optimization","author":"K Miettinen","year":"1998","unstructured":"Miettinen, K.: Nonlinear Multiobjective Optimization. Springer, Berlin (1998)"},{"key":"984_CR26","doi-asserted-by":"publisher","first-page":"794","DOI":"10.1137\/18M1169680","volume":"29","author":"J Niebling","year":"2019","unstructured":"Niebling, J., Eichfelder, G.: A branch-and-bound based algorithm for nonconvex optimization problems. SIAM J. Optim. 29, 794\u2013821 (2019)","journal-title":"SIAM J. Optim."},{"key":"984_CR27","volume-title":"Interval Methods for Systems of Equations","author":"A Neumaier","year":"1990","unstructured":"Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)"},{"key":"984_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-61007-8","volume-title":"Non-convex Multi-Objective Optimization","author":"P Pardalos","year":"2017","unstructured":"Pardalos, P., \u017dilinskas, A., \u017dilinskas, J.: Non-convex Multi-Objective Optimization. Springer, Berlin (2017)"},{"key":"984_CR29","doi-asserted-by":"crossref","unstructured":"Rump, S.M.: INTLAB-interval laboratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp 77\u2013104. Springer, Dordrecht (1999)","DOI":"10.1007\/978-94-017-1247-7_7"},{"key":"984_CR30","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s10957-005-5494-4","volume":"126","author":"S Ruzika","year":"2005","unstructured":"Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126, 473\u2013501 (2005)","journal-title":"J. Optim. Theory Appl."},{"key":"984_CR31","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1007\/s11750-009-0105-4","volume":"18","author":"D Scholz","year":"2010","unstructured":"Scholz, D.: The multicriteria big cube small cube method. TOP 18, 286\u2013302 (2010)","journal-title":"TOP"},{"key":"984_CR32","volume-title":"Theory of Multiobjective Optimization","author":"Y Sawaragi","year":"1985","unstructured":"Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Elsevier, Amsterdam (1985)"},{"key":"984_CR33","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1016\/S0098-1354(97)00146-4","volume":"21","author":"E Smith","year":"1997","unstructured":"Smith, E., Pantelides, C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, 791\u2013796 (1997)","journal-title":"Comput. Chem. Eng."},{"key":"984_CR34","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1016\/S0098-1354(98)00286-5","volume":"23","author":"E Smith","year":"1999","unstructured":"Smith, E., Pantelides, C.: A symbolic reformulation\/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23, 457\u2013478 (1999)","journal-title":"Comput. Chem. Eng."},{"key":"984_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3532-1","volume-title":"Convexification and Global Optimization in Continuous Mixed-Integer Nonlinear Programming","author":"M Tawarmalani","year":"2002","unstructured":"Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous Mixed-Integer Nonlinear Programming. Springer, Dordrecht (2002)"},{"key":"984_CR36","doi-asserted-by":"publisher","first-page":"1921","DOI":"10.1007\/s11590-012-0547-8","volume":"7","author":"A \u017dilinskas","year":"2013","unstructured":"\u017dilinskas, A.: On the worst-case optimal multi-objective global optimization. Optim. Lett. 7, 1921\u20131928 (2013)","journal-title":"Optim. Lett."},{"key":"984_CR37","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.cnsns.2014.08.025","volume":"21","author":"A \u017dilinskas","year":"2015","unstructured":"\u017dilinskas, A., \u017dilinskas, J.: Adaptation of a one-step worst-case optimal univariate algorithm of bi-objective Lipschitz optimization to multidimensional problems. Commun. Nonlinear Sci. Numer. Simul. 21, 89\u201398 (2015)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"updated-by":[{"DOI":"10.1007\/s10898-021-00998-0","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T00:00:00Z","timestamp":1614038400000}}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-020-00984-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-020-00984-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-020-00984-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T19:34:09Z","timestamp":1627673649000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-020-00984-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,19]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["984"],"URL":"https:\/\/doi.org\/10.1007\/s10898-020-00984-y","relation":{"correction":[{"id-type":"doi","id":"10.1007\/s10898-021-00998-0","asserted-by":"object"}]},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,19]]},"assertion":[{"value":"20 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 December 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2021","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The publisher introduced an error in the last sentence of the proof for Lemma 3.2. Its beginning \u201cThe supremum of \u201d must be replaced by \u201cThe supremum of \u201d.","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2021","order":7,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":8,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":9,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s10898-021-00998-0","URL":"https:\/\/doi.org\/10.1007\/s10898-021-00998-0","order":10,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}