{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T05:57:03Z","timestamp":1776491823208,"version":"3.51.2"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Karlsruher Institut f\u00fcr Technologie (KIT)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures.<\/jats:p>","DOI":"10.1007\/s10957-023-02285-2","type":"journal-article","created":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T13:02:26Z","timestamp":1693573346000},"page":"1736-1766","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization"],"prefix":"10.1007","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1938-6316","authenticated-orcid":false,"given":"Gabriele","family":"Eichfelder","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9514-6317","authenticated-orcid":false,"given":"Oliver","family":"Stein","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2177-8466","authenticated-orcid":false,"given":"Leo","family":"Warnow","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,9,1]]},"reference":[{"issue":"9","key":"2285_CR1","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1016\/S0098-1354(98)00218-X","volume":"22","author":"CS Adjiman","year":"1998","unstructured":"Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, $$\\alpha $$BB, for general twice-differentiable constrained NLPs\u2014II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159\u20131179 (1998)","journal-title":"Comput. Chem. Eng."},{"issue":"9","key":"2285_CR2","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, $$\\alpha $$BB, for general twice-differentiable constrained NLPs\u2013I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137\u20131158 (1998)","journal-title":"Comput. Chem. Eng."},{"key":"2285_CR3","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 Science, Berlin (2012)"},{"key":"2285_CR4","volume-title":"Heuristic Algorithms in Global MINLP Solvers","author":"T Berthold","year":"2015","unstructured":"Berthold, T.: Heuristic Algorithms in Global MINLP Solvers. Verlag Dr, Hut (2015)"},{"issue":"8","key":"2285_CR5","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1080\/0305215X.2021.1939695","volume":"54","author":"R Burachik","year":"2021","unstructured":"Burachik, R., Kaya, C.Y., Rizvi, M.M.: Algorithms for generating pareto fronts of multi-objective integer and mixed-integer programming problems. Eng. Optim. 54(8), 1413\u20131425 (2021)","journal-title":"Eng. Optim."},{"issue":"2, Ser. A","key":"2285_CR6","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/s10107-008-0223-z","volume":"120","author":"S Burer","year":"2009","unstructured":"Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Prog. 120(2, Ser. A), 479\u2013495 (2009)","journal-title":"Math. Prog."},{"key":"2285_CR7","doi-asserted-by":"crossref","unstructured":"Burer, S., Letchford, A.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17, 97\u2013106 (2012)","DOI":"10.1016\/j.sorms.2012.08.001"},{"issue":"2","key":"2285_CR8","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1007\/s10479-020-03910-3","volume":"319","author":"Guillermo Cabrera-Guerrero","year":"2022","unstructured":"Cabrera-Guerrero, Guillermo, Ehrgott, Matthias, Mason, Andrew J., Raith, Andrea: Bi-objective optimisation over a set of convex sub-problems. Ann. Oper. Res. 319(2), 1507\u20131532 (2022)","journal-title":"Ann. Oper. Res."},{"key":"2285_CR9","volume-title":"Vector Optimization: Set-Valued and Variational Analysis","author":"G-Y Chen","year":"2006","unstructured":"Chen, G.-Y., Huang, X., Yang, X.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2006)"},{"key":"2285_CR10","doi-asserted-by":"crossref","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(3), 841\u2013855 (2017)","DOI":"10.1016\/j.ejor.2016.05.029"},{"issue":"4","key":"2285_CR11","doi-asserted-by":"publisher","first-page":"3122","DOI":"10.1137\/19M1264709","volume":"30","author":"Marianna De Santis","year":"2020","unstructured":"De Santis, Marianna, Eichfelder, Gabriele, Niebling, Julia, Rockt\u00e4schel, Stefan: Solving multiobjective mixed integer convex optimization problems. SIAM J. Optim. 30(4), 3122\u20133145 (2020)","journal-title":"SIAM J. Optim."},{"issue":"15","key":"2285_CR12","doi-asserted-by":"publisher","first-page":"4321","DOI":"10.1080\/02331934.2021.1939699","volume":"71","author":"E Diessel","year":"2022","unstructured":"Diessel, E.: An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems. Optimization 71(15), 4321\u20134366 (2022)","journal-title":"Optimization"},{"key":"2285_CR13","volume-title":"Multicriteria Optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)"},{"key":"2285_CR14","doi-asserted-by":"crossref","unstructured":"Eichfelder, G., Groetzner, P.: A note on completely positive relaxations of quadratic problems in a multiobjective framework. J. Glob. Optim. 82, 615\u2013626 (2022)","DOI":"10.1007\/s10898-021-01091-2"},{"key":"2285_CR15","doi-asserted-by":"crossref","unstructured":"Eichfelder, G., Kirst, P., Meng, L., Stein.: A general branch-and-bound framework for continuous global multiobjective optimization. J. Glob. Optim. 80(1), 195\u2013227 (2021)","DOI":"10.1007\/s10898-020-00984-y"},{"key":"2285_CR16","doi-asserted-by":"publisher","unstructured":"Eichfelder, G., Stein.: Limit sets in continuous global multiobjective optimization. Optimization (2022). https:\/\/doi.org\/10.1080\/02331934.2022.2092479","DOI":"10.1080\/02331934.2022.2092479"},{"key":"2285_CR17","unstructured":"Eichfelder, G., Warnow, L.: On implementation details and numerical experiments for the HyPaD algorithm to solve multi-objective mixed-integer convex optimization problems. Preprint 2021-08-8538, Optimization Online (2021)"},{"key":"2285_CR18","doi-asserted-by":"crossref","unstructured":"Eichfelder, G., Warnow, L.: An approximation algorithm for multi-objective optimization problems using a box-coverage. J. Glob. Optim. 83(2), 329\u2013357 (2022)","DOI":"10.1007\/s10898-021-01109-9"},{"key":"2285_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/s00186-023-00828-x","author":"G Eichfelder","year":"2023","unstructured":"Eichfelder, G., Warnow, L.: A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems. Math. Meth. Oper. Res. (2023). https:\/\/doi.org\/10.1007\/s00186-023-00828-x","journal-title":"Math. Meth. Oper. Res."},{"issue":"3","key":"2285_CR20","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/s10589-007-9135-8","volume":"42","author":"J Fernandez","year":"2009","unstructured":"Fernandez, J., Toth, B.: Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods. Comput. Optim. Appl. 42(3), 393\u2013419 (2009)","journal-title":"Comput. Optim. Appl."},{"key":"2285_CR21","volume-title":"Variational Methods in Partially Ordered Spaces","author":"A G\u00f6pfert","year":"2006","unstructured":"G\u00f6pfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2006)"},{"issue":"5\u20136","key":"2285_CR22","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1002\/mcda.1780","volume":"29","author":"Pascal Halffmann","year":"2022","unstructured":"Halffmann, Pascal, Sch\u00e4fer, Luca E., D\u00e4chert, Kerstin, Klamroth, Kathrin, Ruzika, Stefan: Exact algorithms for multiobjective linear optimization problems with integer variables: a state of the art survey. J. Multi-Criteria Decis. Anal. 29(5\u20136), 341\u2013363 (2022)","journal-title":"J. Multi-Criteria Decis. Anal."},{"key":"2285_CR23","doi-asserted-by":"publisher","DOI":"10.1201\/9780203026922","volume-title":"Global Optimization Using Interval Analysis: Revised and Expanded","author":"E Hansen","year":"2003","unstructured":"Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis: Revised and Expanded. CRC Press, Boca Raton (2003)"},{"key":"2285_CR24","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/s10107-017-1149-0","volume":"169","author":"G Kirlik","year":"2018","unstructured":"Kirlik, G., Sayin, S.: Bilevel programming for generating discrete representations in multiobjective optimization. Math. Program. 169, 585\u2013604 (2018)","journal-title":"Math. Program."},{"key":"2285_CR25","doi-asserted-by":"crossref","unstructured":"Kirst, P.r, Stein, O., Steuermann, P.: Deterministic upper bounds for spatial branch-and-bound methods in global minimization with nonconvex constraints. TOP 23(2), 591\u2013616 (2015)","DOI":"10.1007\/s11750-015-0387-7"},{"issue":"3","key":"2285_CR26","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(3), 767\u2013778 (2015)","journal-title":"Eur. J. Oper. Res."},{"issue":"4","key":"2285_CR27","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s10898-013-0136-0","volume":"60","author":"A L\u00f6hne","year":"2014","unstructured":"L\u00f6hne, A., Rudloff, B., Ulus, F.: Primal and dual approximation algorithms for convex vector optimization problems. J. Glob. Optim. 60(4), 713\u2013736 (2014)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"2285_CR28","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/BF00936165","volume":"43","author":"P Loridan","year":"1984","unstructured":"Loridan, P.: $$\\varepsilon $$-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265\u2013276 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"2285_CR29","unstructured":"Gurobi Optimization LLC. Gurobi Reference Manual (2020)"},{"key":"2285_CR30","unstructured":"MATLAB. Matlab Bench Documentation. https:\/\/www.mathworks.com\/help\/matlab\/ref\/bench.html. Accessed 01 Oct 2022"},{"key":"2285_CR31","volume-title":"Nonlinear Multiobjective Optimization","author":"K Miettinen","year":"2012","unstructured":"Miettinen, K.: Nonlinear Multiobjective Optimization. Springer, Berlin (2012)"},{"key":"2285_CR32","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)"},{"issue":"3","key":"2285_CR33","doi-asserted-by":"publisher","first-page":"2396","DOI":"10.1137\/20M131922X","volume":"31","author":"C Neumann","year":"2021","unstructured":"Neumann, C., Stein, O.: Generating feasible points for mixed-integer convex optimization problems by inner parallel cuts. SIAM J. Optim. 31(3), 2396\u20132428 (2021)","journal-title":"SIAM J. Optim."},{"key":"2285_CR34","doi-asserted-by":"crossref","unstructured":"Neumann, C., Stein, O.r, Sudermann-Merx, N.: A feasible rounding approach for mixed-integer optimization problems. Comput. Optim. Appl. 72(2), 309\u2013337 (2019)","DOI":"10.1007\/s10589-018-0042-y"},{"key":"2285_CR35","doi-asserted-by":"crossref","unstructured":"Neumann, C., Stein, O.r, Sudermann-Merx, N.: Granularity in nonlinear mixed-integer optimization. J. Optim. Theory Appl. 184(2), 433\u2013465 (2020)","DOI":"10.1007\/s10957-019-01591-y"},{"issue":"1","key":"2285_CR36","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 multiobjective optimization. SIAM J. Optim. 29(1), 794\u2013821 (2019)","journal-title":"SIAM J. Optim."},{"key":"2285_CR37","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/s11750-016-0430-3","volume":"25","author":"S Nobakhtian","year":"2017","unstructured":"Nobakhtian, S., Shafiei, N.: A Benson type algorithm for nonconvex multiobjective programming problems. TOP 25, 271\u2013287 (2017)","journal-title":"TOP"},{"key":"2285_CR38","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1287\/ijoc.2019.0887","volume":"32","author":"T Perini","year":"2019","unstructured":"Perini, T., Boland, N., Pecin, Diego, Savelsbergh, M.: A criterion space method for biobjective mixed integer programming: the boxed line method. INFORMS J. Comput. 32, 16\u201339 (2019)","journal-title":"INFORMS J. Comput."},{"issue":"3","key":"2285_CR39","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1016\/j.ejor.2017.01.032","volume":"260","author":"A Przybylski","year":"2017","unstructured":"Przybylski, A., Gandibleux, X.: Multi-objective branch and bound. Eur. J. Oper. Res. 260(3), 856\u2013872 (2017)","journal-title":"Eur. J. Oper. Res."},{"key":"2285_CR40","unstructured":"Przybylski, A., Klamroth, K., Lacour, R.: A simple and efficient dichotomic search algorithm for multi-objective mixed integer linear programs. Preprint 1911.08937, arXiv (2019)"},{"key":"2285_CR41","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/978-94-017-1247-7_7","volume-title":"Developments in Reliable Computing","author":"SR Rump","year":"1999","unstructured":"Rump, S.R.: INTLAB\u2014INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77\u2013104. Kluwer Academic Publishers, Dordrecht (1999)"},{"issue":"3","key":"2285_CR42","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(3), 473\u2013501 (2005)","journal-title":"J. Optim. Theory Appl."},{"key":"2285_CR43","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1016\/S0098-1354(97)00146-4","volume":"21","author":"EMB Smith","year":"1997","unstructured":"Smith, E.M.B., Pantelides, C.C.: Global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 21, 791\u2013796 (1997)","journal-title":"Comput. Chem. Eng."},{"issue":"4\u20135","key":"2285_CR44","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1016\/S0098-1354(98)00286-5","volume":"23","author":"EMB Smith","year":"1999","unstructured":"Smith, E.M.B., Pantelides, C.C.: A symbolic reformulation\/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 23(4\u20135), 457\u2013478 (1999)","journal-title":"Comput. Chem. Eng."},{"key":"2285_CR45","unstructured":"Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Springer, Berlin (2013)"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02285-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-023-02285-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02285-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,11]],"date-time":"2024-11-11T10:13:29Z","timestamp":1731320009000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-023-02285-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,1]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["2285"],"URL":"https:\/\/doi.org\/10.1007\/s10957-023-02285-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,1]]},"assertion":[{"value":"26 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 July 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}