{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T09:00:44Z","timestamp":1777712444802,"version":"3.51.4"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T00:00:00Z","timestamp":1634256000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T00:00:00Z","timestamp":1634256000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007826","name":"Technische Universit\u00e4t Ilmenau","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007826","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.<\/jats:p>","DOI":"10.1007\/s10898-021-01091-2","type":"journal-article","created":{"date-parts":[[2021,10,15]],"date-time":"2021-10-15T07:28:18Z","timestamp":1634282898000},"page":"615-626","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A note on completely positive relaxations of quadratic problems in a multiobjective framework"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1938-6316","authenticated-orcid":false,"given":"Gabriele","family":"Eichfelder","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patrick","family":"Groetzner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,10,15]]},"reference":[{"issue":"2","key":"1091_CR1","doi-asserted-by":"crossref","first-page":"543","DOI":"10.3934\/jimo.2014.10.543","volume":"10","author":"Y Bai","year":"2014","unstructured":"Bai, Y., Guo, C.: Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. J. Ind. Manag. Optim. 10(2), 543\u2013556 (2014)","journal-title":"J. Ind. Manag. Optim."},{"key":"1091_CR2","doi-asserted-by":"publisher","DOI":"10.1142\/5273","volume-title":"Completely Positive Matrices","author":"A Berman","year":"2003","unstructured":"Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, River Edge (2003)"},{"issue":"3","key":"1091_CR3","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1016\/j.ejor.2011.04.026","volume":"216","author":"IM Bomze","year":"2012","unstructured":"Bomze, I.M.: Copositive optimization\u2014recent developments and applications. Eur. J. Oper. Res. 216(3), 509\u2013520 (2012)","journal-title":"Eur. J. Oper. Res."},{"issue":"2","key":"1091_CR4","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1023\/A:1020209017701","volume":"24","author":"IM Bomze","year":"2002","unstructured":"Bomze, I.M., De Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24(2), 163\u2013185 (2002)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"1091_CR5","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. Program. 120(2), 479\u2013495 (2009)","journal-title":"Math. Program."},{"key":"1091_CR6","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1057\/s41274-017-0233-4","volume":"69","author":"X Chao","year":"2017","unstructured":"Chao, X., Peng, Y.: A cost-sensitive multi-criteria quadratic programming model for imbalanced data. J. Oper. Res. Soc. 69, 500\u2013516 (2017)","journal-title":"J. Oper. Res. Soc."},{"issue":"4","key":"1091_CR7","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1137\/S1052623401383248","volume":"12","author":"E de Klerk","year":"2002","unstructured":"de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875\u2013892 (2002)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1091_CR8","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10589-013-9594-z","volume":"57","author":"PJC Dickinson","year":"2014","unstructured":"Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57(2), 403\u2013415 (2014)","journal-title":"Comput. Optim. Appl."},{"key":"1091_CR9","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-642-12598-0_1","volume-title":"Recent Advances in Optimization and its Applications in Engineering","author":"M D\u00fcr","year":"2010","unstructured":"D\u00fcr, M.: Copositive programming\u2014a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3\u201320. Springer, Berlin (2010)"},{"key":"1091_CR10","volume-title":"Multicriteria Optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)","edition":"2"},{"key":"1091_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.laa.2019.12.024","volume":"591","author":"P Groetzner","year":"2020","unstructured":"Groetzner, P., D\u00fcr, M.: A factorization method for completely positive matrices. Linear Algebra Appl. 591, 1\u201324 (2020)","journal-title":"Linear Algebra Appl."},{"key":"1091_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24828-6","volume-title":"Vector Optimization","author":"J Jahn","year":"2004","unstructured":"Jahn, J.: Theory, applications, and extensions. In: Vector Optimization. Springer, Berlin (2004)"},{"key":"1091_CR13","doi-asserted-by":"crossref","unstructured":"Maxfield, J. E., Minc, H.: On the matrix equation $$X^{prime } X=A$$. Proc. Edin. Math. Soc. (2)13, 125\u2013129 (1962\/63)","DOI":"10.1017\/S0013091500014681"},{"key":"1091_CR14","volume-title":"Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization","author":"PA Parrilo","year":"2000","unstructured":"Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. California Institute of Technology, California (2000).. (PhD thesis)"},{"key":"1091_CR15","unstructured":"Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Mathematics in Science and Engineering, vol. 176. Academic Press, Orlando (1985)"},{"issue":"1\u20132","key":"1091_CR16","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s10107-012-0601-4","volume":"143","author":"J Sponsel","year":"2014","unstructured":"Sponsel, J., D\u00fcr, M.: Factorization and cutting planes for completely positive matrices by copositive projection. Math. Program. 143(1\u20132), 211\u2013229 (2014)","journal-title":"Math. Program."},{"key":"1091_CR17","first-page":"251","volume":"1","author":"R Steuer","year":"2006","unstructured":"Steuer, R., Qi, Y., Hirschberger, M.: Developments in multi-attribute portfolio selection. Multiple Criteria Decis. Mak. 1, 251\u2013262 (2006)","journal-title":"Multiple Criteria Decis. Mak."},{"key":"1091_CR18","doi-asserted-by":"crossref","unstructured":"Zhang, P., Zhang, J., Shi, Y.: A new multi-criteria quadratic-programming linear classification model for vip e-mail analysis. In International Conference on Computational Science, pp. 499\u2013502. Springer, Berlin (2007)","DOI":"10.1007\/978-3-540-72586-2_73"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-021-01091-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-021-01091-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-021-01091-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,12]],"date-time":"2023-01-12T10:09:00Z","timestamp":1673518140000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-021-01091-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,15]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["1091"],"URL":"https:\/\/doi.org\/10.1007\/s10898-021-01091-2","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,15]]},"assertion":[{"value":"27 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 September 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 October 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Open Access funding enabled and organized by Projekt DEAL.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Funding"}}]}}