{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T11:43:46Z","timestamp":1783770226549,"version":"3.55.0"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T00:00:00Z","timestamp":1671408000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T00:00:00Z","timestamp":1671408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11961001"],"award-info":[{"award-number":["11961001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Construction Project of first-class subjects in Ningxia higher Education","award":["NXYLXK2017B09"],"award-info":[{"award-number":["NXYLXK2017B09"]}]},{"name":"Major proprietary funded project of North Minzu University","award":["ZDZX201901"],"award-info":[{"award-number":["ZDZX201901"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2023,5]]},"DOI":"10.1007\/s10898-022-01255-8","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T10:02:41Z","timestamp":1671444161000},"page":"61-92","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problems"],"prefix":"10.1007","volume":"86","author":[{"given":"Bo","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2021-2097","authenticated-orcid":false,"given":"YueLin","family":"Gao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xia","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"XiaoLi","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,12,19]]},"reference":[{"key":"1255_CR1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/978-1-4615-2025-2_5","volume-title":"Handbook of Global Optimization","author":"C Floudas","year":"1995","unstructured":"Floudas, C., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217\u2013269. Kluwer Academic Publishers, Boston (1995)"},{"key":"1255_CR2","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/978-1-4613-0263-6_8","volume-title":"Trends in Industrial and Applied Mathematics","author":"N Gould","year":"2002","unstructured":"Gould, N., Toint, P.: Numerical methods for large-scale non-convex quadratic programming. In: Siddiqi, A.H., Kocvara, M. (eds.) Trends in Industrial and Applied Mathematics, pp. 149\u2013179. Springer, Berlin (2002)"},{"issue":"1","key":"1255_CR3","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF00120662","volume":"1","author":"P Pardalos","year":"1991","unstructured":"Pardalos, P., Vavasis, S.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15\u201322 (1991)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"1255_CR4","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s10107-014-0836-3","volume":"151","author":"I Bomze","year":"2015","unstructured":"Bomze, I., Overton, M.: Narrowing the difficulty gap for the Celis\u2013Dennis\u2013Tapia problem. Math. Program. 151(2), 459\u2013476 (2015)","journal-title":"Math. Program."},{"issue":"1","key":"1255_CR5","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10556780410001647186","volume":"19","author":"C Fortin","year":"2004","unstructured":"Fortin, C., Wolkowicz, H.: The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1), 41\u201367 (2004)","journal-title":"Optim. Methods Softw."},{"issue":"1\u20132","key":"1255_CR6","first-page":"171","volume":"147","author":"V Jeyakumar","year":"2004","unstructured":"Jeyakumar, V., Li, G.: Trust-region problems with linear inequality constraints: exact sdp relaxation, global optimality and robust optimization. Math. Program. 147(1\u20132), 171\u2013206 (2004)","journal-title":"Math. Program."},{"issue":"1","key":"1255_CR7","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10107-007-0138-0","volume":"115","author":"I Bomze","year":"2008","unstructured":"Bomze, I., Locatelli, M., Tardella, F.: New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. 115(1), 31\u201364 (2008)","journal-title":"Math. Program."},{"issue":"4","key":"1255_CR8","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1023\/A:1026583532263","volume":"18","author":"I Bomze","year":"2000","unstructured":"Bomze, I., D\u00fcr, M., Klerk, E., Quist, A., Roos, C., Terlaky, T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18(4), 301\u2013320 (2000)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"1255_CR9","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s10107-004-0549-0","volume":"120","author":"D Vandenbussche","year":"2005","unstructured":"Vandenbussche, D., Nemhauser, G.: A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 120(3), 531\u2013557 (2005)","journal-title":"Math. Program."},{"issue":"2","key":"1255_CR10","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1002\/nav.3800230212","volume":"23","author":"H Vaish","year":"2010","unstructured":"Vaish, H., Shetty, C.: The bilinear programming problem. Nav. Res. Logist. 23(2), 303\u2013309 (2010)","journal-title":"Nav. Res. Logist."},{"issue":"3","key":"1255_CR11","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s10898-020-00919-7","volume":"78","author":"P Shen","year":"2020","unstructured":"Shen, P., Wang, K., Lu, T.: Outer space branch and bound algorithm for solving linear multiplicative programming problems. J. Glob. Optim. 78(3), 453\u2013482 (2020)","journal-title":"J. Glob. Optim."},{"issue":"1\u20133","key":"1255_CR12","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/S0024-3795(98)10032-0","volume":"284","author":"M Lobo","year":"1998","unstructured":"Lobo, M., Vandenberghe, M., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1\u20133), 193\u2013228 (1998)","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"1255_CR13","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1023\/A:1025794313696","volume":"26","author":"S Kim","year":"2003","unstructured":"Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26(2), 143\u2013154 (2003)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"1255_CR14","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s101070050006","volume":"87","author":"S Zhang","year":"2000","unstructured":"Zhang, S.: Quadratic maximization and semidefinite relaxation. Math. Program. 87(3), 453\u2013465 (2000)","journal-title":"Math. Program."},{"issue":"1","key":"1255_CR15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10107-019-01367-2","volume":"181","author":"S Burer","year":"2020","unstructured":"Burer, S., Ye, Y.: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Program. 181(1), 1\u201317 (2020)","journal-title":"Math. Program."},{"issue":"2","key":"1255_CR16","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10898-021-01071-6","volume":"82","author":"G Azuma","year":"2022","unstructured":"Azuma, G., Fukuda, M., Kim, S., Yamashita, M.: Exact SDP relaxations of quadratically constrained quadratic programs with forest structures. J. Glob. Optim. 82(2), 243\u2013262 (2022)","journal-title":"J. Glob. Optim."},{"key":"1255_CR17","doi-asserted-by":"crossref","unstructured":"Horst, R., Pardalos, P., Thoai, T.: Introduction to Global Optimization, 2nd., p. Chapter 3. Kluwer Academic Publishers, Boston (2000)","DOI":"10.1007\/978-1-4615-0015-5"},{"issue":"2","key":"1255_CR18","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10107-006-0080-6","volume":"113","author":"S Burer","year":"2008","unstructured":"Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2), 259\u2013282 (2008)","journal-title":"Math. Program."},{"issue":"3","key":"1255_CR19","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1007\/s10107-004-0550-7","volume":"102","author":"D Vandenbussche","year":"2005","unstructured":"Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programming with box constraints. Math. Program. 102(3), 559\u2013575 (2005)","journal-title":"Math. Program."},{"issue":"1","key":"1255_CR20","first-page":"973","volume":"250","author":"H Jiao","year":"2015","unstructured":"Jiao, H., Liu, S.: A parametric linear relaxation algorithm for globally solving nonconvex quadratic programming. Appl. Math. Comput. 250(1), 973\u2013985 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1255_CR21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1023\/A:1013886408463","volume":"22","author":"I Bomze","year":"2002","unstructured":"Bomze, I.: Branch-and-bound approaches to standard quadratic optimization problems. J. Glob. Optim. 22(1), 17\u201337 (2002)","journal-title":"J. Glob. Optim."},{"issue":"1","key":"1255_CR22","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10107-011-0462-2","volume":"129","author":"X Bao","year":"2011","unstructured":"Bao, X., Sahinidis, N., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programs: a review and comparisons. Math. Program. 129(1), 129\u2013157 (2011)","journal-title":"Math. Program."},{"key":"1255_CR23","doi-asserted-by":"crossref","DOI":"10.1002\/9781118032701","volume-title":"Interior Point Algorithms: Theory and Analysis","author":"Y Ye","year":"1997","unstructured":"Ye, Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)"},{"key":"1255_CR24","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/978-1-4613-0279-7_6","volume-title":"Proceedings of Advances in Convex Analysis and Global Optimization","author":"C Lemarechal","year":"2001","unstructured":"Lemarechal, C., Oustry, F.: SDP relaxations in combinatorial optimization from a Lagrangian viewpoint. In: Hadjisavvas, N., Pardalos, P. (eds.) Proceedings of Advances in Convex Analysis and Global Optimization, pp. 119\u2013134. Springer, Boston (2001)"},{"issue":"6","key":"1255_CR25","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/BF01070233","volume":"23","author":"N Shor","year":"2011","unstructured":"Shor, N.: Class of global minimum bounds of polynomial functions. Cybernetics 23(6), 731\u2013734 (2011)","journal-title":"Cybernetics"},{"issue":"3","key":"1255_CR26","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1137\/0403036","volume":"3","author":"H Sherali","year":"1990","unstructured":"Sherali, H., Adams, W.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Discrete Math. 3(3), 411\u2013430 (1990)","journal-title":"SIAM Discrete Math."},{"issue":"3","key":"1255_CR27","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.ejor.2011.04.026","volume":"216","author":"I Bomze","year":"2012","unstructured":"Bomze, I.: Copositive optimization\u2014Recent developments and applications. Eur. J. Oper. Res. 216(3), 509\u2013520 (2012)","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"1255_CR28","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.ejor.2013.02.031","volume":"229","author":"Z Deng","year":"2013","unstructured":"Deng, Z., Fang, S., Jin, Q., Xing, W.: Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme. Eur. J. Oper. Res. 229(1), 21\u201328 (2013)","journal-title":"Eur. J. Oper. Res."},{"issue":"4","key":"1255_CR29","doi-asserted-by":"crossref","first-page":"2320","DOI":"10.1137\/120890636","volume":"23","author":"N Arima","year":"2013","unstructured":"Arima, N., Kim, S., Kojima, A.: A quadratically constrained quadratic optimization model for completely positive cone programming. SIAM J. Optim. 23(4), 2320\u20132340 (2013)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"1255_CR30","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1137\/100793955","volume":"21","author":"C Lu","year":"2011","unstructured":"Lu, C., Fang, S., Jin, Q., Wang, Z., Xing, W.: KKT solution and conic relaxation for solving quadratically constrained quadratic programming problems. SIAM J. Optim. 21(4), 1475\u20131490 (2011)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1255_CR31","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1287\/moor.28.2.246.14485","volume":"28","author":"J Sturm","year":"2003","unstructured":"Sturm, J., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246\u2013267 (2003)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"1255_CR32","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s12532-011-0033-9","volume":"4","author":"J Chen","year":"2012","unstructured":"Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4(1), 33\u201352 (2012)","journal-title":"Math. Program. Comput."},{"key":"1255_CR33","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s12532-018-0142-9","volume":"11","author":"H Luo","year":"2018","unstructured":"Luo, H., Bai, X., Lim, G., Peng, J.: New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation. Math. Program. Comput. 11, 119\u2013171 (2018)","journal-title":"Math. Program. Comput."},{"issue":"1","key":"1255_CR34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1021765131316","volume":"103","author":"R Horst","year":"1999","unstructured":"Horst, R., Thoai, N.: DC programming: overview. J. Optim. Theory Appl. 103(1), 1\u201343 (1999)","journal-title":"J. Optim. Theory Appl."},{"key":"1255_CR35","doi-asserted-by":"crossref","unstructured":"Dinh, T., Thi, H.: Recent advances in DC programming and DCA. In: Transactions on Computational Intelligence XIII, pp. 1\u201337. Springer, Berlin (2014)","DOI":"10.1007\/978-3-642-54455-2_1"},{"key":"1255_CR36","first-page":"596","volume":"268","author":"A Ashtiani","year":"2015","unstructured":"Ashtiani, A., Ferreira, P.: A branch-and-cut algorithm for a class of sum-of-ratios problems. Appl. Math. Comput. 268, 596\u2013608 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1255_CR37","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s10898-007-9267-5","volume":"41","author":"R Oliveira","year":"2008","unstructured":"Oliveira, R., Ferreira, P.: A convex analysis approach for convex multiplicative programming. J. Glob. Optim. 41(1), 579\u2013592 (2008)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"1255_CR38","first-page":"457","volume":"67","author":"C Lu","year":"2017","unstructured":"Lu, C., Deng, Z., Jin, Q.: An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints. J. Glob. Optim. 67(3), 457\u2013493 (2017)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"1255_CR39","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/BF01097059","volume":"7","author":"C Maranas","year":"1995","unstructured":"Maranas, C., Floudas, C.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143\u2013182 (1995)","journal-title":"J. Glob. Optim."},{"issue":"1","key":"1255_CR40","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s11075-009-9358-0","volume":"55","author":"J Li","year":"2010","unstructured":"Li, J., Wang, P., Ma, L.: A new algorithm for the general quadratic programming problems with box constraints. Numer. Algorithms 55(1), 79\u201385 (2010)","journal-title":"Numer. Algorithms"},{"key":"1255_CR41","unstructured":"Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.2 (2020). http:\/\/cvxr.com\/cvx\/download"},{"key":"1255_CR42","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/978-1-4615-2025-2_8","volume-title":"Handbook of Global Optimization","author":"H Konno","year":"1995","unstructured":"Konno, H., Kuno, T.: Multiplicative programming problems. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 369\u2013405. Kluwer Academic Publishers, Dordrecht (1995)"},{"issue":"2","key":"1255_CR43","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF00121658","volume":"9","author":"T Matsui","year":"1996","unstructured":"Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9(2), 113\u2013119 (1996)","journal-title":"J. Glob. Optim."},{"key":"1255_CR44","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1007\/s11590-018-1378-z","volume":"14","author":"P Shen","year":"2020","unstructured":"Shen, P., Huang, B.: Global algorithm for solving linear multiplicative programming problems. Optim. Lett. 14, 693\u2013710 (2020)","journal-title":"Optim. Lett."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-022-01255-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-022-01255-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-022-01255-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,20]],"date-time":"2023-04-20T19:53:04Z","timestamp":1682020384000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-022-01255-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,19]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["1255"],"URL":"https:\/\/doi.org\/10.1007\/s10898-022-01255-8","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,19]]},"assertion":[{"value":"5 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 December 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}