{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T02:44:49Z","timestamp":1775702689090,"version":"3.50.1"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:00:00Z","timestamp":1771027200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:00:00Z","timestamp":1771027200000},"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":["12271482"],"award-info":[{"award-number":["12271482"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LZ23A010006"],"award-info":[{"award-number":["LZ23A010006"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s10898-026-01598-6","type":"journal-article","created":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T04:57:31Z","timestamp":1771045051000},"page":"873-890","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An effective branch and bound algorithm for generalized risk parity portfolio optimization"],"prefix":"10.1007","volume":"94","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0291-6464","authenticated-orcid":false,"given":"Jing","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Lei","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Wenxun","family":"Xing","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,14]]},"reference":[{"issue":"3","key":"1598_CR1","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1016\/j.jedc.2007.03.005","volume":"32","author":"GJ Alexander","year":"2008","unstructured":"Alexander, G.J., Baptista, A.M.: Active portfolio management with benchmarking: adding a value-at-risk constraint. J. Econ. Dyn. Control 32(3), 779\u2013820 (2008)","journal-title":"J. Econ. Dyn. Control"},{"key":"1598_CR2","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s10898-008-9372-0","volume":"43","author":"K Anstreicher","year":"2009","unstructured":"Anstreicher, K.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43, 471\u2013484 (2009)","journal-title":"J. Glob. Optim."},{"key":"1598_CR3","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1007\/s10479-023-05797-2","volume":"336","author":"C Ararat","year":"2024","unstructured":"Ararat, C., Cesarone, F., Pinar, M.C., Ricci, J.M.: MAD risk parity portfolios. Ann. Oper. Res. 336, 899\u2013924 (2024)","journal-title":"Ann. Oper. Res."},{"issue":"3","key":"1598_CR4","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1080\/14697688.2015.1031815","volume":"16","author":"X Bai","year":"2016","unstructured":"Bai, X., Scheinberg, K., T\u00fct\u00fcnc\u00fc, R.H.: Least-squares approach to risk parity in portfolio selection. Quant. Financ. 16(3), 357\u2013376 (2016)","journal-title":"Quant. Financ."},{"issue":"1","key":"1598_CR5","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/s10107-011-0462-2","volume":"129","author":"X Bao","year":"2011","unstructured":"Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129(1), 129\u2013157 (2011)","journal-title":"Math. Program."},{"key":"1598_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718829","volume-title":"Lectures On Modern Convex Optimization, Analysis","author":"A Ben-Tal","year":"2001","unstructured":"Ben-Tal, A., Nemirovski, A.: Lectures On Modern Convex Optimization, Analysis. Algorithms and Engineering Applications. SIAM Series on Optimization, Philadelphia (2001)"},{"issue":"3","key":"1598_CR7","doi-asserted-by":"publisher","first-page":"150","DOI":"10.3905\/joi.2012.21.3.150","volume":"21","author":"DB Chaves","year":"2012","unstructured":"Chaves, D.B., Hsu, J.C., Li, F., Shakernia, O.: Efficient algorithms for computing risk parity portfolio weights. J. Invest. 21(3), 150\u2013163 (2012)","journal-title":"J. Invest."},{"key":"1598_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.parco.2023.102999","volume":"116","author":"Y Chen","year":"2023","unstructured":"Chen, Y., Li, C., Hu, Y., Lu, Z.: A parallel non-convex approximation framework for risk parity portfolio design. Parallel Comput. 116, 102999 (2023)","journal-title":"Parallel Comput."},{"key":"1598_CR9","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s10898-020-00915-x","volume":"78","author":"G Costa","year":"2020","unstructured":"Costa, G., Kwon, R.H.: Generalized risk parity portfolio optimization: an ADMM approach. J. Glob. Optim. 78, 207\u2013238 (2020)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"1598_CR10","doi-asserted-by":"publisher","first-page":"40","DOI":"10.3905\/jpm.2007.684751","volume":"33","author":"FJ Fabozzi","year":"2007","unstructured":"Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A., Focardi, S.M.: Robust portfolio optimization. J. Portf. Manag. 33(3), 40 (2007)","journal-title":"J. Portf. Manag."},{"key":"1598_CR11","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s10288-006-0011-7","volume":"5","author":"A Faye","year":"2007","unstructured":"Faye, A., Roupin, F.: Partial Lagrangian relaxation for general quadratic programming. 4OR-Q. J. Oper. Res. 5, 75\u201388 (2007)","journal-title":"J. Oper. Res."},{"issue":"19","key":"1598_CR12","doi-asserted-by":"publisher","first-page":"5285","DOI":"10.1109\/TSP.2015.2452219","volume":"63","author":"Y Feng","year":"2015","unstructured":"Feng, Y., Palomar, D.P.: SCRIP: successive convex optimization methods for risk parity portfolio design. IEEE Trans. Signal Process. 63(19), 5285\u20135300 (2015)","journal-title":"IEEE Trans. Signal Process."},{"key":"1598_CR13","first-page":"1","volume":"19","author":"C Giorgio","year":"2018","unstructured":"Giorgio, C., Kwon, R.H.: Risk parity portfolio optimization under a Markov regime-switching framework. Quant. Financ. 19, 1\u201319 (2018)","journal-title":"Quant. Financ."},{"issue":"2","key":"1598_CR14","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1016\/j.ejor.2013.10.060","volume":"234","author":"PN Kolm","year":"2014","unstructured":"Kolm, P.N., T\u00fct\u00fcnc\u00fc, R., Fabozzi, F.J.: 60 Years of portfolio optimization: practical challenges and current trends. Eur. J. Oper. Res. 234(2), 356\u2013371 (2014)","journal-title":"Eur. J. Oper. Res."},{"key":"1598_CR15","doi-asserted-by":"crossref","unstructured":"Le Thi, H.A., Huynh, V.N., Pham Dinh, T.: DC programming and DCA for general DC programs. In: Advanced Computational Methods for Knowledge Engineering, pp. 15\u201335. Springer International Publishing (2014)","DOI":"10.1007\/978-3-319-06569-4_2"},{"issue":"2","key":"1598_CR16","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s11081-015-9294-x","volume":"17","author":"T Lipp","year":"2016","unstructured":"Lipp, T., Boyd, S.: Variations and extension of the convex-concave procedure. Optim. Eng. 17(2), 263\u2013287 (2016)","journal-title":"Optim. Eng."},{"key":"1598_CR17","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/S0024-3795(98)10032-0","volume":"284","author":"MS Lobo","year":"1998","unstructured":"Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193\u2013228 (1998)","journal-title":"Linear Algebra Appl."},{"key":"1598_CR18","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1007\/s10898-016-0436-2","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. Global Optim. 67, 475\u2013493 (2017)","journal-title":"J. Global Optim."},{"issue":"2","key":"1598_CR19","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s10898-018-0726-y","volume":"73","author":"C Lu","year":"2019","unstructured":"Lu, C., Deng, Z., Zhou, J., Guo, X.: A sensitive-eigenvector based global algorithm for quadratically constrained quadratic programming. J. Global Optim. 73(2), 371\u2013388 (2019)","journal-title":"J. Global Optim."},{"issue":"1","key":"1598_CR20","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1111\/mafi.12383","volume":"34","author":"H Luo","year":"2024","unstructured":"Luo, H., Chen, Y., Zhang, X., Li, D., Wu, H.: Effective algorithms for optimal portfolio deleveraging problem with cross impact. Math. Financ. 34(1), 36\u201389 (2024)","journal-title":"Math. Financ."},{"issue":"4","key":"1598_CR21","doi-asserted-by":"publisher","first-page":"60","DOI":"10.3905\/jpm.2010.36.4.060","volume":"36","author":"S Maillard","year":"2010","unstructured":"Maillard, S., Roncalli, T., Te\u00efletche, J.: The properties of equally weighted risk contribution portfolios. J. Portf. Manag. 36(4), 60\u201370 (2010)","journal-title":"J. Portf. Manag."},{"issue":"1","key":"1598_CR22","first-page":"77","volume":"7","author":"HM Markowitz","year":"1952","unstructured":"Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77\u201391 (1952)","journal-title":"J. Finance"},{"issue":"2","key":"1598_CR23","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1111\/jtsa.12792","volume":"46","author":"MS Paolella","year":"2025","unstructured":"Paolella, M.S., Polak, P., Walker, P.S.: Risk parity portfolio optimization under heavy-tailed returns and dynamic correlations. J. Time Ser. Anal. 46(2), 353\u2013377 (2025)","journal-title":"J. Time Ser. Anal."},{"key":"1598_CR24","unstructured":"Qian E.: Risk parity portfolios: efficient portfolios through true diversication. Panagora Asset Management White Paper (2005)"},{"issue":"3","key":"1598_CR25","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1080\/14697688.2015.1046907","volume":"16","author":"T Roncalli","year":"2016","unstructured":"Roncalli, T., Weisang, G.: Risk parity portfolios with risk factors. Quant. Financ. 16(3), 377\u2013388 (2016)","journal-title":"Quant. Financ."},{"key":"1598_CR26","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/s10898-022-01218-z","volume":"87","author":"H Wu","year":"2023","unstructured":"Wu, H., Luo, H., Zhang, X., Liu, J.: A new global algorithm for factor-risk-constrained mean-variance portfolio selection. J. Global Optim. 87, 503\u2013532 (2023)","journal-title":"J. Global Optim."},{"key":"1598_CR27","doi-asserted-by":"publisher","first-page":"3","DOI":"10.21314\/JCF.2010.213","volume":"14","author":"S Zhu","year":"2010","unstructured":"Zhu, S., Li, D., Sun, X.: Portfolio selection with marginal risk control. J. Comput. Financ. 14, 3\u201328 (2010)","journal-title":"J. Comput. Financ."},{"issue":"2","key":"1598_CR28","doi-asserted-by":"publisher","first-page":"51","DOI":"10.21314\/JOR.2012.240","volume":"14","author":"S Zhu","year":"2011","unstructured":"Zhu, S., Cui, X., Sun, X., Li, D.: Factor-risk-constrained mean-variance portfolio selection: formulation and global optimization solution approach. J. Risk 14(2), 51\u201389 (2011)","journal-title":"J. Risk"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-026-01598-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-026-01598-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-026-01598-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T01:50:35Z","timestamp":1775699435000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-026-01598-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,14]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["1598"],"URL":"https:\/\/doi.org\/10.1007\/s10898-026-01598-6","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,14]]},"assertion":[{"value":"10 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}