{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T15:36:27Z","timestamp":1774280187506,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"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":["61877046"],"award-info":[{"award-number":["61877046"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"key scientific and technological project of henan province","award":["202102210385"],"award-info":[{"award-number":["202102210385"]}]},{"name":"key scientific and technological project of henan province","award":["202102210388"],"award-info":[{"award-number":["202102210388"]}]},{"name":"collaborative education project of industry university cooperation of ministry of education","award":["201901225003"],"award-info":[{"award-number":["201901225003"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s11075-022-01330-x","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T04:04:16Z","timestamp":1653365056000},"page":"1063-1082","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Global algorithm for a class of multiplicative programs using piecewise linear approximation technique"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7415-7073","authenticated-orcid":false,"given":"Zhisong","family":"Hou","sequence":"first","affiliation":[]},{"given":"Sanyang","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"issue":"2","key":"1330_CR1","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/BF00933245","volume":"28","author":"HP Benson","year":"1979","unstructured":"Benson, H.P.: Vector maximization with two objective functions. J. Optim. Theory Appl. 28(2), 253\u2013257 (1979)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1330_CR2","first-page":"421","volume":"44","author":"DF Dennis","year":"1998","unstructured":"Dennis, D.F.: Analyzing public inputs to multiple objective decisions on national forests using conjoint analysis. For. Sci. 44(3), 421\u2013429 (1998)","journal-title":"For. Sci."},{"key":"1330_CR3","doi-asserted-by":"crossref","unstructured":"Hartley, R., Kahl, F.: Optimal algorithms in multiview geometry. In: Yagi, S.B., Kang, Y., Kweon, I.S., Zha, H (eds.) Computer Vision\u2013ACCV 2007, pp. 13-34. Springer, Berlin (2007)","DOI":"10.1007\/978-3-540-76386-4_2"},{"issue":"1","key":"1330_CR4","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10589-012-9488-5","volume":"54","author":"T Kuno","year":"2013","unstructured":"Kuno, T., Masaki, T.: A practical but rigorous approach to sum-of-ratios optimization in geometric applications. Comput. Optim. Appl. 54(1), 93\u2013109 (2013)","journal-title":"Comput. Optim. Appl."},{"key":"1330_CR5","first-page":"156","volume":"26","author":"KP Bennett","year":"1994","unstructured":"Bennett, K.P.: Global tree optimization: a non-greedy decision tree algorithm. Computing Sciences and Statistics 26, 156\u2013160 (1994)","journal-title":"Computing Sciences and Statistics"},{"issue":"2","key":"1330_CR6","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/s10203-020-00288-6","volume":"43","author":"R Cambini","year":"2020","unstructured":"Cambini, R.: Underestimation functions for a rank-two partitioning method. Decisions in Economics and Finance 43(2), 465\u2013489 (2020). https:\/\/doi.org\/10.1007\/s10203-020-00288-6","journal-title":"Decisions in Economics and Finance"},{"issue":"2","key":"1330_CR7","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1093\/imaman\/dpaa001","volume":"32","author":"R Cambini","year":"2020","unstructured":"Cambini, R., Venturi, I.: A new solution method for a class of large dimension rank-two nonconvex programs. IMA Journal of Management Mathematics 32(2), 115\u2013137 (2020). https:\/\/doi.org\/10.1093\/imaman\/dpaa001","journal-title":"IMA Journal of Management Mathematics"},{"issue":"3","key":"1330_CR8","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/s40304-016-0087-0","volume":"4","author":"S Trean\u0162\u0103","year":"2016","unstructured":"Trean\u0162\u0103, S.: Multiobjective fractional variational problem on higher-order jet bundles. Commun. Math. Stat. 4(3), 323\u2013340 (2016)","journal-title":"Commun. Math. Stat."},{"key":"1330_CR9","doi-asserted-by":"crossref","unstructured":"Pei, Y., Zhu, D.: Local convergence of a trust-region algorithm with line search filter technique for nonlinear constrained optimization. Appl. Math. Comput., vol. 273 (2016)","DOI":"10.1016\/j.amc.2015.10.060"},{"issue":"1","key":"1330_CR10","doi-asserted-by":"publisher","first-page":"647","DOI":"10.1007\/s12190-017-1126-z","volume":"57","author":"\u015e Mititelu","year":"2018","unstructured":"Mititelu, \u015e., Trean\u0162\u0103, S.: Efficiency conditions in vector control problems governed by multiple integrals. J. Appl. Math. Comput. 57(1), 647\u2013665 (2018)","journal-title":"J. Appl. Math. Comput."},{"key":"1330_CR11","doi-asserted-by":"crossref","unstructured":"Trean\u0163\u0103, S.: On locally and globally optimal solutions in scalar variational control problems. Mathematics, vol. 7(9) (2019)","DOI":"10.3390\/math7090829"},{"issue":"1","key":"1330_CR12","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.ejor.2010.03.047","volume":"207","author":"R Cambini","year":"2010","unstructured":"Cambini, R., Sodini, C.: A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions. Eur. J. Oper. Res. 207(1), 25\u201329 (2010)","journal-title":"Eur. J. Oper. Res."},{"issue":"10","key":"1330_CR13","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1080\/02331934.2013.852548","volume":"63","author":"R Cambini","year":"2014","unstructured":"Cambini, R., Sodini, C.: On the minimization of a class of generalized linear functions on a flow polytope. Optimization 63(10), 1449\u20131464 (2014)","journal-title":"Optimization"},{"issue":"2","key":"1330_CR14","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1287\/opre.43.2.264","volume":"43","author":"JM Mulvey","year":"1995","unstructured":"Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43(2), 264\u2013281 (1995)","journal-title":"Oper. Res."},{"key":"1330_CR15","first-page":"116","volume":"242","author":"P Shen","year":"2014","unstructured":"Shen, P., Yang, L., Liang, Y.: Range division and contraction algorithm for a class of global optimization problems. Appl. Math. Comput. 242, 116\u2013126 (2014)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1330_CR16","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s10957-005-2655-4","volume":"126","author":"HP Benson","year":"2005","unstructured":"Benson, H.P.: Decomposition branch-and-bound based algorithm for linear programs with additional multiplicative constraints. J. Optim. Theory Appl. 126(1), 41\u201361 (2005)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"1330_CR17","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1023\/A:1008308913266","volume":"15","author":"XJ Liu","year":"1999","unstructured":"Liu, X.J., Umegaki, T., Yamamoto, Y.: Heuristic methods for linear multiplicative programming. J. Glob. Optim. 15(4), 433\u2013447 (1999)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"1330_CR18","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1007\/BF02192573","volume":"87","author":"S Schaible","year":"1995","unstructured":"Schaible, S., Sodini, C.: Finite algorithm for generalized linear multiplicative programming. J. Optim. Theory Appl. 87(2), 441\u2013455 (1995)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1330_CR19","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1023\/A:1004657629105","volume":"104","author":"HP Benson","year":"2000","unstructured":"Benson, H.P., Boger, G.M.: Outcome-space cutting-plane algorithm for linear multiplicative programming. J. Optim. Theory Appl. 104(2), 301\u2013322 (2000)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"1330_CR20","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF00120666","volume":"1","author":"H Konno","year":"1991","unstructured":"Konno, H., Yajima, Y., Matsui, T.: Parametric simplex algorithms for solving a special class of nonconvex minimization problems. J. Glob. Optim. 1 (1), 65\u201381 (1991)","journal-title":"J. Glob. Optim."},{"issue":"1","key":"1330_CR21","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/BF01096534","volume":"4","author":"H Konno","year":"1994","unstructured":"Konno, H., Kuno, T., Yajima, Y.: Global minimization of a generalized convex multiplicative function. J. Glob. Optim. 4(1), 47\u201362 (1994)","journal-title":"J. Glob. Optim."},{"issue":"4","key":"1330_CR22","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/BF00130830","volume":"1","author":"N Van Thoai","year":"1991","unstructured":"Van Thoai, N.: A global optimization approach for solving the convex multiplicative programming problem. J. Glob. Optim. 1(4), 341\u2013357 (1991)","journal-title":"J. Glob. Optim."},{"key":"1330_CR23","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.cam.2015.11.009","volume":"296","author":"S Liu","year":"2016","unstructured":"Liu, S., Zhao, Y.: An efficient algorithm for globally solving generalized linear multiplicative programming. J. Comput. Appl. Math. 296, 840\u2013847 (2016)","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1330_CR24","first-page":"494","volume":"179","author":"Y Gao","year":"2006","unstructured":"Gao, Y., Xu, C., Yang, Y.: An outcome-space finite algorithm for solving linear multiplicative programming. Appl. Math. Comput. 179(2), 494\u2013505 (2006)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1330_CR25","first-page":"1113","volume":"70","author":"H Jiao","year":"2009","unstructured":"Jiao, H.: A branch and bound algorithm for globally solving a class of nonconvex programming problems. Nonlinear Analysis: Theory. Methods Appl. 70 (2), 1113\u20131123 (2009)","journal-title":"Methods Appl."},{"key":"1330_CR26","doi-asserted-by":"publisher","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, 453\u2013482 (2020)","journal-title":"J. Glob. Optim."},{"key":"1330_CR27","doi-asserted-by":"crossref","unstructured":"Jiao, H.-W., Shang, Y.-L.: Two-level linear relaxation method for generalized linear fractional programming. J. Oper. Res. Soc. China, pp. 1\u201326 (2022)","DOI":"10.1007\/s40305-021-00375-4"},{"issue":"2","key":"1330_CR28","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1080\/00207160.2021.1909727","volume":"99","author":"H Jiao","year":"2022","unstructured":"Jiao, H., Shang, Y., Wang, W.: Solving generalized polynomial problem by using new affine relaxed technique. Int. J. Comput. Math. 99(2), 309\u2013331 (2022)","journal-title":"Int. J. Comput. Math."},{"key":"1330_CR29","doi-asserted-by":"crossref","unstructured":"Jiao, H., Shang, Y., Chen, R.: A potential practical algorithm for minimizing the sum of affine fractional functions. Optimization, pp. 1\u201331 (2022)","DOI":"10.1080\/02331934.2022.2032051"},{"issue":"1","key":"1330_CR30","first-page":"195","volume":"18","author":"H Jiao","year":"2022","unstructured":"Jiao, H., Ma, J., Shang, Y.: Image space branch-and-bound algorithm for globally solving minimax linear fractional programming problem. Pacific J. Optim. 18(1), 195\u2013212 (2022)","journal-title":"Pacific J. Optim."},{"issue":"1","key":"1330_CR31","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/BF00120663","volume":"1","author":"H Tuy","year":"1991","unstructured":"Tuy, H.: Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms. J. Glob. Optim. 1(1), 23\u201336 (1991)","journal-title":"J. Glob. Optim."},{"key":"1330_CR32","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.cam.2015.11.009","volume":"296","author":"S Liu","year":"2016","unstructured":"Liu, S., Zhao, Y.: An efficient algorithm for globally solving generalized linear multiplicative programming. J. Comput. Appl. Math. 296, 840\u2013847 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"1330_CR33","doi-asserted-by":"publisher","first-page":"315","DOI":"10.3390\/math8030315","volume":"8","author":"B Zhang","year":"2020","unstructured":"Zhang, B., Gao, Y., Liu, X., Huang, X.: Output-space branch-and-bound reduction algorithm for a class of linear multiplicative programs. Mathematics 8, 315 (2020)","journal-title":"Mathematics"},{"issue":"3","key":"1330_CR34","doi-asserted-by":"publisher","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(3), 693\u2013710 (2020)","journal-title":"Optim. Lett."},{"issue":"3","key":"1330_CR35","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1080\/02331934.2016.1269765","volume":"66","author":"C Wang","year":"2017","unstructured":"Wang, C., Bai, Y., Shen, P.: A practicable branch-and-bound algorithm for globally solving linear multiplicative programming. Optimization 66(3), 397\u2013405 (2017)","journal-title":"Optimization"},{"issue":"7","key":"1330_CR36","doi-asserted-by":"publisher","first-page":"1008","DOI":"10.1016\/j.cor.2010.10.016","volume":"38","author":"C-F Wang","year":"2011","unstructured":"Wang, C.-F., Liu, S.-Y.: A new linearization method for generalized linear multiplicative programming. Comput. Oper. Res. 38(7), 1008\u20131013 (2011)","journal-title":"Comput. Oper. Res."},{"issue":"9","key":"1330_CR37","doi-asserted-by":"publisher","first-page":"2573","DOI":"10.1016\/j.cor.2008.11.002","volume":"36","author":"Y Chen","year":"2009","unstructured":"Chen, Y., Jiao, H.: A nonisolated optimal solution of general linear multiplicative programming problems. Comput. Oper. Res. 36(9), 2573\u20132579 (2009)","journal-title":"Comput. Oper. Res."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01330-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01330-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01330-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,12]],"date-time":"2023-01-12T08:31:51Z","timestamp":1673512311000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01330-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["1330"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01330-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,24]]},"assertion":[{"value":"24 June 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2022","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 declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}