{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T03:05:46Z","timestamp":1772852746130,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,5,4]],"date-time":"2024-05-04T00:00:00Z","timestamp":1714780800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,5,4]],"date-time":"2024-05-04T00:00:00Z","timestamp":1714780800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Research Grant for Faculty","award":["6031"],"award-info":[{"award-number":["6031"]}]},{"name":"CSIR-UGC JRF, New Delhi, India","award":["221610093964"],"award-info":[{"award-number":["221610093964"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s11075-024-01840-w","type":"journal-article","created":{"date-parts":[[2024,5,4]],"date-time":"2024-05-04T06:01:26Z","timestamp":1714802486000},"page":"1467-1480","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimality and duality results for fractional programming problems under E-univexity"],"prefix":"10.1007","volume":"98","author":[{"given":"S. K.","family":"Mishra","sequence":"first","affiliation":[]},{"given":"D.","family":"Singh","sequence":"additional","affiliation":[]},{"family":"Pankaj","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,5,4]]},"reference":[{"key":"1840_CR1","doi-asserted-by":"publisher","first-page":"745","DOI":"10.22436\/jnsa.012.11.06","volume":"12","author":"T Antczak","year":"2019","unstructured":"Antczak, T., Abdulaleem, N.: E-optimality conditions and Wolfe E-duality for E-differentiable vector optimization problems with inequality and equality constraints. J. Nonlinear Sci. Appl 12, 745\u2013764 (2019)","journal-title":"J. Nonlinear Sci. Appl"},{"key":"1840_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13660-019-2237-x","volume":"2019","author":"T Antczak","year":"2019","unstructured":"Antczak, T., Abdulaleem, N.: Optimality and duality results for E-differentiable multiobjective fractional programming problems under E-convexity. Journal of Inequalities and Applications 2019, 1\u201324 (2019)","journal-title":"Journal of Inequalities and Applications"},{"key":"1840_CR3","doi-asserted-by":"publisher","first-page":"2125","DOI":"10.3906\/mat-1705-65","volume":"42","author":"T Antczak","year":"2018","unstructured":"Antczak, T., Verma, R.: Parametric nondifferentiable multiobjective fractional programming under $$(b,\\psi,\\phi,\\rho )$$-univexity. Turk. J. Math. 42, 2125\u20132147 (2018)","journal-title":"Turk. J. Math."},{"key":"1840_CR4","first-page":"183","volume":"17","author":"CR Bector","year":"1973","unstructured":"Bector, C.R.: Duality in nonlinear fractional programming. Z. Oper.-Res. 17, 183\u2013193 (1973)","journal-title":"Z. Oper.-Res."},{"key":"1840_CR5","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/BF00941889","volume":"79","author":"CR Bector","year":"1993","unstructured":"Bector, C.R., Chandra, S., Husain, I.: Optimality condition and duality in subdifferentiable multiobjective fractional programming. J. Optim. Theory Appl. 79, 105\u2013125 (1993)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR6","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1016\/S0022-247X(02)00248-2","volume":"273","author":"X Chen","year":"2002","unstructured":"Chen, X.: Optimality and duality for the multiobjective fractional programming with the generalized (F,$$\\rho $$)-convexity. J. Math. Anal. Appl. 273, 190\u2013205 (2002)","journal-title":"J. Math. Anal. Appl."},{"key":"1840_CR7","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/BF02591908","volume":"27","author":"JP Crouzeix","year":"1983","unstructured":"Crouzeix, J.P., Ferland, J.A., Schaible, S.: Duality in generalized fractional programming. Math. Program. 27, 343\u2013354 (1983)","journal-title":"Math. Program."},{"key":"1840_CR8","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/BF00941314","volume":"47","author":"JP Crouzeix","year":"1985","unstructured":"Crouzeix, J.P., Ferland, J.A., Schaible, S.: An algorithm for generalized fractional programs. J. Optim. Theory Appl. 47, 35\u201349 (1985)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR9","first-page":"492","volume":"13","author":"W Dinkelbach","year":"1967","unstructured":"Dinkelbach, W.: On nonlinear fractional programming. Management Sci. 13, 492\u2013498 (1967)","journal-title":"On nonlinear fractional programming. Management Sci."},{"key":"1840_CR10","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1287\/mnsc.12.7.609","volume":"12","author":"R Jagannathan","year":"1966","unstructured":"Jagannathan, R.: On some properties of programming problems in parametric form pertaining to fractional programming. Management Sci. 12, 609\u2013615 (1966)","journal-title":"Management Sci."},{"key":"1840_CR11","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1023\/A:1017540412396","volume":"110","author":"ZA Liang","year":"2001","unstructured":"Liang, Z.A., Huang, H.X., Pardalos, P.M.: Optimality conditions and duality for a class of nonlinear fractional programming problems. J. Optim. Theory Appl. 110, 611\u2013619 (2001)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR12","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1023\/A:1026041403408","volume":"27","author":"ZA Liang","year":"2003","unstructured":"Liang, Z.A., Huang, H.X., Pardalos, P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 27, 447\u2013471 (2003)","journal-title":"J. Glob. Optim."},{"key":"1840_CR13","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1080\/02331939608844188","volume":"36","author":"JC Liu","year":"1996","unstructured":"Liu, J.C.: Optimality and duality for multiobjective fractional programming involving nonsmooth (F,$$\\rho $$)-convex functions. Optimization 36, 333\u2013346 (1996)","journal-title":"Optimization"},{"key":"1840_CR14","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1080\/02331939608844194","volume":"37","author":"JC Liu","year":"1996","unstructured":"Liu, J.C.: Optimality and duality for multiobjective fractional programming involving nonsmooth pseudoinvex functions. Optimization 37, 27\u201339 (1996)","journal-title":"Optimization"},{"key":"1840_CR15","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/s10957-010-9740-z","volume":"148","author":"XJ Long","year":"2011","unstructured":"Long, X.J.: Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with $$(C,\\alpha,\\rho, d)$$-convexity. J. Optim. Theory Appl. 148, 197\u2013208 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR16","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0022-247X(91)90151-O","volume":"162","author":"RN Mukherjee","year":"1991","unstructured":"Mukherjee, R.N.: Generalized convex duality for multiobjective fractional programs. J. Math. Anal. Appl. 162, 309\u2013316 (1991)","journal-title":"J. Math. Anal. Appl."},{"key":"1840_CR17","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s10898-007-9168-7","volume":"41","author":"S Nobakhtian","year":"2008","unstructured":"Nobakhtian, S.: Optimality and duality for nonsmooth multiobjective fractional programming with mixed constraints. J. Glob. Optim. 41, 103\u2013115 (2008)","journal-title":"J. Glob. Optim."},{"key":"1840_CR18","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF02564830","volume":"8","author":"R Osuna-G\u00f3mez","year":"2000","unstructured":"Osuna-G\u00f3mez, R., Rufi\u00e1n-Lizana, A., Ruiz-Canales, P.: Multiobjective fractional programming with generalized convexity. TOP 8, 97\u2013110 (2000)","journal-title":"TOP"},{"key":"1840_CR19","doi-asserted-by":"crossref","unstructured":"Schaible, S.: Fractional programming. I, duality. Manag. Sci. 22, 858\u2013867 (1976)","DOI":"10.1287\/mnsc.22.8.858"},{"key":"1840_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-0035-6","volume-title":"Fractional Programming: Theory","author":"IM Stancu-Minasian","year":"1997","unstructured":"Stancu-Minasian, I.M.: Fractional Programming: Theory. Methods and Applications. Kluwer Academic, Dordrecht (1997)"},{"key":"1840_CR21","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1080\/02331934.2016.1276179","volume":"66","author":"IM Stancu-Minasian","year":"2017","unstructured":"Stancu-Minasian, I.M.: A eighth bibliography of fractional programming. Optimization 66, 439\u2013470 (2017)","journal-title":"Optimization"},{"key":"1840_CR22","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1023\/A:1021792726715","volume":"102","author":"EA Youness","year":"1999","unstructured":"Youness, E.A.: E-Convex sets, E-convex functions, and E-convex programming. J. Optim. Theory Appl. 102, 439\u2013450 (1999)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR23","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1080\/02331939508844040","volume":"32","author":"GJ Zalmai","year":"1995","unstructured":"Zalmai, G.J.: Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and $$\\rho $$-convex functions. Optimization 32, 95\u2013124 (1995)","journal-title":"Optimization"},{"key":"1840_CR24","first-page":"594","volume":"30","author":"W Zezheng","year":"2007","unstructured":"Zezheng, W., Fenghua, Z.: Optimality and duality for a class of nonlinear fractional programming problems. J Sichuan Norm. Univ. 30, 594\u2013597 (2007)","journal-title":"J Sichuan Norm. Univ."},{"key":"1840_CR25","doi-asserted-by":"crossref","unstructured":"Zhang, X., Wu, Z.: Optimality conditions and duality of three kinds of nonlinear fractional programming problems. Adv. Oper. Res. 2013, Article ID 708979 (2013)","DOI":"10.1155\/2013\/708979"},{"issue":"2","key":"1840_CR26","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","volume":"80","author":"MA Hanson","year":"1981","unstructured":"Hanson, M.A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80(2), 545\u2013550 (1981)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1840_CR27","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF00939919","volume":"71","author":"CR Bector","year":"1991","unstructured":"Bector, C.R., Singh, C.: B-vex functions. J. Optim. Theory Appl. 71(2), 237\u2013253 (1991)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"1840_CR28","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1007\/BF00939383","volume":"76","author":"CR Bector","year":"1993","unstructured":"Bector, C.R., Suneja, S.K., Lalitha, C.S.: Generalized B-vex functions and generalized B-vex programming. J. Optim. Theory Appl. 76(3), 561\u2013576 (1993)","journal-title":"J. Optim. Theory Appl."},{"key":"1840_CR29","doi-asserted-by":"crossref","unstructured":"Bector, C. R., Chandra, S., Gupta, S., and Suneja, S. K. . Univex sets, functions and univex nonlinear programming. In Generalized Convexity: Proceedings of the IVth International Workshop on Generalized Convexity Held at Janus Pannonius University P\u00e9cs, Hungary, August 31\u2013September 2, 1992 (pp. 3\u201318). Springer Berlin Heidelberg, (1994)","DOI":"10.1007\/978-3-642-46802-5_1"},{"issue":"2","key":"1840_CR30","doi-asserted-by":"publisher","first-page":"1513","DOI":"10.3934\/jimo.2022004","volume":"19","author":"N Abdulaleem","year":"2023","unstructured":"Abdulaleem, N.: Optimality and duality for E -differentiable multiobjective programming problems involving E -type I functions. Journal of Industrial and Management Optimization 19(2), 1513\u20131527 (2023)","journal-title":"Journal of Industrial and Management Optimization"},{"issue":"1","key":"1840_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1029-242X-2013-246","volume":"2013","author":"AEMA Megahed","year":"2013","unstructured":"Megahed, A.E.M.A., Gomma, H.G., Youness, E.A., El-Banna, A.Z.H.: Optimality conditions of E-convex programming for an E-differentiable function. Journal of Inequalities and Applications 2013(1), 1\u201311 (2013)","journal-title":"Journal of Inequalities and Applications"},{"issue":"12","key":"1840_CR32","first-page":"5525","volume":"217","author":"SK Mishra","year":"2011","unstructured":"Mishra, S.K., Mohapatra, R.N., Youness, E.A.: Some properties of semi E-b-vex functions. Appl. Math. Comput. 217(12), 5525\u20135530 (2011)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"1840_CR33","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s11590-008-0099-0","volume":"2","author":"SK Mishra","year":"2009","unstructured":"Mishra, S.K., Rautela, J.S.: On nonlinear multiple objective fractional programming involving semilocally type-I univex functions. Optimization Letters 2(3), 171\u2013185 (2009)","journal-title":"Optimization Letters"},{"issue":"1","key":"1840_CR34","first-page":"5","volume":"47","author":"T Antczak","year":"2018","unstructured":"Antczak, T., Mishra, S.K., Upadhyay, B.B.: Optimality conditions and duality for generalized fractional minimax programming involving locally Lipschitz $$(b, \\psi, \\phi, \\rho )$$-univex functions. Control. Cybern. 47(1), 5\u201332 (2018)","journal-title":"Control. Cybern."},{"issue":"1","key":"1840_CR35","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1016\/S0377-2217(03)00439-9","volume":"160","author":"SK Mishra","year":"2005","unstructured":"Mishra, S.K., Wang, S.Y., Lai, K.K.: Nondifferentiable multiobjective programming under generalized d-univexity. Eur. J. Oper. Res. 160(1), 218\u2013226 (2005)","journal-title":"Eur. J. Oper. Res."},{"key":"1840_CR36","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1007\/s40305-019-00263-y","volume":"8","author":"BC Joshi","year":"2020","unstructured":"Joshi, B.C., Mishra, S.K., Kumar, P.: On semi-infinite mathematical programming problems with equilibrium constraints using generalized convexity. Journal of the Operations Research Society of China 8, 619\u2013636 (2020)","journal-title":"Journal of the Operations Research Society of China"},{"issue":"3","key":"1840_CR37","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1023\/A:1017532225395","volume":"109","author":"XM Yang","year":"2001","unstructured":"Yang, X.M.: On E-convex sets, E-convex functions, and E-convex programming. J. Optim. Theory Appl. 109(3), 699 (2001)","journal-title":"J. Optim. Theory Appl."},{"issue":"9","key":"1840_CR38","doi-asserted-by":"publisher","first-page":"1737","DOI":"10.1016\/S0960-0779(00)00036-9","volume":"12","author":"EA Youness","year":"2001","unstructured":"Youness, E.A.: Optimality criteria in E-convex programming. Chaos, Solitons Fractals 12(9), 1737\u20131745 (2001)","journal-title":"Chaos, Solitons Fractals"},{"key":"1840_CR39","doi-asserted-by":"crossref","unstructured":"Abdulaleem, N. E-invexity and generalized E-invexity in E-differentiable multiobjective programming. In ITM Web of Conferences (Vol. 24, p. 01002). EDP Sciences, (2019)","DOI":"10.1051\/itmconf\/20192401002"},{"key":"1840_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.rico.2021.100046","volume":"4","author":"N Abdulaleem","year":"2021","unstructured":"Abdulaleem, N.: EB-invexity in E-differentiable mathematical programming. Results in Control and Optimization 4, 100046 (2021)","journal-title":"Results in Control and Optimization"},{"issue":"1","key":"1840_CR41","first-page":"3","volume":"78","author":"N Abdulaleem","year":"2023","unstructured":"Abdulaleem, N.: E-Univex Sets, E-Univex Functions and E-Differentiable E-Univex Programming. RM 78(1), 3 (2023)","journal-title":"RM"},{"key":"1840_CR42","unstructured":"Stancu, A. M. Mathematical programming with type-I functions. Matrix Rom, (2013)"},{"issue":"2","key":"1840_CR43","first-page":"309","volume":"15","author":"IM Stancu-Minasian","year":"2013","unstructured":"Stancu-Minasian, I.M.: A seventh bibliography of fractional programming. Adv. Model. Optim 15(2), 309\u2013386 (2013)","journal-title":"Adv. Model. Optim"},{"key":"1840_CR44","doi-asserted-by":"crossref","unstructured":"Stancu-Minasian, I. M. A ninth bibliography of fractional programming, (2019)","DOI":"10.1080\/02331934.2019.1632250"},{"issue":"6","key":"1840_CR45","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1007\/s11590-014-0783-1","volume":"11","author":"AM Stancu","year":"2017","unstructured":"Stancu, A.M., Stancu-Minasian, I.M.: Sufficient efficiency criteria in multiobjective fractional programming with generalized F, b, $$\\psi,\\rho,\\theta $$-univex n-set functions. Optimization Letters 11(6), 1029\u20131045 (2017)","journal-title":"Optimization Letters"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01840-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01840-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01840-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,12]],"date-time":"2025-02-12T08:49:34Z","timestamp":1739350174000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01840-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,4]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["1840"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01840-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,4]]},"assertion":[{"value":"11 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 April 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2024","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 that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}