{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T18:08:32Z","timestamp":1759342112391,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T00:00:00Z","timestamp":1635811200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T00:00:00Z","timestamp":1635811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s12190-021-01655-1","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T10:04:55Z","timestamp":1635847495000},"page":"3041-3066","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On sufficiency and duality theorems for nonsmooth semi-infinite mathematical programming problem with equilibrium constraints"],"prefix":"10.1007","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9828-554X","authenticated-orcid":false,"given":"Tran","family":"Van Su","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dinh Dieu","family":"Hang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"key":"1655_CR1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-08114-4","volume-title":"Introduction to Nonsmooth Optimization: Theory, Practice and Software","author":"A Bagirov","year":"2014","unstructured":"Bagirov, A., Karmitsa, N., Makela, M.M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, New York (2014)"},{"issue":"2","key":"1655_CR2","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/j.na.2010.03.026","volume":"73","author":"RI Bot","year":"2010","unstructured":"Bot, R.I., Grad, S.-M.: Wolfe duality and Mond\u2013Weir duality via perturbations. Nonlinear Anal. Theory Methods Appl. 73(2), 374\u2013384 (2010)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"1655_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/s10479-017-2734-6","author":"TD Chuong","year":"2018","unstructured":"Chuong, T.D.: Optimality conditions for nonsmooth multiobjective bilevel optimization problems. Ann. Oper. Res. (2018). https:\/\/doi.org\/10.1007\/s10479-017-2734-6","journal-title":"Ann. Oper. Res."},{"key":"1655_CR4","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s10479-007-0176-2","volume":"153","author":"B Colson","year":"2007","unstructured":"Colson, B., Marcotte, P., Savard, G.: A overview of bilevel optimization. Ann. Oper. Res. 153, 235\u2013256 (2007)","journal-title":"Ann. Oper. Res."},{"key":"1655_CR5","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s10479-011-1023-z","volume":"196","author":"S Dempe","year":"2012","unstructured":"Dempe, S., Zemkoho, A.B.: Bilevel road pricing: theoretical analysis and optimality conditions. Ann. Oper. Res. 196, 223\u2013240 (2012)","journal-title":"Ann. Oper. Res."},{"key":"1655_CR6","first-page":"4119","volume":"218","author":"A Jayswal","year":"2011","unstructured":"Jayswal, A., Stancu-Minasian, I., Ahmad, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218, 4119\u20134127 (2011)","journal-title":"Appl. Math. Comput."},{"key":"1655_CR7","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s12215-015-0215-9","volume":"65","author":"A Jayswal","year":"2016","unstructured":"Jayswal, A., Stancu-Minasian, I., Banerjee, J.: Optimality conditions and duality for interval-valued optimization problems using convexificators. Rend. Cire. Mat. Palermo 65, 17\u201332 (2016)","journal-title":"Rend. Cire. Mat. Palermo"},{"issue":"3","key":"1655_CR8","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1080\/02331930601120516","volume":"57","author":"B Jim\u00e9nez","year":"2008","unstructured":"Jim\u00e9nez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449\u2013471 (2008)","journal-title":"Optimization"},{"key":"1655_CR9","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1016\/j.jmaa.2008.11.045","volume":"352","author":"B Jim\u00e9nez","year":"2009","unstructured":"Jim\u00e9nez, B., Novo, V., Sama, M.: Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives. J. Math. Anal. Appl. 352, 788\u2013798 (2009)","journal-title":"J. Math. Anal. Appl."},{"key":"1655_CR10","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3403-4","volume-title":"Semi-infinite Programming: Recent Advances","author":"MA Goberna","year":"2001","unstructured":"Goberna, M.A., L\u00f3pez, M.A.: Semi-infinite Programming: Recent Advances. Kluwer, Dordrecht (2001)"},{"key":"1655_CR11","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1137\/1035089","volume":"35","author":"R Hettich","year":"1993","unstructured":"Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods and applications. SIAM Rev. 35, 380\u2013429 (1993)","journal-title":"SIAM Rev."},{"key":"1655_CR12","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.ejor.2006.08.045","volume":"180","author":"MA L\u00f3pez","year":"2007","unstructured":"L\u00f3pez, M.A., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491\u2013518 (2007)","journal-title":"Eur. J. Oper. Res."},{"key":"1655_CR13","series-title":"Lect. notes in Eco. and Math. systems","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-50280-4","volume-title":"Theory of Vector Optimization","author":"DT Luc","year":"1989","unstructured":"Luc, D.T.: Theory of Vector Optimization. Lect. notes in Eco. and Math. systems, vol. 319. Springer, Berlin (1989)"},{"key":"1655_CR14","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01594928","volume":"50","author":"DT Luc","year":"1991","unstructured":"Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99\u2013111 (1991)","journal-title":"Math. Program."},{"key":"1655_CR15","volume-title":"Mathematical Problems with Equilibrium Constraints","author":"ZQ Luo","year":"1996","unstructured":"Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Problems with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)"},{"key":"1655_CR16","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10288-017-0369-8","volume":"16","author":"D.V Luu","year":"2018","unstructured":"Luu, D..V., Mai, T..V.: Optimality and duality in constrained interval-valued optimization. 4OR-Q. J. Oper. Res 16, 311\u2013327 (2018)","journal-title":"4OR-Q. J. Oper. Res"},{"key":"1655_CR17","doi-asserted-by":"crossref","first-page":"1622","DOI":"10.1080\/01630563.2015.1078813","volume":"36","author":"DV Luu","year":"2015","unstructured":"Luu, D.V., Hang, D.D.: On efficiency conditions for nonsmooth vector equilibrium problems with equilibrium constraints. Numer. Funct. Anal. Optim. 36, 1622\u20131642 (2015)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"1655_CR18","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1051\/ro\/2017042","volume":"52","author":"DV Luu","year":"2018","unstructured":"Luu, D.V., Su, T.V.: Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints. RAIRO Oper. Res. 52, 543\u2013559 (2018)","journal-title":"RAIRO Oper. Res."},{"key":"1655_CR19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13660-016-1272-0","volume":"2017","author":"S-M Guu","year":"2017","unstructured":"Guu, S.-M., Singh, Y., Mishra, S.K.: On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints. J. Inequal. Appl. 2017, 1 (2017)","journal-title":"J. Inequal. Appl."},{"issue":"5","key":"1655_CR20","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1007\/s11590-013-0707-5","volume":"8","author":"V Laha","year":"2014","unstructured":"Laha, V., Al-Shamary, B., Mishra, S.K.: On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel-Penot subdifferentials. Optim. Lett. 8(5), 1675\u20131690 (2014)","journal-title":"Optim. Lett."},{"key":"1655_CR21","volume-title":"Generallized Concavity and Duality, Generallized Concavity in Optimization and Economics","author":"M Mond","year":"1981","unstructured":"Mond, M., Weir, T.: Generallized Concavity and Duality, Generallized Concavity in Optimization and Economics. Academic Press, New York (1981)"},{"key":"1655_CR22","doi-asserted-by":"crossref","first-page":"2694","DOI":"10.1016\/j.na.2009.11.015","volume":"72","author":"N Movahedian","year":"2010","unstructured":"Movahedian, N., Nabakhtian, S.: Necessary and sufficient conditions for nonsmooth mathematical problems with equilibrium constraints. Nonlinear Anal. 72, 2694\u20132705 (2010)","journal-title":"Nonlinear Anal."},{"key":"1655_CR23","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2868-2","volume-title":"Semi-infinite Programming","author":"R Reemtsen","year":"1998","unstructured":"Reemtsen, R., Ruckmann, J.J.: Semi-infinite Programming. Kluwer, Dordrecht (1998)"},{"key":"1655_CR24","doi-asserted-by":"crossref","first-page":"640","DOI":"10.1080\/01630563.2016.1155158","volume":"37","author":"TV Su","year":"2016","unstructured":"Su, T.V.: Optimality conditions for vector equilibrium problems in terms of contingent epiderivatives. Numer. Funct. Anal. Optim. 37, 640\u2013665 (2016)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"1655_CR25","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10288-017-0360-4","volume":"16","author":"TV Su","year":"2018","unstructured":"Su, T.V.: New optimality conditions for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces. 4OR- Q. J. Oper. Res. 16, 173\u2013198 (2018)","journal-title":"4OR- Q. J. Oper. Res."},{"key":"1655_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-011-9819-1","volume":"150","author":"SK Suneja","year":"2011","unstructured":"Suneja, S.K., Kohli, B.: Optimality and duality results for bilevel programming problem using convexificators. J. Optim. Theory Appl. 150, 1\u201319 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"1655_CR27","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.jmaa.2004.10.032","volume":"307","author":"JJ Ye","year":"2005","unstructured":"Ye, J.J.: Necessary and sufficient optimality conditions for mathematical program with equilibrium constraints. J. Math. Anal. Appl. 307, 350\u2013369 (2005)","journal-title":"J. Math. Anal. Appl."},{"key":"1655_CR28","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1090\/qam\/135625","volume":"19","author":"P Wolfe","year":"1961","unstructured":"Wolfe, P.: A duality theorem for nonlinear programming. Q. J. Appl. Math. 19, 239\u2013244 (1961)","journal-title":"Q. J. Appl. Math."},{"key":"1655_CR29","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/j.jmaa.2007.05.023","volume":"338","author":"HC Wu","year":"2008","unstructured":"Wu, H.C.: On interval-valued nonlinear programming problems. J. Math. Anal. Appl. 338, 299\u2013316 (2008)","journal-title":"J. Math. Anal. Appl."},{"key":"1655_CR30","volume-title":"Set-Valued Analysis","author":"J-P Aubin","year":"1990","unstructured":"Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)"},{"key":"1655_CR31","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1016\/S0893-9659(03)90114-5","volume":"16","author":"J Jahn","year":"2013","unstructured":"Jahn, J., Khan, A.A.: The existence of contingent epiderivatives for set-valued maps. Appl. Math. Lett. 16, 1179\u20131185 (2013)","journal-title":"Appl. Math. Lett."},{"issue":"2","key":"1655_CR32","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/0233193031000079865","volume":"52","author":"J Jahn","year":"2003","unstructured":"Jahn, J., Khan, A.A.: Some calculus rules for contingent epiderivatives. Optimization 52(2), 113\u2013125 (2003)","journal-title":"Optimization"},{"key":"1655_CR33","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01217690","volume":"46","author":"J Jahn","year":"1997","unstructured":"Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Meth. Oper. Res. 46, 193\u2013211 (1997)","journal-title":"Math. Meth. Oper. Res."},{"key":"1655_CR34","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.jmaa.2006.04.060","volume":"327","author":"L Rodr\u00edguez-Mar\u00edn","year":"2007","unstructured":"Rodr\u00edguez-Mar\u00edn, L., Sama, M.: About contingent epiderivatives. J. Math. Anal. Appl. 327, 745\u2013762 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"1655_CR35","doi-asserted-by":"crossref","first-page":"1374","DOI":"10.1016\/j.jmaa.2007.01.110","volume":"335","author":"L Rodr\u00edguez-Mar\u00edn","year":"2007","unstructured":"Rodr\u00edguez-Mar\u00edn, L.: Variational characterization of the contingent epiderivative. J. Math. Anal. Appl. 335, 1374\u20131382 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"1655_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/s11117-021-00842-5","author":"TV Su","year":"2021","unstructured":"Su, T.V., Hang, D.D.: Optimality conditions in terms of contingent epiderivatives for strict local Pareto minima in vector optimization problems with constraints. Positivity (2021). https:\/\/doi.org\/10.1007\/s11117-021-00842-5","journal-title":"Positivity"},{"issue":"127","key":"1655_CR37","first-page":"1","volume":"39","author":"TV Su","year":"2020","unstructured":"Su, T.V., Hang, D.D.: Duality results for interval-valued pseudoconvex optimization problem with equilibrium constraints with applications. Comput. Appl. Math. 39(127), 1\u201324 (2020)","journal-title":"Comput. Appl. Math."},{"key":"1655_CR38","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1007\/s10479-017-2422-6","volume":"269","author":"Y Pandey","year":"2018","unstructured":"Pandey, Y., Mishra, S.K.: Optimality conditions and duality for semi-infinite mathematical programming problems with equilibrium constraints, using convexificators. Ann. Oper. Res. 269, 549\u2013564 (2018)","journal-title":"Ann. Oper. Res."},{"key":"1655_CR39","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1007\/s10957-016-0885-2","volume":"17","author":"Y Pandey","year":"2016","unstructured":"Pandey, Y., Mishra, S.K.: Duality for nonsmooth optimization problems with equilibrium constraints, using convexificators. J. Optim. Theory Appl. 17, 694\u2013707 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"1655_CR40","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s11590-010-0240-8","volume":"6","author":"SK Mishra","year":"2012","unstructured":"Mishra, S.K., Jayswal, M., An, L.T.M.: Duality for nonsmooth semi-infinite programming problems. Optim. Lett. 6, 261\u2013271 (2012)","journal-title":"Optim. Lett."},{"key":"1655_CR41","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1186\/s13660-016-0969-4","volume":"2016","author":"S-M Guu","year":"2016","unstructured":"Guu, S.-M., Mishra, S.K., Pandey, Y.: Duality for nonsmooth mathematical programming problems with equilibrium constraints. J. Inequal. Appl. 2016, 28 (2016)","journal-title":"J. Inequal. Appl."},{"key":"1655_CR42","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1080\/01630563.2015.1013552","volume":"36","author":"SK Mishra","year":"2015","unstructured":"Mishra, S.K., Jayswal, M.: Optimality conditions and duality for semi-infinite mathematical programming problem with equilibrium constraints. Numer. Func. Anal. Optim. 36, 460\u2013480 (2015)","journal-title":"Numer. Func. Anal. Optim."},{"issue":"37","key":"1655_CR43","first-page":"1","volume":"40","author":"TV Su","year":"2021","unstructured":"Su, T.V., Hang, D.D., Dieu, N.C.: Optimality conditions and duality in terms of convexificators for multiobjective bilevel programming problem with equilibrium constraints. Comput. Appl. Math. 40(37), 1\u201326 (2021)","journal-title":"Comput. Appl. Math."},{"key":"1655_CR44","volume-title":"Nonlinear Programming","author":"OL Mangasarian","year":"1969","unstructured":"Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)"},{"key":"1655_CR45","doi-asserted-by":"publisher","DOI":"10.1007\/s10288-021-00482-1","author":"T Antczak","year":"2021","unstructured":"Antczak, T.: Optimality conditions and Mond\u2013Weir duality for a class of differentiable semi-infinitemultiobjective programming problems with vanishing constraints. 4OR-Q. J. Oper. Res. (2021). https:\/\/doi.org\/10.1007\/s10288-021-00482-1","journal-title":"4OR-Q. J. Oper. Res."},{"key":"1655_CR46","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-021-01655-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-021-01655-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-021-01655-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T04:05:52Z","timestamp":1699761952000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-021-01655-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,2]]},"references-count":46,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1655"],"URL":"https:\/\/doi.org\/10.1007\/s12190-021-01655-1","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2021,11,2]]},"assertion":[{"value":"7 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 November 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This manuscript does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by both authors. The authors read and approved the final manuscript.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed consent"}}]}}