{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T06:10:14Z","timestamp":1744179014506},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,11,28]],"date-time":"2016-11-28T00:00:00Z","timestamp":1480291200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"National Natural Science Foundation of China (CN)","award":["11171362","11571055"],"award-info":[{"award-number":["11171362","11571055"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2017,11]]},"DOI":"10.1007\/s10957-016-1029-4","type":"journal-article","created":{"date-parts":[[2016,11,28]],"date-time":"2016-11-28T20:31:53Z","timestamp":1480365113000},"page":"527-544","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Separation Functions and Optimality Conditions in Vector Optimization"],"prefix":"10.1007","volume":"175","author":[{"given":"Manxue","family":"You","sequence":"first","affiliation":[]},{"given":"Shengjie","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,11,28]]},"reference":[{"key":"1029_CR1","unstructured":"Castellani, G., Giannessi, F.: Decomposition of mathematical programs by means of theorems of alternative for linear and nonlinear systems. In: Proceedings of the Ninth International Mathematical Programming Symposium, Budapest, Survey of Mathematical Programming, vol. 2, pp. 423-439. North-Holland, Amsterdam (1979)"},{"key":"1029_CR2","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF00935321","volume":"42","author":"F Giannessi","year":"1984","unstructured":"Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 42, 331\u2013365 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR3","doi-asserted-by":"crossref","DOI":"10.1007\/0-387-28020-0","volume-title":"Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions","author":"F Giannessi","year":"2005","unstructured":"Giannessi, F.: Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, Berlin (2005)"},{"key":"1029_CR4","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/978-1-4613-0299-5_11","volume-title":"Vector Variational Inequalities and Vector Equilibria","author":"F Giannessi","year":"2000","unstructured":"Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 153\u2013215. Kluwer, Dordrech (2000)"},{"key":"1029_CR5","doi-asserted-by":"crossref","first-page":"1901","DOI":"10.1080\/00036811.2011.614603","volume":"91","author":"G Mastroeni","year":"2012","unstructured":"Mastroeni, G.: Nonlinear separation in the image space with applications to penalty methods. Appl. Anal. 91, 1901\u20131914 (2012)","journal-title":"Appl. Anal."},{"key":"1029_CR6","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s10957-009-9598-0","volume":"144","author":"HZ Luo","year":"2010","unstructured":"Luo, H.Z., Mastroeni, G., Wu, H.X.: Separation approach for augmented Lagrangians in constrained nonconvex optimization. J. Optim. Theory Appl. 144, 275\u2013290 (2010)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR7","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1007\/s10957-013-0358-9","volume":"159","author":"HZ Luo","year":"2013","unstructured":"Luo, H.Z., Wu, H.X., Liu, J.Z.: Some results on augmented Lagrangians in constrained global optimization via image space analysis. J. Optim. Theory Appl. 159, 360\u2013385 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR8","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1007\/s10957-012-0027-4","volume":"154","author":"SJ Li","year":"2012","unstructured":"Li, S.J., Xu, Y.D., Zhu, S.K.: Nonlinear separation approach to constrained extremum problems. J. Optim. Theory Appl. 154, 842\u2013856 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR9","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s10957-013-0276-x","volume":"159","author":"J Li","year":"2013","unstructured":"Li, J., Feng, S.Q., Zhang, Z.: A unified approach for constrained extremum problems: image space analysis. J. Optim. Theory Appl. 159, 69\u201392 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR10","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1007\/s11590-013-0644-3","volume":"8","author":"YD Xu","year":"2014","unstructured":"Xu, Y.D., Li, S.J.: Nonlinear separation functions and constrained extremum problems. Optim. Lett. 8, 1149\u20131160 (2014)","journal-title":"Optim. Lett."},{"key":"1029_CR11","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s10898-008-9301-2","volume":"42","author":"F Giannessi","year":"2008","unstructured":"Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401\u2013412 (2008)","journal-title":"J. Glob. Optim."},{"key":"1029_CR12","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1007\/s10898-009-9445-8","volume":"46","author":"G Mastroeni","year":"2010","unstructured":"Mastroeni, G.: Some applications of the image space analysis to the duality theory for constrained extremum problems. J. Glob. Optim. 46, 603\u2013614 (2010)","journal-title":"J. Glob. Optim."},{"key":"1029_CR13","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s10957-009-9518-3","volume":"142","author":"A Moldovan","year":"2009","unstructured":"Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 1: sufficient optimality conditions. J. Optim. Theory Appl. 142, 147\u2013163 (2009)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR14","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10957-009-9521-8","volume":"142","author":"A Moldovan","year":"2009","unstructured":"Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2: necessary optimality conditions. J. Optim. Theory Appl. 142, 165\u2013183 (2009)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR15","doi-asserted-by":"crossref","first-page":"738","DOI":"10.1007\/s10957-013-0468-4","volume":"161","author":"SK Zhu","year":"2014","unstructured":"Zhu, S.K., Li, S.J.: United duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738\u2013762 (2014)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR16","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s10957-013-0467-5","volume":"161","author":"SK Zhu","year":"2014","unstructured":"Zhu, S.K., Li, S.J.: United duality theory for constrained extremum problems. Part II: special duality schemes. J. Optim. Theory Appl. 161, 763\u2013782 (2014)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR17","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1016\/0022-247X(68)90201-1","volume":"22","author":"AM Geoffrion","year":"1968","unstructured":"Geoffrion, A.M.: Proper efficiency and theory of vector maximization. J. Math. Anal. Appl. 22, 618\u2013630 (1968)","journal-title":"J. Math. Anal. Appl."},{"key":"1029_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71333-3","volume-title":"Nonsmooth Analysis","author":"W Schirotzek","year":"2007","unstructured":"Schirotzek, W.: Nonsmooth Analysis. Springer, Berlin (2007)"},{"key":"1029_CR19","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/BF00940478","volume":"67","author":"C Gerth","year":"1990","unstructured":"Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297\u2013320 (1990)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-54265-7","volume-title":"Set-Valued Optimization. An Introduction with Applications","author":"AA Khan","year":"2015","unstructured":"Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015)"},{"key":"1029_CR21","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s10957-012-0078-6","volume":"155","author":"RS Burachik","year":"2012","unstructured":"Burachik, R.S., Rizvi, M.M.: On weak and strong Kuhn\u2013Tucker conditions for smooth multiobjective optimization. J. Optim. Theory Appl. 155, 477\u2013491 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"1029_CR22","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1080\/02331930701761615","volume":"58","author":"M Dureaa","year":"2009","unstructured":"Dureaa, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449\u2013467 (2009)","journal-title":"Optimization"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-016-1029-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-016-1029-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-016-1029-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T02:51:04Z","timestamp":1692586264000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-016-1029-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,28]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,11]]}},"alternative-id":["1029"],"URL":"https:\/\/doi.org\/10.1007\/s10957-016-1029-4","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,11,28]]}}}