{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T15:52:11Z","timestamp":1762444331795},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2013,2,17]],"date-time":"2013-02-17T00:00:00Z","timestamp":1361059200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2014,3]]},"DOI":"10.1007\/s11590-013-0620-y","type":"journal-article","created":{"date-parts":[[2013,2,16]],"date-time":"2013-02-16T06:14:34Z","timestamp":1360995274000},"page":"1047-1061","source":"Crossref","is-referenced-by-count":14,"title":["$$\\epsilon $$ -Optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces"],"prefix":"10.1007","volume":"8","author":[{"given":"Zhi-Ang","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin-Min","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian-Wen","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2013,2,17]]},"reference":[{"key":"620_CR1","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/BF01584336","volume":"13","author":"JM Borwein","year":"1977","unstructured":"Borwein, J.M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13, 183\u2013199 (1977)","journal-title":"Math. Program."},{"key":"620_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 alterative and optimality conditions. J. Optim. Theory Appl. 42, 331\u2013365 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR3","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s001860050026","volume":"48","author":"WD Rong","year":"1998","unstructured":"Rong, W.D., Wu, Y.N.: Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps. Math. Method. Oper. Res. 48, 247\u2013258 (1998)","journal-title":"Math. Method. Oper. Res."},{"key":"620_CR4","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1023\/A:1022676013609","volume":"98","author":"ZF Li","year":"1998","unstructured":"Li, Z.F.: Benson proper efficiency in the vector optimization of set-valued maps. J. Optim. Theory Appl. 98, 623\u2013649 (1998)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR5","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1023\/A:1026407517917","volume":"107","author":"XM Yang","year":"2000","unstructured":"Yang, X.M., Yang, X.Q., Chen, G.Y.: Theorems of the alternative and optimization with set-valued maps. J. Optim. Theory Appl. 107, 627\u2013640 (2000)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR6","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s10957-004-1716-4","volume":"125","author":"PH Sach","year":"2005","unstructured":"Sach, P.H.: New generalized convexity notion for set-valued maps and application to vector optimization. J. Optim. Theory Appl. 125, 157\u2013179 (2005)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR7","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1023\/A:1017535631418","volume":"110","author":"XM Yang","year":"2001","unstructured":"Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413\u2013427 (2001)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR8","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1023\/A:1004657412928","volume":"106","author":"WD Rong","year":"2000","unstructured":"Rong, W.D., Wu, Y.N.: $\\epsilon $-Weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569\u2013579 (2000)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR9","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s00186-005-0007-7","volume":"62","author":"A Taa","year":"2005","unstructured":"Taa, A.: $\\epsilon $-Subdifferential of set-valued maps and $\\epsilon $-weak Pareto optimality for multiobjective optimization. Math. Method. Oper. Res. 62, 187\u2013209 (2005)","journal-title":"Math. Method. Oper. Res."},{"key":"620_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10898-009-9452-9","volume":"47","author":"QS Qiu","year":"2010","unstructured":"Qiu, Q.S., Yang, X.M.: Some properties of approximate solutions for vector optimization problem with set-valued functions. J. Glob. Optim. 47, 1\u201312 (2010)","journal-title":"J. Glob. Optim."},{"key":"620_CR11","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s10898-010-9546-4","volume":"49","author":"C Guti\u00e9rrez","year":"2011","unstructured":"Guti\u00e9rrez, C., Jim\u00e9nez, B., Novo, V.: A generic approach to approximate efficiency and applications to vector optimization with set-valued maps. J. Glob. Optim. 49, 313\u2013342 (2011)","journal-title":"J. Glob. Optim."},{"key":"620_CR12","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.ejor.2009.02.007","volume":"201","author":"C Guti\u00e9rrez","year":"2010","unstructured":"Guti\u00e9rrez, C., Jim\u00e9nez, B., Novo, V.: Optimality conditions via scalarization for a new $\\epsilon $-efficiency concept in vector optimization problems. Eur. J. Oper. Res. 201, 11\u201322 (2010)","journal-title":"Eur. J. Oper. Res."},{"key":"620_CR13","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1142\/S0217595907001577","volume":"24","author":"TY Li","year":"2007","unstructured":"Li, T.Y., Xu, Y.H., Zhu, C.X.: $\\epsilon $-Strictly efficient solutions of vector optimization problems with set-valued maps. Asia. Pac. J. Oper. Res. 24, 841\u2013854 (2007)","journal-title":"Asia. Pac. J. Oper. Res."},{"key":"620_CR14","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1080\/01630560903499845","volume":"31","author":"LA Tuan","year":"2010","unstructured":"Tuan, L.A.: $\\epsilon $-Optimality conditions for vector optimization problems with set-valued maps. Numer. Func. Anal. Optim. 31, 78\u201395 (2010)","journal-title":"Numer. Func. Anal. Optim."},{"key":"620_CR15","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1006\/jmaa.1999.6426","volume":"237","author":"ZM Li","year":"1999","unstructured":"Li, Z.M.: The optimality conditions for vector optimization of set-valued maps. J. Math. Anal. Appl. 237, 413\u2013424 (1999)","journal-title":"J. Math. Anal. Appl."},{"key":"620_CR16","first-page":"275","volume":"14","author":"E Hern\u00e1ndez","year":"2007","unstructured":"Hern\u00e1ndez, E., Jim\u00e9nez, B., Novo, V.: Weak and proper efficiency in set-valued optimization on real linear spaces. J. Convex. Anal. 14, 275\u2013296 (2007)","journal-title":"J. Convex. Anal."},{"key":"620_CR17","doi-asserted-by":"crossref","first-page":"830","DOI":"10.1007\/s10957-012-0045-2","volume":"154","author":"ZA Zhou","year":"2012","unstructured":"Zhou, Z.A., Peng, J.W.: Scalarization of set-valued optimization problems with generalized cone subconvexlikeness in real ordered linear spaces. J. Optim. Theory Appl. 154, 830\u2013841 (2012)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-17005-8","volume-title":"Vector Optimization: Theory, Applications and Extensions","author":"J Jahn","year":"2011","unstructured":"Jahn, J.: Vector Optimization: Theory, Applications and Extensions, 2nd edn. Springer, Berlin (2011)","edition":"2"},{"key":"620_CR19","doi-asserted-by":"crossref","unstructured":"Zhou, Z.A., Yang, X.M., Peng, J.W.: Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces. Optimization. doi: 10.1080\/02331934.2012.656117","DOI":"10.1080\/02331934.2012.656117"},{"key":"620_CR20","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/S0377-2217(02)00444-7","volume":"149","author":"M Ad\u00e1n","year":"2003","unstructured":"Ad\u00e1n, M., Novo, V.: Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness. Eur. J. Oper. Res. 149, 641\u2013653 (2003)","journal-title":"Eur. J. Oper. Res."},{"key":"620_CR21","volume-title":"Convex Analysis","author":"SZ Shi","year":"1990","unstructured":"Shi, S.Z.: Convex Analysis. Shanghai Science and Technology Press, Shanghai (1990)"},{"key":"620_CR22","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1023\/B:JOTA.0000037602.13941.ed","volume":"121","author":"M Ad\u00e1n","year":"2004","unstructured":"Ad\u00e1n, M., Novo, V.: Proper efficiency in vector optimization on real linear spaces. J. Optim. Theory Appl. 121, 515\u2013540 (2004)","journal-title":"J. Optim. Theory Appl."},{"key":"620_CR23","first-page":"63","volume":"5","author":"YW Huang","year":"2001","unstructured":"Huang, Y.W., Li, Z.M.: Optimality condition and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces. Oper. Res. Tran. 5, 63\u201369 (2001)","journal-title":"Oper. Res. Tran."},{"key":"620_CR24","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s11590-011-0301-7","volume":"6","author":"M Maghri El","year":"2012","unstructured":"El Maghri, M.: Pareto-Fenchel $\\epsilon $-subdifferential sum rule and $\\epsilon $-efficiency. Optim. Lett. 6, 763\u2013781 (2012)","journal-title":"Optim. Lett."},{"key":"620_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9369-6","volume-title":"Geometric Functional Analysis and its Applications","author":"RB Holmes","year":"1975","unstructured":"Holmes, R.B.: Geometric Functional Analysis and its Applications. Springer, New York (1975)"},{"issue":"11","key":"620_CR26","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1007\/s10114-009-6144-9","volume":"25","author":"GL Yu","year":"2009","unstructured":"Yu, G.L., Liu, S.Y.: Globally proper saddle point in ic-cone convexlike set-valued optimization problems. Acta. Math. Sin. 25(11), 1921\u20131928 (2009)","journal-title":"Acta. Math. Sin."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-013-0620-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11590-013-0620-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-013-0620-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T17:11:35Z","timestamp":1559409095000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11590-013-0620-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,17]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,3]]}},"alternative-id":["620"],"URL":"https:\/\/doi.org\/10.1007\/s11590-013-0620-y","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,17]]}}}