{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T06:22:38Z","timestamp":1773469358437,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:00:00Z","timestamp":1773446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100014718","name":"Innovative Research Group Project of the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271071"],"award-info":[{"award-number":["12271071"]}],"id":[{"id":"10.13039\/100014718","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10957-026-02957-9","type":"journal-article","created":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T05:46:05Z","timestamp":1773467165000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Characterization of Approximate Efficient Solutions of Nonconvex Vector Optimization Problems"],"prefix":"10.1007","volume":"209","author":[{"given":"Wen-Bin","family":"Wei","sequence":"first","affiliation":[]},{"given":"Jian-Wen","family":"Peng","sequence":"additional","affiliation":[]},{"given":"Elisabeth","family":"K\u00f6bis","sequence":"additional","affiliation":[]},{"given":"Jen-Chih","family":"Yao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,14]]},"reference":[{"key":"2957_CR1","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1016\/0022-247X(79)90226-9","volume":"71","author":"HP Benson","year":"1979","unstructured":"Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232\u2013241 (1979)","journal-title":"J. Math. Anal. Appl."},{"key":"2957_CR2","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1137\/0315004","volume":"15","author":"JM Borwein","year":"1977","unstructured":"Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control. Optim. 15, 57\u201363 (1977)","journal-title":"SIAM J. Control. Optim."},{"key":"2957_CR3","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1007\/s10957-011-9851-1","volume":"150","author":"M Chicco","year":"2011","unstructured":"Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150, 516\u2013529 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"2957_CR4","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/s00186-023-00818-z","volume":"97","author":"F Garc\u00eda-Casta\u00f1o","year":"2023","unstructured":"Garc\u00eda-Casta\u00f1o, F., Melguizo-Padial, M.\u00c1., Parzanese, G.: Sublinear scalarizations for proper and approximate proper efficient points in nonconvex vector optimization. Math. Methods Oper. Res. 97, 367\u2013382 (2023)","journal-title":"Math. Methods Oper. Res."},{"key":"2957_CR5","doi-asserted-by":"publisher","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 the theory of vector maximization. J. Math. Anal. Appl. 22, 618\u2013630 (1968)","journal-title":"J. Math. Anal. Appl."},{"key":"2957_CR6","doi-asserted-by":"publisher","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":"2957_CR7","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1137\/22M1542003","volume":"34","author":"C G\u00fcnther","year":"2024","unstructured":"G\u00fcnther, C., Khazayel, B., Tammer, C.: Nonlinear cone separation theorems in real topological linear spaces. SIAM J. Optim. 34, 225\u2013250 (2024)","journal-title":"SIAM J. Optim."},{"key":"2957_CR8","unstructured":"G\u00fcnther, C., Khazayel, B., Strugariu, R., Tammer, C.: Bishop-Phelps type scalarization for vector optimization in real topological-linear spaces. arXiv e-prints arXiv:2410.10026v3 (2025)"},{"key":"2957_CR9","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s40305-016-0139-x","volume":"5","author":"H Guo","year":"2017","unstructured":"Guo, H., Zhang, W.L.: Conic scalarizations for approximate efficient solutions in nonconvex vector optimization problems. J. Oper. Res. Soc. China 5, 419\u2013430 (2017)","journal-title":"J. Oper. Res. Soc. China"},{"key":"2957_CR10","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.ejor.2012.05.050","volume":"223","author":"C Guti\u00e9rrez","year":"2012","unstructured":"Guti\u00e9rrez, C., Jim\u00e9nez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304\u2013311 (2012)","journal-title":"Eur. J. Oper. Res."},{"key":"2957_CR11","doi-asserted-by":"publisher","first-page":"875","DOI":"10.1007\/s10898-017-0593-y","volume":"70","author":"C Guti\u00e9rrez","year":"2018","unstructured":"Guti\u00e9rrez, C., Huerga, L., Jim\u00e9nez, B., Novo, V.: Approximate solutions of vector optimization problems via improvement sets in real linear spaces. J. Glob. Optim. 70, 875\u2013901 (2018)","journal-title":"J. Glob. Optim."},{"key":"2957_CR12","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1137\/0134018","volume":"34","author":"R Hartley","year":"1978","unstructured":"Hartley, R.: On cone efficiency, cone convexity, and cone compactness. SIAM J. Appl. Math. 34, 211\u2013222 (1978)","journal-title":"SIAM J. Appl. Math."},{"key":"2957_CR13","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/BF00934353","volume":"36","author":"MI Henig","year":"1982","unstructured":"Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387\u2013407 (1982)","journal-title":"J. Optim. Theory Appl."},{"key":"2957_CR14","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1287\/moor.4.1.79","volume":"4","author":"JB Hiriart-Urruty","year":"1979","unstructured":"Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79\u201397 (1979)","journal-title":"Math. Oper. Res."},{"key":"2957_CR15","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1137\/070694089","volume":"20","author":"R Kasimbeyli","year":"2010","unstructured":"Kasimbeyli, R.: A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591\u20131619 (2010)","journal-title":"SIAM J. Optim."},{"key":"2957_CR16","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s10898-011-9789-8","volume":"56","author":"R Kasimbeyli","year":"2013","unstructured":"Kasimbeyli, R.: A conic scalarization method in multiobjective optimization. J. Glob. Optim. 56, 279\u2013297 (2013)","journal-title":"J. Glob. Optim."},{"key":"2957_CR17","first-page":"1397","volume":"53","author":"CP Liu","year":"2023","unstructured":"Liu, C.P., Yang, X.M.: E-proper efficient elements of vector optimization problems and their scalarization (in Chinese). Sci. Sin. Math. 53, 1397\u20131408 (2023)","journal-title":"Sci. Sin. Math."},{"key":"2957_CR18","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1007\/BF00934564","volume":"42","author":"A Pascoletti","year":"1984","unstructured":"Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499\u2013524 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"2957_CR19","first-page":"659","volume":"8","author":"JW Peng","year":"2024","unstructured":"Peng, J.W., Wei, W.B., Ghosh, D., Yao, J.C.: Characterization of $$\\cal{E} $$-Benson proper efficient solutions of vector optimization problems with variable ordering structures in linear spaces. J. Nonlinear Var. Anal. 8, 659\u2013680 (2024)","journal-title":"J. Nonlinear Var. Anal."},{"key":"2957_CR20","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1007\/s10898-016-0495-4","volume":"68","author":"A Sayadi-Bander","year":"2017","unstructured":"Sayadi-Bander, A., Pourkarimi, L., Kasimbeyli, R.: Coradiant sets and $$\\varepsilon $$-efficiency in multiobjective optimization. J. Glob. Optim. 68, 587\u2013600 (2017)","journal-title":"J. Glob. Optim."},{"key":"2957_CR21","doi-asserted-by":"publisher","first-page":"1071","DOI":"10.1137\/S0363012902411532","volume":"42","author":"A Zaffaroni","year":"2003","unstructured":"Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control. Optim. 42, 1071\u20131086 (2003)","journal-title":"SIAM J. Control. Optim."},{"key":"2957_CR22","doi-asserted-by":"publisher","first-page":"643","DOI":"10.3934\/naco.2013.3.643","volume":"3","author":"KQ Zhao","year":"2013","unstructured":"Zhao, K.Q., Yang, X.M.: Characterizations of the E-Benson proper efficiency in vector optimization problems. Numer. Algebra Control Optim. 3, 643\u2013653 (2013)","journal-title":"Numer. Algebra Control Optim."},{"key":"2957_CR23","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.11650\/tjm.17.2013.2721","volume":"17","author":"KQ Zhao","year":"2013","unstructured":"Zhao, K.Q., Yang, X.M., Peng, J.W.: Weak E-optimal solution in vector optimization. Taiwan. J. Math. 17, 1287\u20131302 (2013)","journal-title":"Taiwan. J. Math."},{"key":"2957_CR24","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1080\/02331934.2013.798321","volume":"64","author":"KQ Zhao","year":"2015","unstructured":"Zhao, K.Q., Yang, X.M.: E-Benson proper efficiency in vector optimization. Optimization 64, 739\u2013752 (2015)","journal-title":"Optimization"},{"key":"2957_CR25","doi-asserted-by":"publisher","first-page":"944","DOI":"10.1007\/s10957-019-01554-3","volume":"183","author":"ZA Zhou","year":"2019","unstructured":"Zhou, Z.A., Chen, W., Yang, X.M.: Scalarizations and optimality of constrained set-valued optimization using improvement sets and image space analysis. J. Optim. Theory Appl. 183, 944\u2013962 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"2957_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00186-025-00887-2","volume":"101","author":"ZA Zhou","year":"2025","unstructured":"Zhou, Z.A., Liang, K.H., Ansari, Q.H.: Optimality conditions for Benson proper efficiency of set-valued equilibrium problems. Math. Methods Oper. Res. 101, 1\u201324 (2025)","journal-title":"Math. Methods Oper. Res."},{"key":"2957_CR27","doi-asserted-by":"crossref","unstructured":"Zhou, Z.A., Feng, B., K\u00f6bis, E.: New type of Benson properly efficient solutions in set-valued optimization: scalarization results and optimality conditions. Optimization, 1\u201321 (2025)","DOI":"10.1080\/02331934.2025.2517323"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02957-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-026-02957-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02957-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T05:46:07Z","timestamp":1773467167000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-026-02957-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,14]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["2957"],"URL":"https:\/\/doi.org\/10.1007\/s10957-026-02957-9","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,14]]},"assertion":[{"value":"22 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"8"}}