{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T22:43:02Z","timestamp":1779230582519,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T00:00:00Z","timestamp":1668556800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971078"],"award-info":[{"award-number":["11971078"]}]},{"name":"National Natural Science Foundation of China","award":["CXQT21021"],"award-info":[{"award-number":["CXQT21021"]}]},{"name":"National Natural Science Foundation of China","award":["2021ST004"],"award-info":[{"award-number":["2021ST004"]}]},{"name":"Group Building Project for Scientifc Innovation for Universities in Chongqing","award":["11971078"],"award-info":[{"award-number":["11971078"]}]},{"name":"Group Building Project for Scientifc Innovation for Universities in Chongqing","award":["CXQT21021"],"award-info":[{"award-number":["CXQT21021"]}]},{"name":"Group Building Project for Scientifc Innovation for Universities in Chongqing","award":["2021ST004"],"award-info":[{"award-number":["2021ST004"]}]},{"name":"Graduate Student Science and Technology Innovation Project","award":["11971078"],"award-info":[{"award-number":["11971078"]}]},{"name":"Graduate Student Science and Technology Innovation Project","award":["CXQT21021"],"award-info":[{"award-number":["CXQT21021"]}]},{"name":"Graduate Student Science and Technology Innovation Project","award":["2021ST004"],"award-info":[{"award-number":["2021ST004"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce a second-order strong subdifferential of set-valued maps, and discuss some properties, such as convexity, sum rule and so on. By the new subdifferential and its properties, we establish a necessary and sufficient optimality condition of set-based robust efficient solutions for the uncertain set-valued optimization problem. We also introduce a Wolfe type dual problem of the uncertain set-valued optimization problem. Finally, we establish the robust weak duality theorem and the robust strong duality theorem between the uncertain set-valued optimization problem and its robust dual problem. Several main results extend to the corresponding ones in the literature.<\/jats:p>","DOI":"10.3390\/axioms11110648","type":"journal-article","created":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T03:27:44Z","timestamp":1668655664000},"page":"648","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Optimality Conditions and Dualities for Robust Efficient Solutions of Uncertain Set-Valued Optimization with Set-Order Relations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3170-1630","authenticated-orcid":false,"given":"Yuwen","family":"Zhai","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7968-051X","authenticated-orcid":false,"given":"Qilin","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tian","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.orl.2008.09.010","article-title":"Duality in robust optimization: Primal worst equals dual best","volume":"37","author":"Beck","year":"2009","journal-title":"Oper. 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