{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:43:53Z","timestamp":1760237033509,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T00:00:00Z","timestamp":1581379200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A new mixed type nondifferentiable higher-order symmetric dual programs over cones is formulated. As of now, in the literature, either Wolfe-type or Mond\u2013Weir-type nondifferentiable symmetric duals have been studied. However, we present a unified dual model and discuss weak, strong, and converse duality theorems for such programs under higher-order    F   - convexity\/higher-order    F   - pseudoconvexity. Self-duality is also discussed. Our dual programs and results generalize some dual formulations and results appeared in the literature. Two non-trivial examples are given to show the uniqueness of higher-order    F   - convex\/higher-order    F   - pseudoconvex functions and existence of higher-order symmetric dual programs.<\/jats:p>","DOI":"10.3390\/sym12020274","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T11:45:30Z","timestamp":1581421530000},"page":"274","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Mixed Type Nondifferentiable Higher-Order Symmetric Duality over Cones"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3482-8634","authenticated-orcid":false,"given":"Izhar","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4314-3664","authenticated-orcid":false,"given":"Khushboo","family":"Verma","sequence":"additional","affiliation":[{"name":"Department of Applied Sciences and Humanities, Faculty of Engineering, University of Lucknow, New Campus, Lucknow 226 021, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5998-0635","authenticated-orcid":false,"given":"Suliman","family":"Al-Homidan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/0022-247X(75)90111-0","article-title":"Second and higher-order duality in nonlinear programming","volume":"51","author":"Mangasarian","year":"1975","journal-title":"J. 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