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Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>For a general set-valued problem with set and inclusion constraints we establish higher-order Karush--Kuhn--Tucker multiplier rules with additional complementarity slackness conditions in terms of contingent-type derivatives of index $\\gamma \\in \\{0,1\\}$ for weak and firm solutions. Our multiplier rules are in a nonclassical form with a supremum expression on the right-hand side (instead of zero). An example is provided to show a case in which the fourth-order envelope-like effect occurs. We also prove Karush--Kuhn--Tucker rules in terms of Studniarski's derivatives under directional higher-order H\u00f6lder metric subregularity. The results are novel even in classical finite-dimensional scalar problems of nonlinear programming. As applications, we consider vector nonlinear programming to have detailed comparisons with known results.<\/jats:p>","DOI":"10.1137\/16m1079920","type":"journal-article","created":{"date-parts":[[2018,3,15]],"date-time":"2018-03-15T15:21:59Z","timestamp":1521127319000},"page":"820-848","source":"Crossref","is-referenced-by-count":19,"title":["Higher-Order Karush--Kuhn--Tucker Conditions in Nonsmooth Optimization"],"prefix":"10.1137","volume":"28","author":[{"given":"Phan Quoc","family":"Khanh","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nguyen Minh","family":"Tung","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,3,15]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/060651616"},{"key":"atypb2","unstructured":"J. P. Aubin and H. 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