{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T08:21:47Z","timestamp":1775636507198,"version":"3.50.1"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"1-3","license":[{"start":{"date-parts":[[1988,5,1]],"date-time":"1988-05-01T00:00:00Z","timestamp":578448000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Programming"],"published-print":{"date-parts":[[1988,5]]},"DOI":"10.1007\/bf01580771","type":"journal-article","created":{"date-parts":[[2005,4,28]],"date-time":"2005-04-28T09:52:50Z","timestamp":1114681970000},"page":"327-339","source":"Crossref","is-referenced-by-count":27,"title":["The upper and lower second order directional derivatives of a sup-type function"],"prefix":"10.1007","volume":"41","author":[{"given":"Hidefumi","family":"Kawasaki","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"CR1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/BF01585110","volume":"24","author":"A. Auslender","year":"1982","unstructured":"A. Auslender, \u201cOn the differential properties of the support function of the\u03b5-subdifferential of a convex function,\u201dMathematical Programming 24 (1982) 257\u2013268.","journal-title":"Mathematical Programming"},{"key":"CR2","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BFb0120982","volume":"19","author":"A. Ben-Tal","year":"1982","unstructured":"A. Ben-Tal and J. Zowe, \u201cA unified theory of first and second order conditions for extremum problems in topological vector spaces,\u201dMathematical Programming Study 19 (1982) 39\u201376.","journal-title":"Mathematical Programming Study"},{"key":"CR3","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/BF00942193","volume":"47","author":"A. Ben-Tal","year":"1985","unstructured":"A. Ben-Tal and J. Zowe, \u201cDirectional derivatives in nonsmooth optimization,\u201dJournal of Optimization Theory and Applications 47 (1985) 483\u2013490.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"CR4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/0041-5553(74)90193-1","volume":"14","author":"V.F. Dem'yanov","year":"1975","unstructured":"V.F. Dem'yanov and A.B. Pevnyi, \u201cExpansion with respect to a parameter of the extremal values of game problems,\u201dU.S.S.R. Computational Mathematics and Mathematical Physics 14 (1975) 33\u201345.","journal-title":"U.S.S.R. Computational Mathematics and Mathematical Physics"},{"key":"CR5","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1137\/0320057","volume":"20","author":"J.B. Hiriart-Urruty","year":"1982","unstructured":"J.B. Hiriart-Urruty, \u201cApproximating a second-order directional derivative for nonsmooth convex functions,\u201dSIAM Journal on Control and Optimization 20 (1982) 783\u2013807.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"CR6","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1016\/0362-546X(82)90106-7","volume":"6","author":"J.B. Hiriart-Urruty","year":"1982","unstructured":"J.B. Hiriart-Urruty, \u201cLimiting behavior of the approximate first order and second order directional derivatives for a convex function,\u201dNonlinear Analysis: Theory, Methods, and Applications 6 (1982) 1309\u20131326.","journal-title":"Nonlinear Analysis: Theory, Methods, and Applications"},{"key":"CR7","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF00938593","volume":"48","author":"J.B. Hiriart-Urruty","year":"1986","unstructured":"J.B. Hiriart-Urruty, \u201cApproximate first-order and second-order directional derivatives of a marginal function in convex optimization,\u201dJournal of Optimization Theory and Applications 48 (1986) 127\u2013140.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"CR8","unstructured":"H. Kawasaki, \u201cAn envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems,\u201dMathematical Programming."},{"key":"CR9","unstructured":"H. Kawasaki, \u201cSecond order necessary optimality conditions for minimizing a sup-type function,\u201d submitted toMathematical Programming."},{"key":"CR10","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1287\/moor.10.2.207","volume":"10","author":"A. Shapiro","year":"1985","unstructured":"A. Shapiro, \u201cSecond-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs,\u201dMathematics of Operations Research 10 (1985) 207\u2013219.","journal-title":"Mathematics of Operations Research"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF01580771.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF01580771\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF01580771","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,3]],"date-time":"2019-05-03T15:12:11Z","timestamp":1556896331000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF01580771"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,5]]},"references-count":10,"journal-issue":{"issue":"1-3","published-print":{"date-parts":[[1988,5]]}},"alternative-id":["BF01580771"],"URL":"https:\/\/doi.org\/10.1007\/bf01580771","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,5]]}}}