{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T05:26:36Z","timestamp":1787289996360,"version":"build-2736575974"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100007224","name":"National Foundation for Science and Technology Development","doi-asserted-by":"publisher","award":["101.01-2025.14"],"award-info":[{"award-number":["101.01-2025.14"]}],"id":[{"id":"10.13039\/100007224","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":[[2026,8]]},"DOI":"10.1007\/s10957-026-03064-5","type":"journal-article","created":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T03:44:26Z","timestamp":1786074266000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Clarke-Type Directional Derivatives and their Applications to Generalized Convexity and Optimality Conditions in Non-Lipschitz Settings"],"prefix":"10.1007","volume":"210","author":[{"given":"Minh Tung","family":"Nguyen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-3767-9306","authenticated-orcid":false,"given":"Xuan Duy Bao","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,7]]},"reference":[{"key":"3064_CR1","unstructured":"Aubin, J.P., Frankowska, H.: Set-Valued Analysis, Birkh\u00e4user Boston (1990)"},{"issue":"1","key":"3064_CR2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF01584551","volume":"2","author":"M Avriel","year":"1972","unstructured":"Avriel, M.: r-convex functions. Math. Program. 2(1), 309\u2013323 (1972)","journal-title":"Math. Program."},{"issue":"6","key":"3064_CR3","doi-asserted-by":"publisher","first-page":"1693","DOI":"10.3934\/dcdsb.2012.17.1693","volume":"17","author":"EN Barron","year":"2012","unstructured":"Barron, E.N., Goebel, R., Jensen, R.R.: The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete Contin. Dyn. Syst. Ser. B 17(6), 1693\u20131706 (2012)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"8","key":"3064_CR4","doi-asserted-by":"publisher","first-page":"4229","DOI":"10.1090\/S0002-9947-2013-05760-1","volume":"365","author":"EN Barron","year":"2013","unstructured":"Barron, E.N., Goebel, R., Jensen, R.R.: Quasiconvex functions and nonlinear PDEs. Trans. Amer. Math. Soc. 365(8), 4229\u20134255 (2013)","journal-title":"Trans. Amer. Math. Soc."},{"key":"3064_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1394-9","volume-title":"Perturbation Analysis of Optimization Problems","author":"JF Bonnans","year":"2000","unstructured":"Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)"},{"key":"3064_CR6","volume-title":"Generalized Convexity and Optimization","author":"A Cambini","year":"2009","unstructured":"Cambini, A., Martein, L.: Generalized Convexity and Optimization. Theory and Applications. Springer, Berlin (2009)"},{"issue":"3","key":"3064_CR7","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1137\/S0363012992227423","volume":"32","author":"WL Chan","year":"1994","unstructured":"Chan, W.L., Huang, L.R., Ng, K.F.: On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control. Optim. 32(3), 591\u2013611 (1994)","journal-title":"SIAM J. Control. Optim."},{"key":"3064_CR8","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1090\/S0002-9947-1975-0367131-6","volume":"205","author":"FH Clarke","year":"1975","unstructured":"Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247\u2013262 (1975)","journal-title":"Trans. Amer. Math. Soc."},{"issue":"1","key":"3064_CR9","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/0001-8708(81)90032-3","volume":"40","author":"FH Clarke","year":"1981","unstructured":"Clarke, F.H.: Generalized gradients of Lipschitz functionals. Adv. Math. 40(1), 52\u201367 (1981)","journal-title":"Adv. Math."},{"key":"3064_CR10","doi-asserted-by":"crossref","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, (1990)","DOI":"10.1137\/1.9781611971309"},{"issue":"2","key":"3064_CR11","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF01390337","volume":"29","author":"L Cromme","year":"1978","unstructured":"Cromme, L.: Strong uniqueness: A far-reaching criterion for the convergence analysis of iterative procedures. Numer. Math. 29(2), 179\u2013193 (1978)","journal-title":"Numer. Math."},{"issue":"1","key":"3064_CR12","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/BF01588330","volume":"18","author":"JP Crouzeix","year":"1980","unstructured":"Crouzeix, J.P.: On second order conditions for quasiconvexity. Math. Program. 18(1), 349\u2013352 (1980)","journal-title":"Math. Program."},{"key":"3064_CR13","doi-asserted-by":"crossref","unstructured":"Crouzeix, J.P.: Characterizations of Generalized Convexity and Generalized Monotonicity, A Survey, pp. 237\u2013256. Springer (1998)","DOI":"10.1007\/978-1-4613-3341-8_10"},{"issue":"3","key":"3064_CR14","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1023\/A:1017519922669","volume":"109","author":"A Guerraggio","year":"2001","unstructured":"Guerraggio, A., Luc, D.T.: Optimality conditions for $$\\text{ C}^{1,1}$$ vector optimization problems. J. Optim. Theory Appl. 109(3), 615\u2013629 (2001)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"3064_CR15","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s10107-009-0318-1","volume":"123","author":"C Guti\u00e9rrez","year":"2010","unstructured":"Guti\u00e9rrez, C., Jim\u00e9nez, B., Novo, V.: On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. 123(1), 199\u2013223 (2010)","journal-title":"Math. Program."},{"issue":"1","key":"3064_CR16","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF01442169","volume":"11","author":"JB Hiriart-Urruty","year":"1984","unstructured":"Hiriart-Urruty, J.B., Strodiot, J.J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with $$\\text{ C}^{1,1}$$ data. Appl. Math. Optim. 11(1), 43\u201356 (1984)","journal-title":"Appl. Math. Optim."},{"issue":"2","key":"3064_CR17","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s00245-017-9432-2","volume":"79","author":"NQ Huy","year":"2019","unstructured":"Huy, N.Q., Kim, D.S., Tuyen, N.V.: New second-order Karush-Kuhn-Tucker optimality conditions for vector optimization. Appl. Math. Optim. 79(2), 279\u2013307 (2019)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"3064_CR18","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s10957-016-0980-4","volume":"171","author":"NQ Huy","year":"2016","unstructured":"Huy, N.Q., Tuyen, N.V.: New second-order optimality conditions for a class of differentiable optimization problems. J. Optim. Theory Appl. 171(1), 27\u201344 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"5","key":"3064_CR19","doi-asserted-by":"publisher","first-page":"1815","DOI":"10.1137\/S0363012996311745","volume":"36","author":"V Jeyakumar","year":"1998","unstructured":"Jeyakumar, V., Luc, D.T.: Approximate Jacobian matrices for nonsmooth continuous maps and $$\\text{ C}^1$$-optimization. SIAM J. Control. Optim. 36(5), 1815\u20131832 (1998)","journal-title":"SIAM J. Control. Optim."},{"key":"3064_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-54265-7","volume-title":"Set-Valued Optimization","author":"AA Khan","year":"2015","unstructured":"Khan, A.A., Tammer, C., Z\u0103linescu, C.: Set-Valued Optimization. Springer, Berlin-Heidelberg (2015)"},{"issue":"3","key":"3064_CR21","doi-asserted-by":"publisher","first-page":"2756","DOI":"10.1137\/23M1604667","volume":"34","author":"PD Khanh","year":"2024","unstructured":"Khanh, P.D., Khoa, V.V.H., Mordukhovich, B.S., Phat, V.T.: Variational and strong variational convexity in infinite-dimensional variational analysis. SIAM J. Optim. 34(3), 2756\u20132787 (2024)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3064_CR22","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1137\/23M1627237","volume":"35","author":"PD Khanh","year":"2025","unstructured":"Khanh, P.D., Khoa, V.V.H., Mordukhovich, B.S., Phat, V.T.: Second-order subdifferential optimality conditions in nonsmooth optimization. SIAM J. Optim. 35(2), 678\u2013711 (2025)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3064_CR23","doi-asserted-by":"publisher","first-page":"1121","DOI":"10.1137\/22M1519250","volume":"33","author":"PD Khanh","year":"2023","unstructured":"Khanh, P.D., Mordukhovich, B.S., Phat, V.T.: Variational convexity of functions and variational sufficiency in optimization. SIAM J. Optim. 33(2), 1121\u20131158 (2023)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"3064_CR24","doi-asserted-by":"publisher","first-page":"568","DOI":"10.1016\/j.orl.2018.09.003","volume":"46","author":"PD Khanh","year":"2018","unstructured":"Khanh, P.D., Phat, V.T.: Second-order characterizations of $$\\text{ C}^1$$-smooth robustly quasiconvex functions. Oper. Res. Lett. 46(6), 568\u2013572 (2018)","journal-title":"Oper. Res. Lett."},{"issue":"8","key":"3064_CR25","doi-asserted-by":"publisher","first-page":"2413","DOI":"10.1007\/s11590-020-01563-6","volume":"14","author":"PD Khanh","year":"2020","unstructured":"Khanh, P.D., Phat, V.T.: Second-order characterizations of quasiconvexity and pseudoconvexity for differentiable functions with Lipschitzian derivatives. Optim. Lett. 14(8), 2413\u20132427 (2020)","journal-title":"Optim. Lett."},{"issue":"3","key":"3064_CR26","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/s10957-007-9169-1","volume":"133","author":"PQ Khanh","year":"2007","unstructured":"Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133(3), 341\u2013357 (2007)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"3064_CR27","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1137\/0805032","volume":"5","author":"DT Luc","year":"1995","unstructured":"Luc, D.T.: Taylor\u2019s formula for $$\\text{ C}^{k, 1}$$ functions. SIAM J. Optim. 5(3), 659\u2013669 (1995)","journal-title":"SIAM J. Optim."},{"key":"3064_CR28","volume-title":"Variational Analysis and Generalized Differentiation I: Basic Theory, II: Applications","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory, II: Applications. Springer, Berlin-Heidelberg (2006)"},{"key":"3064_CR29","doi-asserted-by":"crossref","unstructured":"Mordukhovich, B.S.: Variational Analysis and Applications, Springer International Publishing (2018)","DOI":"10.1007\/978-3-319-92775-6"},{"issue":"1","key":"3064_CR30","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/BF02592147","volume":"74","author":"Z P\u00e1les","year":"1996","unstructured":"P\u00e1les, Z., Zeidan, V.: Generalized Hessian for $$\\text{ C}^{1,1}$$ functions in infinite-dimensional normed spaces. Math. Program. 74(1), 59\u201378 (1996)","journal-title":"Math. Program."},{"issue":"1","key":"3064_CR31","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1137\/S0363012996311095","volume":"37","author":"JP Penot","year":"1998","unstructured":"Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control. Optim. 37(1), 303\u2013318 (1998)","journal-title":"SIAM J. Control. Optim."},{"key":"3064_CR32","volume-title":"Introduction to Optimization","author":"BT Polyak","year":"1987","unstructured":"Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)"},{"issue":"2","key":"3064_CR33","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1112\/plms\/s3-39.2.331","volume":"39","author":"RT Rockafellar","year":"1979","unstructured":"Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. Lond. Math. Soc. (3) 39(2), 331\u2013355 (1979)","journal-title":"Proc. Lond. Math. Soc. (3)"},{"issue":"2","key":"3064_CR34","doi-asserted-by":"publisher","first-page":"257","DOI":"10.4153\/CJM-1980-020-7","volume":"32","author":"RT Rockafellar","year":"1980","unstructured":"Rockafellar, R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Canad. J. Math. 32(2), 257\u2013280 (1980)","journal-title":"Canad. J. Math."},{"issue":"5\u20136","key":"3064_CR35","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1080\/01630569308816543","volume":"14","author":"X Yang","year":"1993","unstructured":"Yang, X.: Second-order conditions in $$\\text{ C}^{1,1}$$ optimization with applications. Numer. Funct. Anal. Optim. 14(5\u20136), 621\u2013632 (1993)","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"3\u20134","key":"3064_CR36","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1080\/02331939208843851","volume":"26","author":"X Yang","year":"1992","unstructured":"Yang, X., Jeyakumar, V.: Generalized second-order directional derivatives and optimization with $$\\text{ C}^{1,1}$$ functions. Optimization 26(3\u20134), 165\u2013185 (1992)","journal-title":"Optimization"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-03064-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-026-03064-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-03064-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T05:08:55Z","timestamp":1787288935000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-026-03064-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["3064"],"URL":"https:\/\/doi.org\/10.1007\/s10957-026-03064-5","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"24 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":1,"name":"Ethics","label":"Competing interests","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"41"}}