{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T11:16:05Z","timestamp":1771413365336,"version":"3.50.1"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:00:00Z","timestamp":1771372800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T00:00:00Z","timestamp":1771372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s10957-026-02941-3","type":"journal-article","created":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T10:46:52Z","timestamp":1771411612000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Projectional Subdifferentials and Projectional Coderivatives with Respect to Smooth Manifolds"],"prefix":"10.1007","volume":"208","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0524-0498","authenticated-orcid":false,"given":"Phan Quoc","family":"Khanh","sequence":"first","affiliation":[]},{"given":"Tran Tuan","family":"Minh","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,18]]},"reference":[{"issue":"2","key":"2941_CR1","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.jfa.2004.10.008","volume":"220","author":"D Azagra","year":"2005","unstructured":"Azagra, D., Ferrera, J., L\u00f3pez-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220(2), 304\u2013361 (2005)","journal-title":"J. Funct. Anal."},{"issue":"4","key":"2941_CR2","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1007\/s00009-005-0056-4","volume":"2","author":"D Azagra","year":"2005","unstructured":"Azagra, D., Ferrera, J.: Proximal calculus on Riemannian manifolds. Mediterr. J. Math. 2(4), 437\u2013450 (2005)","journal-title":"Mediterr. J. Math."},{"issue":"4","key":"2941_CR3","doi-asserted-by":"publisher","first-page":"2423","DOI":"10.1137\/18M1181602","volume":"29","author":"R Bergmann","year":"2019","unstructured":"Bergmann, R., Herzog, R.: Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM J. Optim. 29(4), 2423\u20132444 (2019)","journal-title":"SIAM J. Optim."},{"key":"2941_CR4","doi-asserted-by":"crossref","unstructured":"Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge Univ, Press (2023)","DOI":"10.1017\/9781009166164"},{"key":"2941_CR5","doi-asserted-by":"crossref","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)","DOI":"10.1137\/1.9781611971309"},{"issue":"12","key":"2941_CR6","doi-asserted-by":"publisher","first-page":"3884","DOI":"10.1016\/j.na.2011.02.023","volume":"74","author":"S Hosseini","year":"2011","unstructured":"Hosseini, S., Pouryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 74(12), 3884\u20133895 (2011)","journal-title":"Nonlinear Anal."},{"key":"2941_CR7","doi-asserted-by":"crossref","unstructured":"Ioffe, A.D.: On the theory of subdifferentials. Adv. Nonlinear Anal. 1(1) (2012)","DOI":"10.1515\/ana-2011-0003"},{"issue":"8","key":"2941_CR8","doi-asserted-by":"publisher","first-page":"3687","DOI":"10.1090\/S0002-9947-07-04075-5","volume":"359","author":"YS Ledyaev","year":"2007","unstructured":"Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Amer. Math. Soc. 359(8), 3687\u20133732 (2007)","journal-title":"Trans. Amer. Math. Soc."},{"key":"2941_CR9","doi-asserted-by":"crossref","unstructured":"Lee, J.M.: Introduction to Smooth Manifolds. Grad. Texts Math, Springer, New York, NY (2012)","DOI":"10.1007\/978-1-4419-9982-5"},{"issue":"1\u20132","key":"2941_CR10","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1007\/s10107-020-01568-0","volume":"189","author":"KW Meng","year":"2021","unstructured":"Meng, K.W., Li, M.H., Yao, W.F., Yang, X.Q.: Lipschitz-like property relative to a set and the generalized Mordukhovich criterion. Math. Program. 189(1\u20132), 455\u2013489 (2021)","journal-title":"Math. Program."},{"key":"2941_CR11","doi-asserted-by":"crossref","unstructured":"Mordukhovich, B.S.: Variational Analysis and Applications. Springer Monogr. Math, Springer, Cham (2018)","DOI":"10.1007\/978-3-319-92775-6"},{"issue":"1","key":"2941_CR12","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/0022-247X(82)90180-9","volume":"88","author":"D Motreanu","year":"1982","unstructured":"Motreanu, D., Pavel, N.H.: Quasi-tangent vectors in flow-invariance and optimization problems on Banach manifolds. J. Math. Anal. Appl. 88(1), 116\u2013132 (1982)","journal-title":"J. Math. Anal. Appl."},{"key":"2941_CR13","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton Math. Ser., Princeton Univ. Press, Princeton (1972)"},{"key":"2941_CR14","doi-asserted-by":"crossref","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Math. Wiss, Springer, Berlin-Heidelberg (1998)","DOI":"10.1007\/978-3-642-02431-3"},{"key":"2941_CR15","doi-asserted-by":"crossref","unstructured":"Yao, W.F., Meng, K.W., Li, M.H., Yang, X.Q: Projectional coderivatives and calculus rules. Set-Valued Var. Anal. 31(4), 36 (2023)","DOI":"10.1007\/s11228-023-00698-9"},{"issue":"3","key":"2941_CR16","doi-asserted-by":"publisher","first-page":"2021","DOI":"10.1137\/22M151296X","volume":"33","author":"WF Yao","year":"2023","unstructured":"Yao, W.F., Yang, X.Q.: Relative Lipschitz-like property of parametric systems via projectional coderivative. SIAM J. Optim. 33(3), 2021\u20132040 (2023)","journal-title":"SIAM J. Optim."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02941-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-026-02941-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-026-02941-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T10:46:55Z","timestamp":1771411615000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-026-02941-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,18]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2941"],"URL":"https:\/\/doi.org\/10.1007\/s10957-026-02941-3","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,18]]},"assertion":[{"value":"18 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"My research has no potential Conflict of Interest, including potential Financial Interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}},{"value":"My research has no Human Participants and\/or Animals.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"value":"My co-author confirms his consent with the submission.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed Consent"}}],"article-number":"113"}}