{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T02:40:12Z","timestamp":1728355212003},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T00:00:00Z","timestamp":1668211200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T00:00:00Z","timestamp":1668211200000},"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":[[2023,1]]},"DOI":"10.1007\/s10957-022-02130-y","type":"journal-article","created":{"date-parts":[[2022,11,12]],"date-time":"2022-11-12T20:02:39Z","timestamp":1668283359000},"page":"240-265","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["S-Derivative of the Extremum Multifunction to a Multi-objective Parametric Discrete Optimal Control Problem"],"prefix":"10.1007","volume":"196","author":[{"given":"Nguyen Thi","family":"Toan","sequence":"first","affiliation":[]},{"given":"Le Quang","family":"Thuy","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,11,12]]},"reference":[{"key":"2130_CR1","first-page":"201","volume":"43","author":"DTV An","year":"2018","unstructured":"An, D.T.V., Toan, N.T.: Differential stability of convex discrete optimal control problem. Acta Math. Vietnam 43, 201\u2013217 (2018)","journal-title":"Acta Math. Vietnam"},{"key":"2130_CR2","first-page":"159","volume-title":"Mathematical Analysis and Applications","author":"J-P Aubin","year":"1981","unstructured":"Aubin, J.-P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, pp. 159\u2013229. Academic Press, New York (1981)"},{"key":"2130_CR3","first-page":"375","volume":"27","author":"EM Bednarczuk","year":"1998","unstructured":"Bednarczuk, E.M., Song, W.: Contingent epiderivate and its applications to set-valued maps. Control. Cybern. 27, 375\u2013386 (1998)","journal-title":"Control. Cybern."},{"key":"2130_CR4","doi-asserted-by":"crossref","first-page":"2823","DOI":"10.1016\/j.automatica.2009.09.032","volume":"45","author":"A Bemporad","year":"2009","unstructured":"Bemporad, A., Mu\u00f1oz de la Pe\u00f1a, D.: Multiobjective model predictive control. Automatica 45, 2823\u20132830 (2009)","journal-title":"Automatica"},{"key":"2130_CR5","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1002\/aic.690460516","volume":"46","author":"V Bhaskar","year":"2000","unstructured":"Bhaskar, V., Gupta, S.K., Ray, A.K.: Multiobjective optimization of an industrial wiped-film pet reactor. Am. Inst. Chem. Eng. J. 46, 1046\u20131058 (2000)","journal-title":"Am. Inst. Chem. Eng. J."},{"key":"2130_CR6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/REVCE.2000.16.1.1","volume":"16","author":"V Bhaskar","year":"2000","unstructured":"Bhaskar, V., Gupta, S.K., Ray, A.K.: Applications of multiobjective optimization in chemical engineering. Rev. Chem. Eng. 16, 1\u201354 (2000)","journal-title":"Rev. Chem. Eng."},{"key":"2130_CR7","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-9038-8","volume-title":"Infinite-Horizon Optimal Control in the Discrete-Time Framework","author":"J Blot","year":"2014","unstructured":"Blot, J., Hayek, N.: Infinite-Horizon Optimal Control in the Discrete-Time Framework. Springer, New York (2014)"},{"key":"2130_CR8","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s001860050021","volume":"48","author":"GY Chen","year":"1998","unstructured":"Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Meth. Oper. Res. 48, 187\u2013200 (1998)","journal-title":"Math. Meth. Oper. Res."},{"key":"2130_CR9","first-page":"303","volume":"3","author":"L Chen","year":"2002","unstructured":"Chen, L.: Generalized tangent epiderivative and applications to set-valued map optimization. J. Nonlinear Convex Anal. 3, 303\u2013313 (2002)","journal-title":"J. Nonlinear Convex Anal."},{"key":"2130_CR10","doi-asserted-by":"crossref","first-page":"401","DOI":"10.3934\/jimo.2010.6.401","volume":"6","author":"NH Chieu","year":"2010","unstructured":"Chieu, N.H., Yao, J.-C.: Subgradients of the optimal value function in a parametric discrete optimal control problem. J. Ind. Manag. Optim. 6, 401\u2013410 (2010)","journal-title":"J. Ind. Manag. Optim."},{"key":"2130_CR11","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10957-010-9646-9","volume":"146","author":"TD Chuong","year":"2010","unstructured":"Chuong, T.D., Yao, J.-C.: Generalized Clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl. 146, 77\u201394 (2010)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR12","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s10957-012-0099-1","volume":"156","author":"TD Chuong","year":"2013","unstructured":"Chuong, T.D.: Derivatives of the efficient point multifunction in parametric vector optimization problems. J. Optim. Theory Appl. 156, 247\u2013265 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR13","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1007\/s10898-012-0025-y","volume":"57","author":"TD Chuong","year":"2013","unstructured":"Chuong, T.D.: Fr\u00e9chet subdifferentials of efficient point multifunctions in parametric vector optimization. J. Glob. Optim. 57, 1229\u20131243 (2013)","journal-title":"J. Glob. Optim."},{"key":"2130_CR14","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1007\/s11590-012-0463-y","volume":"7","author":"TD Chuong","year":"2013","unstructured":"Chuong, T.D.: Normal subdifferentials of efficient point multifunctions in parametric vector optimization. Optim. Lett. 7, 1087\u20131117 (2013)","journal-title":"Optim. Lett."},{"key":"2130_CR15","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1006\/jeem.1993.1023","volume":"25","author":"E Dockner","year":"1993","unstructured":"Dockner, E., Long, N.V.: International pollution control: cooperative versus non-cooperative strategies. J. Environ. Econ. Manag. 25, 13\u201329 (1993)","journal-title":"J. Environ. Econ. Manag."},{"key":"2130_CR16","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511805127","volume-title":"Differential Games in Economics and Management Science","author":"E Dockner","year":"2000","unstructured":"Dockner, E., Jorgensen, S., Long, N.V., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000)"},{"key":"2130_CR17","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1080\/1023619042000193667","volume":"10","author":"EJ Dockner","year":"2004","unstructured":"Dockner, E.J., Nishimura, K.: Strategic growth. J. Differ. Equ. Appl. 10, 515\u2013527 (2004)","journal-title":"J. Differ. Equ. Appl."},{"key":"2130_CR18","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1080\/02331930903480352","volume":"60","author":"N Hayek","year":"2011","unstructured":"Hayek, N.: Infinite horizon multiobjective optimal control problems in the discrete time case. Optimization 60, 509\u2013529 (2011)","journal-title":"Optimization"},{"key":"2130_CR19","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1007\/s10957-011-9850-2","volume":"150","author":"N Hayek","year":"2011","unstructured":"Hayek, N.: A generalization of mixed problems with an application to multiobjective optimal control. J. Optim. Theory Appl. 150, 498\u2013515 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR20","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01217690","volume":"46","author":"J Jahn","year":"1997","unstructured":"Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Methods Oper. Res. 46, 193\u2013211 (1997)","journal-title":"Math. Methods Oper. Res."},{"key":"2130_CR21","volume-title":"Vector Optimization. Theory, Applications and Extensions","author":"J Jahn","year":"2004","unstructured":"Jahn, J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2004)"},{"key":"2130_CR22","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1007\/s10589-013-9603-2","volume":"57","author":"CY Kaya","year":"2014","unstructured":"Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57, 685\u2013702 (2014)","journal-title":"Comput. Optim. Appl."},{"key":"2130_CR23","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1006\/jmaa.1996.0331","volume":"202","author":"H Kuk","year":"1996","unstructured":"Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in parametrized convex vector optimization. J. Math. Anal. Appl. 202, 511\u2013522 (1996)","journal-title":"J. Math. Anal. Appl."},{"key":"2130_CR24","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/BF02275356","volume":"89","author":"H Kuk","year":"1996","unstructured":"Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in vector optimization. J. Optim. Theory Appl. 89, 713\u2013730 (1996)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR25","volume-title":"Lecture Notes in Economics and Mathematical Systems","author":"DT Luc","year":"1989","unstructured":"Luc, D.T.: Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1989)"},{"key":"2130_CR26","volume-title":"Variational Analysis and Generalized Differentiation I. Basis Theory","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Basis Theory. Springer, Berlin (2006)"},{"key":"2130_CR27","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1137\/0804038","volume":"4","author":"M Moussaoui","year":"1994","unstructured":"Moussaoui, M., Seeger, A.: Sensitivity analysis of optimal value functions of convex parametric programs with possibly empty solution sets. SIAM J. Optim. 4, 65\u201375 (1994)","journal-title":"SIAM J. Optim."},{"key":"2130_CR28","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/02331934.2016.1261349","volume":"66","author":"T-N Ngo","year":"2017","unstructured":"Ngo, T.-N., Hayek, N.: Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints. Optimization 66, 149\u2013177 (2017)","journal-title":"Optimization"},{"key":"2130_CR29","doi-asserted-by":"publisher","DOI":"10.3390\/mca23020030","author":"S Peitz","year":"2018","unstructured":"Peitz, S., Dellnitz, M.: A Survey of recent trends in multiobjective optimal control: surrogate models, feedback control and objective reduction. Math. Comput. Appl. (2018). https:\/\/doi.org\/10.3390\/mca23020030","journal-title":"Math. Comput. Appl."},{"key":"2130_CR30","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"key":"2130_CR31","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1023\/B:JOGO.0000026459.51919.0e","volume":"248","author":"RT Rockafellar","year":"2004","unstructured":"Rockafellar, R.T.: Hamilton-Jacobi theory and parametric analysis in fully convex problems of optimal control. J. Glob. Optim. 248, 419\u2013431 (2004)","journal-title":"J. Glob. Optim."},{"key":"2130_CR32","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/BF00940634","volume":"70","author":"DS Shi","year":"1991","unstructured":"Shi, D.S.: Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl. 70, 385\u2013396 (1991)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR33","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/BF00940783","volume":"77","author":"DS Shi","year":"1993","unstructured":"Shi, D.S.: Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl. 77, 145\u2013159 (1993)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR34","first-page":"198","volume":"22","author":"W Song","year":"2005","unstructured":"Song, W., Wan, L.-J.: Contingent epidifferentiability of the value map in vector optimization. Heilongjiang Daxue Ziran Kexue Xuebao 22, 198\u2013203 (2005)","journal-title":"Heilongjiang Daxue Ziran Kexue Xuebao"},{"key":"2130_CR35","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/j.1742-7363.2005.00002.x","volume":"1","author":"G Sorger","year":"2005","unstructured":"Sorger, G.: A dynamic common property resource problem with amenity value and extraction costs. Int. J. Econ. Theory 1, 3\u201319 (2005)","journal-title":"Int. J. Econ. Theory"},{"key":"2130_CR36","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF00939554","volume":"56","author":"T Tanino","year":"1988","unstructured":"Tanino, T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479\u2013499 (1988)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR37","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1137\/0326031","volume":"26","author":"T Tanino","year":"1988","unstructured":"Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control. Optim. 26, 521\u2013536 (1988)","journal-title":"SIAM J. Control. Optim."},{"key":"2130_CR38","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10957-016-0921-2","volume":"170","author":"LQ Thuy","year":"2016","unstructured":"Thuy, L.Q., Toan, N.T.: Subgradients of the value function in a parametric convex optimal control problem. J. Optim. Theory Appl. 170, 43\u201364 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"2130_CR39","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s10898-013-0062-1","volume":"58","author":"NT Toan","year":"2014","unstructured":"Toan, N.T., Yao, J.-C.: Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J. Glob. Optim. 58, 595\u2013612 (2014)","journal-title":"J. Glob. Optim."},{"key":"2130_CR40","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s10898-020-00935-7","volume":"79","author":"NT Toan","year":"2021","unstructured":"Toan, N.T., Thuy, L.Q., Tuyen, N.V., Xiao, Y.-B.: On the no-gap second-order optimality conditions for a multi-objective discrete optimal control problem with mixed constraints. J. Glob. Optim. 79, 203\u2013231 (2021)","journal-title":"J. Glob. Optim."},{"key":"2130_CR41","doi-asserted-by":"crossref","first-page":"2705","DOI":"10.3934\/jimo.2021088","volume":"18","author":"NT Toan","year":"2022","unstructured":"Toan, N.T.: Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. J. Ind. Manag. Optim. 18, 2705\u20132720 (2022)","journal-title":"J. Ind. Manag. Optim."},{"key":"2130_CR42","doi-asserted-by":"crossref","first-page":"3517","DOI":"10.1007\/s00245-021-09755-x","volume":"84","author":"NT Toan","year":"2021","unstructured":"Toan, N.T., Thuy, L.Q.: Sensitivity analysis of multi-objective optimal control problems. Appl. Math. Optim. 84, 3517\u20133545 (2021)","journal-title":"Appl. Math. Optim."},{"key":"2130_CR43","volume-title":"Optimal Control","author":"RB Vinter","year":"2000","unstructured":"Vinter, R.B.: Optimal Control. Birkh\u00e4user, Boston (2000)"},{"key":"2130_CR44","doi-asserted-by":"crossref","unstructured":"Vroemen, B., De Jager, B.: Multiobjective control: an overview. In: Proceeding of the 36th IEEE Conference on Decision and Control, San Diego CA, pp. 440\u2013445 (1997)","DOI":"10.1109\/CDC.1997.650664"},{"key":"2130_CR45","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1016\/j.na.2010.04.042","volume":"73","author":"J Wu","year":"2010","unstructured":"Wu, J.: Tangent cone and contingent cone to the intersection of two closed sets. Nonlinear Anal. 73, 1203\u20131220 (2010)","journal-title":"Nonlinear Anal."},{"key":"2130_CR46","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1017\/S0334270000010973","volume":"40","author":"XQ Yang","year":"1999","unstructured":"Yang, X.Q., Teo, K.L.: Necessary optimality conditions for bicriterion discrete time optimal control problems. J. Aust. Math. Soc. Ser. B. 40, 392\u2013402 (1999)","journal-title":"J. Aust. Math. Soc. Ser. B."},{"key":"2130_CR47","doi-asserted-by":"crossref","first-page":"2627","DOI":"10.1016\/j.automatica.2012.06.066","volume":"48","author":"VM Zavala","year":"2012","unstructured":"Zavala, V.M., Flores-Tlacuahuac, A.: Stability of multiobjective predictive control: a utopia-tracking approach. Automatica 48, 2627\u20132632 (2012)","journal-title":"Automatica"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02130-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-022-02130-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02130-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T01:59:50Z","timestamp":1728352790000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-022-02130-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,12]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["2130"],"URL":"https:\/\/doi.org\/10.1007\/s10957-022-02130-y","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2022,11,12]]},"assertion":[{"value":"22 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 October 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}