{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T10:09:45Z","timestamp":1770977385090,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T00:00:00Z","timestamp":1770940800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T00:00:00Z","timestamp":1770940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["307294\/2023-4"],"award-info":[{"award-number":["307294\/2023-4"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002923","name":"Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas","doi-asserted-by":"publisher","award":["403197\/2025-2"],"award-info":[{"award-number":["403197\/2025-2"]}],"id":[{"id":"10.13039\/501100002923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>This technical note proves that, for a smooth vector optimization problem on a closed convex feasible set ordered by a pointed cone, the projected gradient direction depends continuously on the decision variable. Our argument is based on a simple and direct proof via a fixed-domain reformulation of the subproblem. We then give a necessary and sufficient dual characterization of this direction and show that its associated set-valued dual variable mapping is outer semicontinuous.<\/jats:p>","DOI":"10.1007\/s40314-025-03611-2","type":"journal-article","created":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T09:14:44Z","timestamp":1770974084000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The projected gradient direction in vector optimization: continuity and dual characterization"],"prefix":"10.1007","volume":"45","author":[{"given":"N.","family":"Fazzio","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1098-4317","authenticated-orcid":false,"given":"L. F.","family":"Prudente","sequence":"additional","affiliation":[]},{"given":"M. L.","family":"Schuverdt","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,13]]},"reference":[{"issue":"3","key":"3611_CR1","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1007\/s10589-020-00260-5","volume":"78","author":"PB Assun\u00e7\u00e3o","year":"2021","unstructured":"Assun\u00e7\u00e3o PB, Ferreira OP, Prudente LF (2021) Conditional gradient method for multiobjective optimization. Comput Optim Appl 78(3):741\u2013768","journal-title":"Comput Optim Appl"},{"issue":"4","key":"3611_CR2","doi-asserted-by":"publisher","first-page":"2169","DOI":"10.1137\/120866415","volume":"23","author":"JY Bello Cruz","year":"2013","unstructured":"Bello Cruz JY (2013) A subgradient method for vector optimization problems. SIAM J. Optim. 23(4):2169\u20132182","journal-title":"SIAM J. Optim."},{"issue":"16","key":"3611_CR3","doi-asserted-by":"publisher","first-page":"5268","DOI":"10.1016\/j.na.2011.04.067","volume":"74","author":"JY Bello Cruz","year":"2011","unstructured":"Bello Cruz JY, Lucambio P\u00e9rez LR, Melo JG (2011) Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. 74(16):5268\u20135273","journal-title":"Nonlinear Anal."},{"key":"3611_CR4","volume-title":"Topological Spaces: Including a Treatment of Multi-valued Functions","author":"C Berge","year":"1963","unstructured":"Berge C (1963) Topological Spaces: Including a Treatment of Multi-valued Functions. Vector Spaces and Convexity. Macmillan, New York"},{"key":"3611_CR5","volume-title":"Convex Analysis and Optimization","author":"D Bertsekas","year":"2003","unstructured":"Bertsekas D, Nedic A, Ozdaglar A (2003) Convex Analysis and Optimization. Athena Scientific, Athena Scientific optimization and computation series"},{"issue":"1","key":"3611_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/REVCE.2000.16.1.1","volume":"16","author":"V Bhaskar","year":"2000","unstructured":"Bhaskar V, Gupta SK, Ray AK (2000) Applications of multiobjective optimization in chemical engineering. Rev Chem Eng 16(1):1\u201354","journal-title":"Rev Chem Eng"},{"issue":"4","key":"3611_CR7","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/S1052623403429093","volume":"15","author":"H Bonnel","year":"2005","unstructured":"Bonnel H, Iusem AN, Svaiter BF (2005) Proximal methods in vector optimization. SIAM J Optim 15(4):953\u2013970","journal-title":"SIAM J Optim"},{"issue":"3","key":"3611_CR8","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1007\/s10589-024-00605-4","volume":"89","author":"GA Carrizo","year":"2024","unstructured":"Carrizo GA, Fazzio NS, S\u00e1nchez MD, Schuverdt ML (2024) Scaled-PAKKT sequential optimality condition for multiobjective problems and its application to an augmented Lagrangian method. Comput Optim Appl 89(3):769\u2013803","journal-title":"Comput Optim Appl"},{"issue":"2","key":"3611_CR9","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1007\/s40305-022-00410-y","volume":"12","author":"GA Carrizo","year":"2024","unstructured":"Carrizo GA, Fazzio NS, Schuverdt ML (2024) A nonmonotone projected gradient method for multiobjective problems on convex sets. J Oper Res Soc China 12(2):410\u2013427","journal-title":"J Oper Res Soc China"},{"issue":"1","key":"3611_CR10","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.ejor.2023.04.022","volume":"311","author":"J Chen","year":"2023","unstructured":"Chen J, Tang L, Yang X (2023) A Barzilai-Borwein descent method for multiobjective optimization problems. Eur J Oper Res 311(1):196\u2013209","journal-title":"Eur J Oper Res"},{"issue":"3","key":"3611_CR11","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s10589-012-9495-6","volume":"54","author":"TD Chuong","year":"2013","unstructured":"Chuong TD (2013) Newton-like methods for efficient solutions in vector optimization. Comput Optim Appl 54(3):495\u2013516","journal-title":"Comput Optim Appl"},{"issue":"1","key":"3611_CR12","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10589-020-00204-z","volume":"77","author":"G Cocchi","year":"2020","unstructured":"Cocchi G, Lapucci M (2020) An augmented Lagrangian algorithm for multi-objective optimization. Comput Optim Appl 77(1):29\u201356","journal-title":"Comput Optim Appl"},{"key":"3611_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-79159-1","volume-title":"Adaptive Scalarization Methods in Multiobjective Optimization","author":"G Eichfelder","year":"2008","unstructured":"Eichfelder G (2008) Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin Heidelberg"},{"issue":"6","key":"3611_CR14","doi-asserted-by":"publisher","first-page":"1365","DOI":"10.1007\/s11590-018-1353-8","volume":"13","author":"NS Fazzio","year":"2019","unstructured":"Fazzio NS, Schuverdt ML (2019) Convergence analysis of a nonmonotone projected gradient method for multiobjective optimization problems. Optim Lett 13(6):1365\u20131379","journal-title":"Optim Lett"},{"issue":"4","key":"3611_CR15","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1287\/moor.1060.0221","volume":"31","author":"J Fliege","year":"2006","unstructured":"Fliege J (2006) An efficient interior-point method for convex multicriteria optimization problems. Math Oper Res 31(4):825\u2013845","journal-title":"Math Oper Res"},{"issue":"3","key":"3611_CR16","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/s001860000043","volume":"51","author":"J Fliege","year":"2000","unstructured":"Fliege J, Svaiter BF (2000) Steepest descent methods for multicriteria optimization. Math Methods Oper Res 51(3):479\u2013494","journal-title":"Math Methods Oper Res"},{"issue":"4","key":"3611_CR17","doi-asserted-by":"publisher","first-page":"2091","DOI":"10.1137\/15M1016424","volume":"26","author":"J Fliege","year":"2016","unstructured":"Fliege J, Vaz AIF (2016) A method for constrained multiobjective optimization based on SQP techniques. SIAM J Optim 26(4):2091\u20132119","journal-title":"SIAM J Optim"},{"issue":"2","key":"3611_CR18","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1137\/08071692X","volume":"20","author":"J Fliege","year":"2009","unstructured":"Fliege J, Gra\u00f1a Drummond LM, Svaiter BF (2009) Newton\u2019s method for multiobjective optimization. SIAM J. Optim. 20(2):602\u2013626","journal-title":"SIAM J. Optim."},{"issue":"8\u20139","key":"3611_CR19","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1080\/02331934.2010.522710","volume":"60","author":"EH Fukuda","year":"2011","unstructured":"Fukuda EH, Gra\u00f1a Drummond LM (2011) On the convergence of the projected gradient method for vector optimization. Optimization 60(8\u20139):1009\u20131021","journal-title":"Optimization"},{"issue":"3","key":"3611_CR20","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s10589-012-9501-z","volume":"54","author":"EH Fukuda","year":"2013","unstructured":"Fukuda EH, Gra\u00f1a Drummond LM (2013) Inexact projected gradient method for vector optimization. Comput. Optim. Appl. 54(3):473\u2013493","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"3611_CR21","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1007\/s10589-019-00146-1","volume":"76","author":"MLN Gon\u00e7alves","year":"2020","unstructured":"Gon\u00e7alves MLN, Prudente LF (2020) On the extension of the Hager-Zhang conjugate gradient method for vector optimization. Comput Optim Appl 76(3):889\u2013916","journal-title":"Comput Optim Appl"},{"issue":"2","key":"3611_CR22","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10589-022-00414-7","volume":"83","author":"MLN Gon\u00e7alves","year":"2022","unstructured":"Gon\u00e7alves MLN, Lima FS, Prudente LF (2022) Globally convergent Newton-type methods for multiobjective optimization. Comput Optim Appl 83(2):403\u2013434","journal-title":"Comput Optim Appl"},{"issue":"1","key":"3611_CR23","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/B:COAP.0000018877.86161.8b","volume":"28","author":"LM Gra\u00f1a Drummond","year":"2004","unstructured":"Gra\u00f1a Drummond LM, Iusem AN (2004) A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28(1):5\u201329","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"3611_CR24","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.cam.2004.06.018","volume":"175","author":"LM Gra\u00f1a Drummond","year":"2005","unstructured":"Gra\u00f1a Drummond LM, Svaiter BF (2005) A steepest descent method for vector optimization. J. Comput. Appl. Math. 175(2):395\u2013414","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"3611_CR25","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1080\/02331934.2012.693082","volume":"63","author":"LM Gra\u00f1a Drummond","year":"2014","unstructured":"Gra\u00f1a Drummond LM, Raupp FMP, Svaiter BF (2014) A quadratically convergent Newton method for vector optimization. Optimization 63(5):661\u2013677","journal-title":"Optimization"},{"issue":"3","key":"3611_CR26","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1137\/1015073","volume":"15","author":"WW Hogan","year":"1973","unstructured":"Hogan WW (1973) Point-to-set maps in mathematical programming. SIAM Rev 15(3):591\u2013603","journal-title":"SIAM Rev"},{"issue":"1","key":"3611_CR27","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10589-023-00454-7","volume":"85","author":"M Lapucci","year":"2023","unstructured":"Lapucci M, Mansueto P (2023) A limited memory quasi-Newton approach for multi-objective optimization. Comput Optim Appl 85(1):33\u201373","journal-title":"Comput Optim Appl"},{"issue":"3","key":"3611_CR28","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1162\/106365602760234108","volume":"10","author":"M Laumanns","year":"2002","unstructured":"Laumanns M, Thiele L, Deb K, Zitzler E (2002) Combining convergence and diversity in evolutionary multiobjective optimization. Evol Comput 10(3):263\u2013282","journal-title":"Evol Comput"},{"issue":"3","key":"3611_CR29","doi-asserted-by":"publisher","first-page":"2690","DOI":"10.1137\/17M1126588","volume":"28","author":"LR Lucambio P\u00e9rez","year":"2018","unstructured":"Lucambio P\u00e9rez LR, Prudente LF (2018) Nonlinear conjugate gradient methods for vector optimization. SIAM J. Optim. 28(3):2690\u20132720","journal-title":"SIAM J. Optim."},{"key":"3611_CR30","doi-asserted-by":"crossref","unstructured":"Miettinen K (1999) Nonlinear multiobjective optimization, volume\u00a012. Springer Science & Business Media","DOI":"10.1007\/978-1-4615-5563-6"},{"issue":"3","key":"3611_CR31","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/s11075-015-0058-7","volume":"72","author":"V Morovati","year":"2016","unstructured":"Morovati V, Pourkarimi L, Basirzadeh H (2016) Barzilai and Borwein\u2019s method for multiobjective optimization problems. Numer Algorithms 72(3):539\u2013604","journal-title":"Numer Algorithms"},{"key":"3611_CR32","doi-asserted-by":"crossref","unstructured":"Morovati V, Basirzadeh H, Pourkarimi L (2017) Quasi-Newton methods for multiobjective optimization problems. 4OR, 16(3):261\u2013294","DOI":"10.1007\/s10288-017-0363-1"},{"key":"3611_CR33","doi-asserted-by":"publisher","first-page":"765","DOI":"10.1016\/j.cam.2013.06.045","volume":"255","author":"Z Povalej","year":"2014","unstructured":"Povalej Z (2014) Quasi-Newton method for multiobjective optimization. J Comput Appl Math 255:765\u2013777","journal-title":"J Comput Appl Math"},{"issue":"3","key":"3611_CR34","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1007\/s10957-022-02072-5","volume":"194","author":"LF Prudente","year":"2022","unstructured":"Prudente LF, Souza DR (2022) A quasi-Newton method with wolfe line searches for multiobjective optimization. J Optim Theory Appl 194(3):1107\u20131140","journal-title":"J Optim Theory Appl"},{"key":"3611_CR35","unstructured":"Rockafellar RT, Wets RJ-B (2009) Variational analysis volume 317. Springer Science & Business Media"},{"key":"3611_CR36","unstructured":"Rudin W (1976) Principles of Mathematical Analysis. International series in pure and applied mathematics. McGraw-Hill"},{"key":"3611_CR37","doi-asserted-by":"crossref","unstructured":"Stewart T et al (2008) Real-world applications of multiobjective optimization. In J.\u00a0Branke, K.\u00a0Deb, K.\u00a0Miettinen, and R.\u00a0S\u0142owi\u0144ski, editors, Multiobjective Optimization: Interactive and Evolutionary Approaches, pages 285\u2013327. Springer, Berlin, Heidelberg. ISBN 978-3-540-88908-3","DOI":"10.1007\/978-3-540-88908-3_11"},{"issue":"3","key":"3611_CR38","doi-asserted-by":"publisher","first-page":"2388","DOI":"10.1137\/18M1191737","volume":"29","author":"J Wang","year":"2019","unstructured":"Wang J, Hu Y, Wai Yu CK, Li C, Yang X (2019) Extended Newton methods for multiobjective optimization: majorizing function technique and convergence analysis. SIAM J. Optim. 29(3):2388\u20132421","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3611_CR39","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10898-023-01349-x","volume":"89","author":"J-J Wang","year":"2024","unstructured":"Wang J-J, Tang L-P, Yang X-M (2024) Spectral projected subgradient method with a 1-memory momentum term for constrained multiobjective optimization problem. J Glob Optim 89(2):277\u2013302","journal-title":"J Glob Optim"},{"issue":"6","key":"3611_CR40","first-page":"929","volume":"5","author":"X Zhao","year":"2021","unstructured":"Zhao X, Sun Q, Liu L, Cho SY (2021) Convergence analysis of a projected gradient method for multiobjective optimization problems. J Nonlinear Var Anal 5(6):929\u2013938","journal-title":"J Nonlinear Var Anal"},{"issue":"6","key":"3611_CR41","first-page":"693","volume":"6","author":"X Zhao","year":"2022","unstructured":"Zhao X, Yao J, Yao Y (2022) A nonmonotone gradient method for constrained multiobjective optimization problems. J Nonlinear Var Anal 6(6):693\u2013706","journal-title":"J Nonlinear Var Anal"},{"issue":"4","key":"3611_CR42","first-page":"517","volume":"8","author":"X Zhao","year":"2024","unstructured":"Zhao X, Zhang H, Yao Y (2024) An inexact nonmonotone projected gradient method for constrained multiobjective optimization. J Nonlinear Var Anal 8(4):517\u2013531","journal-title":"J Nonlinear Var Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03611-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03611-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03611-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T09:14:47Z","timestamp":1770974087000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03611-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,13]]},"references-count":42,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["3611"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03611-2","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,13]]},"assertion":[{"value":"26 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 December 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"261"}}