{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T07:08:05Z","timestamp":1773472085360,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T00:00:00Z","timestamp":1686268800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T00:00:00Z","timestamp":1686268800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["20J21961"],"award-info":[{"award-number":["20J21961"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["21K11769"],"award-info":[{"award-number":["21K11769"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["19K11840"],"award-info":[{"award-number":["19K11840"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2023,11]]},"DOI":"10.1007\/s10589-023-00497-w","type":"journal-article","created":{"date-parts":[[2023,6,9]],"date-time":"2023-06-09T13:01:53Z","timestamp":1686315713000},"page":"421-455","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["An accelerated proximal gradient method for multiobjective optimization"],"prefix":"10.1007","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0042-9221","authenticated-orcid":false,"given":"Hiroki","family":"Tanabe","sequence":"first","affiliation":[]},{"given":"Ellen H.","family":"Fukuda","sequence":"additional","affiliation":[]},{"given":"Nobuo","family":"Yamashita","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,9]]},"reference":[{"key":"497_CR1","doi-asserted-by":"crossref","first-page":"2088","DOI":"10.1109\/TSMCB.2004.834438","volume":"34","author":"S Bandyopadhyay","year":"2004","unstructured":"Bandyopadhyay, S., Pal, S., Aruna, B.: Multiobjective GAs, quantitative indices, and pattern classification. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 34, 2088\u20132099 (2004)","journal-title":"IEEE Trans. Syst. Man Cybern. Part B (Cybern.)"},{"key":"497_CR2","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611974997","volume-title":"First-Order Methods in Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-Order Methods in Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2017)"},{"key":"497_CR3","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2, 183\u2013202 (2009)","journal-title":"SIAM J. Imag. Sci."},{"key":"497_CR4","volume-title":"Nonlinear Programming","author":"DP Bertsekas","year":"1999","unstructured":"Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont, Massachusetts (1999)","edition":"2"},{"key":"497_CR5","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1080\/02331934.2018.1440553","volume":"67","author":"RI Bo\u0163","year":"2018","unstructured":"Bo\u0163, R.I., Grad, S.M.: Inertial forward\u2013backward methods for solving vector optimization problems. Optimization 67, 959\u2013974 (2018)","journal-title":"Optimization"},{"key":"497_CR6","doi-asserted-by":"crossref","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":"497_CR7","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1137\/S1052623403429093","volume":"15","author":"H Bonnel","year":"2005","unstructured":"Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15, 953\u2013970 (2005)","journal-title":"SIAM J. Optim."},{"key":"497_CR8","first-page":"605","volume":"6","author":"G Bouza","year":"2022","unstructured":"Bouza, G., Tammer, C.: A steepest descent-like method for vector optimization problems with variable domination structure. J. Nonlinear Var. Anal. 6, 605\u2013618 (2022)","journal-title":"J. Nonlinear Var. Anal."},{"key":"497_CR9","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex Optimization","author":"S Boyd","year":"2004","unstructured":"Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)"},{"key":"497_CR10","volume-title":"Algorithms for Minimization Without Derivatives","author":"RP Brent","year":"1973","unstructured":"Brent, R.P.: Algorithms for Minimization Without Derivatives. Prentice-Hall, New Jersey (1973)"},{"key":"497_CR11","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/S1052623497325107","volume":"9","author":"RH Byrd","year":"1999","unstructured":"Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9, 877\u2013900 (1999)","journal-title":"SIAM J. Optim."},{"key":"497_CR12","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s10107-015-0962-6","volume":"159","author":"GA Carrizo","year":"2016","unstructured":"Carrizo, G.A., Lotito, P.A., Maciel, M.C.: Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem. Math. Program. 159, 339\u2013369 (2016)","journal-title":"Math. Program."},{"key":"497_CR13","doi-asserted-by":"crossref","first-page":"968","DOI":"10.1007\/s10957-015-0746-4","volume":"166","author":"A Chambolle","year":"2015","unstructured":"Chambolle, A., Dossal, C.: On the convergence of the iterates of the \u201cfast iterative shrinkage\/thresholding algorithm\u2019\u2019. J. Optim. Theory Appl. 166, 968\u2013982 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"497_CR14","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1137\/10079731X","volume":"21","author":"AL Cust\u00f3dio","year":"2011","unstructured":"Cust\u00f3dio, A.L., Madeira, J.F., Vaz, A.I., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21, 1109\u20131140 (2011)","journal-title":"SIAM J. Optim."},{"key":"497_CR15","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. B 91, 201\u2013213 (2002)","journal-title":"Math. Program. Ser. B"},{"key":"497_CR16","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1007\/s10957-020-01785-9","volume":"188","author":"M El Moudden","year":"2021","unstructured":"El Moudden, M., El Mouatasim, A.: Accelerated diagonal steepest descent method for unconstrained multiobjective optimization. J. Optim. Theory Appl. 188, 220\u2013242 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"497_CR17","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1137\/08071692X","volume":"20","author":"J Fliege","year":"2009","unstructured":"Fliege, J., Gra\u00f1a Drummond, L.M., Svaiter, B.F.: Newton\u2019s method for multiobjective optimization. SIAM J. Optim. 20, 602\u2013626 (2009)","journal-title":"SIAM J. Optim."},{"key":"497_CR18","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s001860000043","volume":"51","author":"J Fliege","year":"2000","unstructured":"Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479\u2013494 (2000)","journal-title":"Math. Methods Oper. Res."},{"key":"497_CR19","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1080\/10556788.2018.1510928","volume":"34","author":"J Fliege","year":"2019","unstructured":"Fliege, J., Vaz, A.I.F., Vicente, L.N.: Complexity of gradient descent for multiobjective optimization. Optim. Methods Softw. 34, 949\u2013959 (2019)","journal-title":"Optim. Methods Softw."},{"key":"497_CR20","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/s10589-012-9501-z","volume":"54","author":"EH Fukuda","year":"2013","unstructured":"Fukuda, E.H., Gra\u00f1a Drummond, L.M.: Inexact projected gradient method for vector optimization. Comput. Optim. Appl. 54, 473\u2013493 (2013)","journal-title":"Comput. Optim. Appl."},{"key":"497_CR21","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1590\/0101-7438.2014.034.03.0585","volume":"34","author":"EH Fukuda","year":"2014","unstructured":"Fukuda, E.H., Gra\u00f1a Drummond, L.M.: A survey on multiobjective descemt methods. Pesquisa Operacional 34, 585\u2013620 (2014)","journal-title":"Pesquisa Operacional"},{"key":"497_CR22","doi-asserted-by":"crossref","unstructured":"Gandibleux,\u00a0X., Sevaux,\u00a0M., S\u00f6rensen,\u00a0K., T\u2019kindt,\u00a0V.: Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and Mathematical Systems, vol. 535. Springer, Berlin (2004)","DOI":"10.1007\/978-3-642-17144-4"},{"key":"497_CR23","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1002\/nav.3800020106","volume":"2","author":"S Gass","year":"1955","unstructured":"Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2, 39\u201345 (1955)","journal-title":"Naval Res. Logist. Q."},{"key":"497_CR24","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1016\/0022-247X(68)90201-1","volume":"22","author":"AM Geoffrion","year":"1968","unstructured":"Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618\u2013630 (1968)","journal-title":"J. Math. Anal. Appl."},{"key":"497_CR25","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s10589-022-00414-7","volume":"83","author":"MLN Gon\u00e7alves","year":"2022","unstructured":"Gon\u00e7alves, M.L.N., Lima, F.S., Prudente, L.F.: Globally convergent Newton-type methods for multiobjective optimization. Comput. Optim. Appl. 83, 403\u2013434 (2022)","journal-title":"Comput. Optim. Appl."},{"key":"497_CR26","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/B:COAP.0000018877.86161.8b","volume":"28","author":"LM Gra\u00f1a Drummond","year":"2004","unstructured":"Gra\u00f1a Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28, 5\u201329 (2004)","journal-title":"Comput. Optim. Appl."},{"key":"497_CR27","unstructured":"Jin,\u00a0Y., Olhofer,\u00a0M., Sendhoff,\u00a0B.: Dynamic weighted aggregation for evolutionary multi-objective optimization: Why does it work and how? In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation, GECCO\u201901, San Francisco, CA, USA, (2001), Morgan Kaufmann Publishers Inc"},{"key":"497_CR28","first-page":"725","volume":"6","author":"E K\u00f6bis","year":"2022","unstructured":"K\u00f6bis, E., K\u00f6bis, M.A., Tammer, C.: A first bibliography on set and vector optimization problems with respect to variable domination structures. J. Nonlinear Var. Anal. 6, 725\u2013735 (2022)","journal-title":"J. Nonlinear Var. Anal."},{"key":"497_CR29","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1137\/17M1126588","volume":"28","author":"LR Lucambio P\u00e9rez","year":"2018","unstructured":"Lucambio P\u00e9rez, L.R., Prudente, L.F.: Nonlinear conjugate gradient methods for vector optimization. SIAM J. Optim. 28, 2690\u20132720 (2018)","journal-title":"SIAM J. Optim."},{"key":"497_CR30","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1007\/s00158-009-0460-7","volume":"41","author":"RT Marler","year":"2010","unstructured":"Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41, 853\u2013862 (2010)","journal-title":"Struct. Multidiscip. Optim."},{"key":"497_CR31","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10898-019-00802-0","volume":"75","author":"K Mita","year":"2019","unstructured":"Mita, K., Fukuda, E.H., Yamashita, N.: Nonmonotone line searches for unconstrained multiobjective optimization problems. J. Global Optim. 75, 63\u201390 (2019)","journal-title":"J. Global Optim."},{"key":"497_CR32","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1145\/355934.355936","volume":"7","author":"JJ Mor\u00e9","year":"1981","unstructured":"Mor\u00e9, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17\u201341 (1981)","journal-title":"ACM Trans. Math. Softw."},{"key":"497_CR33","doi-asserted-by":"crossref","first-page":"273","DOI":"10.24033\/bsmf.1625","volume":"93","author":"J-J Moreau","year":"1965","unstructured":"Moreau, J.-J.: Proximit\u00e9 et dualit\u00e9 dans un espace hilbertien. Bull. Soc. Math. France 93, 273\u2013299 (1965)","journal-title":"Bull. Soc. Math. France"},{"key":"497_CR34","first-page":"543","volume":"269","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method for solving the convex programming problem with convergence rate $$O(1\/k^2)$$. Dokl. Akad. Nauk SSSR 269, 543\u2013547 (1983)","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"497_CR35","doi-asserted-by":"crossref","DOI":"10.1561\/9781601987174","volume-title":"Proximal Algorithms","author":"N Parikh","year":"2014","unstructured":"Parikh, N., Boyd, S.: Proximal Algorithms, vol. 1. Now Publishers Inc, Boston (2014)"},{"key":"497_CR36","volume-title":"Variational Analysis. Grundlehren der mathematischen Wissenschaften","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Grundlehren der mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)"},{"key":"497_CR37","doi-asserted-by":"crossref","first-page":"171","DOI":"10.2140\/pjm.1958.8.171","volume":"8","author":"M Sion","year":"1958","unstructured":"Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171\u2013176 (1958)","journal-title":"Pac. J. Math."},{"key":"497_CR38","volume-title":"Progress in Aeronautics and Astronautics: Structural Optimization: Status and Promise","author":"W Stadler","year":"1992","unstructured":"Stadler, W., Dauer, J.: Multicriteria optimization in engineering: a tutorial and survey. In: Kamat, M.P. (ed.) Progress in Aeronautics and Astronautics: Structural Optimization: Status and Promise, vol. 150. American Institute of Aeronautics and Astronautics, Washington DC (1992)"},{"key":"497_CR39","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/j.orl.2018.05.008","volume":"46","author":"BF Svaiter","year":"2018","unstructured":"Svaiter, B.F.: The multiobjective steepest descent direction is not Lipschitz continuous, but is H\u00f6lder continuous. Oper. Res. Lett. 46, 430\u2013433 (2018)","journal-title":"Oper. Res. Lett."},{"key":"497_CR40","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s10589-018-0043-x","volume":"72","author":"H Tanabe","year":"2019","unstructured":"Tanabe, H., Fukuda, E.H., Yamashita, N.: Proximal gradient methods for multiobjective optimization and their applications. Comput. Optim. Appl. 72, 339\u2013361 (2019)","journal-title":"Comput. Optim. Appl."},{"key":"497_CR41","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s11590-022-01877-7","volume":"17","author":"H Tanabe","year":"2023","unstructured":"Tanabe, H., Fukuda, E.H., Yamashita, N.: Convergence rates analysis of a multiobjective proximal gradient method. Optim. Lett. 17, 333\u2013350 (2023)","journal-title":"Optim. Lett."},{"key":"497_CR42","doi-asserted-by":"crossref","unstructured":"Tanabe,\u00a0H., Fukuda,\u00a0E.\u00a0H., Yamashita,\u00a0N.: New merit functions for multiobjective optimization and their properties. arXiv:2010.09333 (2023)","DOI":"10.1080\/02331934.2023.2232794"},{"key":"497_CR43","unstructured":"Toint, P.L.: Test problems for partially separable optimization and results for the routine PSPMIN, Namur Report (1983)"},{"key":"497_CR44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/jimo.2018136","volume":"16","author":"X Wang","year":"2020","unstructured":"Wang, X., Wang, Y., Wang, G.: An accelerated augmented Lagrangian method for multi-criteria optimization problem. J. Ind. Manag. Optim. 16, 1\u20139 (2020)","journal-title":"J. Ind. Manag. Optim."},{"key":"497_CR45","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61859-8","volume-title":"Functional Analysis, Classics in Mathematics","author":"K Yosida","year":"1995","unstructured":"Yosida, K.: Functional Analysis, Classics in Mathematics, vol. 123, 6th edn. Springer, Berlin (1995)","edition":"6"},{"key":"497_CR46","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1109\/TAC.1963.1105511","volume":"8","author":"LA Zadeh","year":"1963","unstructured":"Zadeh, L.A.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8, 59\u201360 (1963)","journal-title":"IEEE Trans. Autom. Control"},{"key":"497_CR47","first-page":"441","volume":"5","author":"X Zhao","year":"2021","unstructured":"Zhao, X., Jolaoso, L.O., Shehu, Y., Yao, J.-C.: Convergence of a nonmonotone projected gradient method for nonconvex multiobjective optimization. J. Nonlinear Var. Anal. 5, 441\u2013457 (2021)","journal-title":"J. Nonlinear Var. Anal."},{"key":"497_CR48","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1162\/106365600568202","volume":"8","author":"E Zitzler","year":"2000","unstructured":"Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8, 173\u2013195 (2000)","journal-title":"Evol. Comput."},{"key":"497_CR49","unstructured":"Zitzler,\u00a0E., Technische,\u00a0E., Z\u00fcrich,\u00a0H.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications, Phd thesis, Swiss Federal Institute of Technology Zurich (1999)"},{"key":"497_CR50","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1111\/j.1467-9868.2005.00503.x","volume":"67","author":"H Zou","year":"2005","unstructured":"Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67, 301\u2013320 (2005)","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00497-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-023-00497-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00497-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T16:06:13Z","timestamp":1695830773000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-023-00497-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,9]]},"references-count":50,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,11]]}},"alternative-id":["497"],"URL":"https:\/\/doi.org\/10.1007\/s10589-023-00497-w","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,9]]},"assertion":[{"value":"20 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 June 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}