{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T11:02:32Z","timestamp":1780052552636,"version":"3.53.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T00:00:00Z","timestamp":1777766400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T00:00:00Z","timestamp":1777766400000},"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":["Ann Oper Res"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s10479-026-07241-7","type":"journal-article","created":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T05:36:21Z","timestamp":1777786581000},"page":"837-871","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A proximal algorithm for constrained multiobjective minimization with quasiconvex functions"],"prefix":"10.1007","volume":"361","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8678-6918","authenticated-orcid":false,"given":"E. A. Papa","family":"Quiroz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"O. A.","family":"Bermeo Carrasco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Ichpas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,3]]},"reference":[{"key":"7241_CR1","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10957-018-1223-7","volume":"177","author":"SN Al-Homidan","year":"2018","unstructured":"Al-Homidan, S. N., & Hadjisavvas, L. (2018). Shaalan: Transformation of quasiconvex functions to eliminate local minima. Journal of Optimization Theory and Applications, 177, 93\u2013105.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"1","key":"7241_CR2","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/s10898-015-0367-3","volume":"64","author":"HCF Apolin\u00e1rio","year":"2016","unstructured":"Apolin\u00e1rio, H. C. F., Papa Quiroz, E., & Oliveira, P. R. (2016). A scalarization proximal point method for quasiconvex multiobjective minimization. Journal of Global Optimization, 64(1), 79\u201396.","journal-title":"Journal of Global Optimization"},{"key":"7241_CR3","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/s10107-004-0568-x","volume":"104","author":"A Auslender","year":"2005","unstructured":"Auslender, A., & Teboulle, M. (2005). Interior projection-like methods for monotone variational inequalities. Mathematical Programming, 104, 39\u201368.","journal-title":"Mathematical Programming"},{"issue":"3","key":"7241_CR4","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1137\/S1052623403427823","volume":"16","author":"A Auslender","year":"2006","unstructured":"Auslender, A., & Teboulle, M. (2006). Interior gradient and proximal methods for convex and conic optimization. SIAM Journal on Optimization, 16(3), 697\u2013725.","journal-title":"SIAM Journal on Optimization"},{"key":"7241_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-25939-6","volume-title":"Splitting algorithms, modern operator theory, and applications","author":"H Bauscke","year":"2019","unstructured":"Bauscke, H., Burachik, R., & Russell Luke, D. (2019). Splitting algorithms, modern operator theory, and applications (Vol. 1). Springer."},{"issue":"1","key":"7241_CR6","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1080\/02331934.2020.1800699","volume":"71","author":"Y Bello Cruz","year":"2022","unstructured":"Bello Cruz, Y., Melo, Y., & Serra, N. (2022). A proximal gradient splitting method for solving convex vector optimization problems. Optimization, 71(1), 33\u201353.","journal-title":"Optimization"},{"issue":"2","key":"7241_CR7","doi-asserted-by":"publisher","first-page":"1104","DOI":"10.1137\/16M107534X","volume":"28","author":"GC Bento","year":"2018","unstructured":"Bento, G. C., Cruz Neto, J., Lopez, G., Soubeyran, A., & Souza, J. (2018). The proximal point method for locally lipschitz functions in multiobjective optimization with application to the compromise problem. SIAM J. Optim., 28(2), 1104\u20131120.","journal-title":"SIAM J. Optim."},{"key":"7241_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10479-022-04719-y","volume":"316","author":"GC Bento","year":"2022","unstructured":"Bento, G. C., Cruz Neto, J., Meireles, L., & Soubeyran, A. (2022). Pareto solutions as limits of collective traps: An inexact multiobjective proximal point algorithm. Annals of Operations Research, 316, 1\u201319.","journal-title":"Annals of Operations Research"},{"issue":"3","key":"7241_CR9","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1007\/s11228-014-0279-2","volume":"22","author":"GC Bento","year":"2014","unstructured":"Bento, G. C., Cruz Neto, J., & Soubeyran, A. (2014). A proximal point-type method for multicriteria optimization. Set-Valued and Variational Analysis, 22(3), 557\u2013573.","journal-title":"Set-Valued and Variational Analysis"},{"issue":"1","key":"7241_CR10","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s10957-018-1258-9","volume":"177","author":"GC Bento","year":"2018","unstructured":"Bento, G. C., Ferreira, O. P., Soubeyran, A., & Sousa J\u00fanior, V. L. (2018). Inexact multi-objective local search proximal algorithms: Application to group dynamic and distributive justice problems. Journal of Optimization Theory and Applications, 177(1), 181\u2013200.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"4","key":"7241_CR11","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1137\/S1052623403429093","volume":"15","author":"H Bonnel","year":"2005","unstructured":"Bonnel, H., Iusem, A. N., & Svaiter, B. F. (2005). Proximal methods in vector optimization. SIAM Journal on Optimization, 15(4), 953\u2013970.","journal-title":"SIAM Journal on Optimization"},{"issue":"7","key":"7241_CR12","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1080\/02331934.2018.1440553","volume":"67","author":"R Bot","year":"2018","unstructured":"Bot, R., & Grad, S. (2018). Inertial forward-backward methods for solving vector optimization problems. Optimization, 67(7), 959\u2013974.","journal-title":"Optimization"},{"key":"7241_CR13","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1287\/opre.22.2.350","volume":"22","author":"SP Bradley","year":"1974","unstructured":"Bradley, S. P., & Frey, S. C. (1974). Fractional programming with homogeneus function. Operations Research, 22, 350\u2013357.","journal-title":"Operations Research"},{"key":"7241_CR14","doi-asserted-by":"publisher","first-page":"2653","DOI":"10.1137\/080733437","volume":"20","author":"R Burachik","year":"2010","unstructured":"Burachik, R., & Dutta, J. (2010). Inexact proximal point methods for variational inequality problems. SIAM Journal on Optimization, 20, 2653\u20132678.","journal-title":"SIAM Journal on Optimization"},{"issue":"1","key":"7241_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ejor.2006.09.070","volume":"183","author":"L-C Ceng","year":"2007","unstructured":"Ceng, L.-C., & Yao, J.-C. (2007). Approximate proximal methods in vector optimization. European Journal of Operational Research, 183(1), 1\u201319.","journal-title":"European Journal of Operational Research"},{"key":"7241_CR16","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971309","volume-title":"Optimization and nonsmooth analysis","author":"F Clarke","year":"1990","unstructured":"Clarke, F. (1990). Optimization and nonsmooth analysis. Philadelphia: SIAM."},{"key":"7241_CR17","unstructured":"De\u00a0Weck, O. L. (2004). Multiobjective optimization: History and promise. In Invited keynote paper, GL2-2, the third China-Japan-Korea joint symposium on optimization of structural and mechanical systems (vol. 2, pp. 34). Kanazawa, Japan."},{"issue":"1","key":"7241_CR18","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/BF02680553","volume":"83","author":"J Eckstein","year":"1998","unstructured":"Eckstein, J. (1998). Approximate iterations in bregman-function-based proximal algorithms. Mathematical Programming, 83(1), 113\u2013123.","journal-title":"Mathematical Programming"},{"issue":"1","key":"7241_CR19","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., & Bertsekas, D. P. (1992). On the douglas\u2013rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55(1), 293\u2013318.","journal-title":"Mathematical Programming"},{"key":"7241_CR20","volume-title":"Mathematical psychics","author":"FY Edgeworth","year":"1881","unstructured":"Edgeworth, F. Y. (1881). Mathematical psychics (Vol. 10). London: CK Paul."},{"key":"7241_CR21","volume-title":"Multicriteria optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M. (2005). Multicriteria optimization (Vol. 491). Heidelberg: Springer."},{"issue":"3","key":"7241_CR22","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s10898-022-01226-z","volume":"85","author":"S-M Grad","year":"2023","unstructured":"Grad, S.-M., Lara, F., & Marcavillaca, R. (2023). Relaxed-inertial proximal point type algorithms for quasiconvex minimization. Journal of Global Optimization, 85(3), 615\u2013635.","journal-title":"Journal of Global Optimization"},{"issue":"1","key":"7241_CR23","doi-asserted-by":"publisher","first-page":"1259","DOI":"10.1080\/02331931003603133","volume":"59","author":"AN Iusem","year":"2010","unstructured":"Iusem, A. N., & Sosa, W. (2010). A proximal point method for quasi-equilibrium problems in hilbert spaces. Optimization, 59(1), 1259\u20131274.","journal-title":"Optimization"},{"key":"7241_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-24828-6","volume-title":"Vector optimization, theory, application, and extensions","author":"J Jhan","year":"2004","unstructured":"Jhan, J. (2004). Vector optimization, theory, application, and extensions. Springer."},{"key":"7241_CR25","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1080\/02331930410001661217","volume":"53","author":"A Kaplan","year":"2004","unstructured":"Kaplan, A., & Tichatschke, R. (2004). On inexact generalized proximal methods with a weakened error tolerance criterion. Optimization, 53, 3\u201317.","journal-title":"Optimization"},{"issue":"1","key":"7241_CR26","doi-asserted-by":"publisher","first-page":"51","DOI":"10.7151\/dmdico.1111","volume":"30","author":"A Kaplan","year":"2007","unstructured":"Kaplan, A., & Tichatschke, R. (2007). Interior proximal method for variational inequalities on non-polyhedral sets. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, 30(1), 51\u201359.","journal-title":"Discussiones Mathematicae, Differential Inclusions, Control and Optimization"},{"issue":"1","key":"7241_CR27","doi-asserted-by":"publisher","first-page":"51","DOI":"10.7151\/dmdico.1111","volume":"30","author":"A Kaplan","year":"2010","unstructured":"Kaplan, A., & Tichatschke, R. (2010). Note on the paper: Interior proximal method for variational inequalities on non-polyhedral sets. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, 30(1), 51\u201359.","journal-title":"Discussiones Mathematicae, Differential Inclusions, Control and Optimization"},{"issue":"4","key":"7241_CR28","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1137\/S0363012995281742","volume":"35","author":"KC Kiwiel","year":"1997","unstructured":"Kiwiel, K. C. (1997). Proximal minimization methods with generalized bregman functions. SIAM Journal on Control and Optimization, 35(4), 1142\u20131168.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"3","key":"7241_CR29","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1007\/s10898-011-9752-8","volume":"52","author":"N Langenberg","year":"2012","unstructured":"Langenberg, N., & Tichatschke, R. (2012). Interior proximal methods for quasiconvex optimization. Journal of Global Optimization, 52(3), 641\u2013661.","journal-title":"Journal of Global Optimization"},{"key":"7241_CR30","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s10957-021-01996-8","volume":"192","author":"F Lara","year":"2022","unstructured":"Lara, F. (2022). On strongly quasiconvex functions: Existence results and proximal point algorithms. Journal of Optimization Theory and Applications, 192, 891\u2013911.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"7241_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7091-2438-3","volume-title":"Multicriteria decision making, courses and lectures","author":"G Leitmann","year":"1975","unstructured":"Leitmann, G., & Marzollo, A. (1975). Multicriteria decision making, courses and lectures (Vol. 211). Springer."},{"key":"7241_CR32","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet, B. (1970). Regularisation, d\u2019in\u00e9quations variationelles par approximations succesives. Revue Francaise d\u2019informatique et de Recherche Operationelle, 4, 154\u2013159.","journal-title":"Revue Francaise d\u2019informatique et de Recherche Operationelle"},{"key":"7241_CR33","volume-title":"Microeconomic theory","author":"A Mas-Colell","year":"1995","unstructured":"Mas-Colell, A., Whinston, M. D., & Green, J. (1995). Microeconomic theory (Vol. 1). New York, New York: Oxford University Press."},{"key":"7241_CR34","volume-title":"Nonlinear multiobjective optimization","author":"K Miettinen","year":"2012","unstructured":"Miettinen, K. (2012). Nonlinear multiobjective optimization (Vol. 12). New York: Springer."},{"key":"7241_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-31246-3","volume-title":"Variational analysis and generalized differentiation i: basic theory","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation i: basic theory (Vol. 330). Heidelberg: Springer."},{"issue":"3","key":"7241_CR36","doi-asserted-by":"publisher","first-page":"1028","DOI":"10.1007\/s10957-019-01582-z","volume":"183","author":"EA Papa Quiroz","year":"2019","unstructured":"Papa Quiroz, E. A., Apolin\u00e1rio, H. C. F., Villacorta, K. D., & Oliveira, P. R. (2019). A linear scalarization proximal point method for quasiconvex multiobjective minimization. Journal of Optimization Theory and Applications, 183(3), 1028\u20131052.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"7241_CR37","doi-asserted-by":"crossref","unstructured":"Papa Quiroz E. A., & Cruzado, S. (2022). An inexact scalarization proximal point method for multiobjective quasiconvex minimization. Annals of Operations Research,316, 1445\u20131470.","DOI":"10.1007\/s10479-020-03622-8"},{"key":"7241_CR38","doi-asserted-by":"crossref","unstructured":"Papa Quiroz E. A., & Cruzado, S. (2022). Linear and superlinear convergence of an inexact algorithm with proximal distances for variational inequality problems. Fixed Point Theory,23(1), 311\u2013330.","DOI":"10.24193\/fpt-ro.2022.1.20"},{"key":"7241_CR39","doi-asserted-by":"crossref","unstructured":"Papa Quiroz E. A., Ramirez, L. M., & Oliveira, P. R. (2015). An inexact proximal method for quasiconvex minimization. European Journal of Operational Research,246(3), 721\u2013729.","DOI":"10.1016\/j.ejor.2015.05.041"},{"key":"7241_CR40","doi-asserted-by":"crossref","unstructured":"Papa Quiroz E. A., Ramirez, L. M., & Oliveira, P. R. (2018). An inexact algorithm with proximal distances for variational inequalities. RAIRO-Operations Research,52(1), 159\u2013176.","DOI":"10.1051\/ro\/2017078"},{"key":"7241_CR41","unstructured":"Pareto, V. (1906). Manuale di economia politica."},{"key":"7241_CR42","volume-title":"Introduction to optimization","author":"BT Polyak","year":"2018","unstructured":"Polyak, B. T. (2018). Introduction to optimization. New York: Optimization Software Inc."},{"issue":"5","key":"7241_CR43","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R. T. (1976). Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization, 14(5), 877\u2013898.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"7241_CR44","volume-title":"Variational analysis","author":"RT Rockafellar","year":"2009","unstructured":"Rockafellar, R. T., & Wets, R.J.-B. (2009). Variational analysis (Vol. 317). Heidelberg: Grundlehren der Mathematischen, Wissenschaften, Springer."},{"issue":"2","key":"7241_CR45","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1080\/02331934.2015.1062009","volume":"65","author":"O Sarmiento","year":"2016","unstructured":"Sarmiento, O., & Papa Quiroz, P. R. E. A. (2016). Oliveira: A proximal multiplier method for separable convex minimization. Optimization, 65(2), 501\u2013537.","journal-title":"Optimization"},{"key":"7241_CR46","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10589-005-3065-0","volume":"33","author":"P Silva","year":"2006","unstructured":"Silva, P., & Eckstein, J. (2006). Double-regularization proximal methods, with complementarity applications. Computational Optimization and Applications, 33, 115\u2013156.","journal-title":"Computational Optimization and Applications"},{"issue":"1","key":"7241_CR47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00932634","volume":"29","author":"W Stadler","year":"1979","unstructured":"Stadler, W. (1979). A survey of multicriteria optimization or the vector maximum problem. Journal of Optimization Theory and Applications, 29(1), 1\u201352.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"7241_CR48","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-0035-6","volume-title":"Fractional programming: theory, methods and applications","author":"IM Stancu-Minassian","year":"1997","unstructured":"Stancu-Minassian, I. M. (1997). Fractional programming: theory, methods and applications. The Netherlands: Kluwer Academic Publisher."},{"key":"7241_CR49","volume-title":"Multiple criteria optimization; theory, computation, and application","author":"RE Steuer","year":"1985","unstructured":"Steuer, R. E. (1985). Multiple criteria optimization; theory, computation, and application. Wiley Online Library."},{"issue":"4","key":"7241_CR50","doi-asserted-by":"publisher","first-page":"1069","DOI":"10.1137\/S1052623495292130","volume":"7","author":"M Teboulle","year":"1997","unstructured":"Teboulle, M. (1997). Convergence of proximal-like algorithms. SIAM Journal on Optimization, 7(4), 1069\u20131083.","journal-title":"SIAM Journal on Optimization"},{"issue":"3","key":"7241_CR51","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1016\/j.ejor.2011.05.006","volume":"214","author":"KD Villacorta","year":"2011","unstructured":"Villacorta, K. D., & Oliveira, P. R. (2011). An interior proximal method in vector optimization. European Journal of Operational Research, 214(3), 485\u2013492.","journal-title":"European Journal of Operational Research"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-026-07241-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-026-07241-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-026-07241-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T10:01:53Z","timestamp":1780048913000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-026-07241-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,3]]},"references-count":51,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["7241"],"URL":"https:\/\/doi.org\/10.1007\/s10479-026-07241-7","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,3]]},"assertion":[{"value":"6 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 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":"The authors have not relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}},{"value":"This work was not involving human participants and\/or animals.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"value":"The authors obtained consent from the responsible authorities at the institute where the work has been carried out.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed Consent"}}]}}