{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T19:44:01Z","timestamp":1771530241476,"version":"3.50.1"},"reference-count":87,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T00:00:00Z","timestamp":1736035200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T00:00:00Z","timestamp":1736035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001843","name":"Science and Engineering Research Board","doi-asserted-by":"crossref","award":["SIR\/2022\/001358"],"award-info":[{"award-number":["SIR\/2022\/001358"]}],"id":[{"id":"10.13039\/501100001843","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Prime Minister Research Fellowship","award":["PMRF ID-2703573"],"award-info":[{"award-number":["PMRF ID-2703573"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s10479-024-06461-z","type":"journal-article","created":{"date-parts":[[2025,1,5]],"date-time":"2025-01-05T06:52:52Z","timestamp":1736059972000},"page":"417-466","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Inexact proximal point method with a Bregman regularization for quasiconvex multiobjective optimization problems via limiting subdifferentials"],"prefix":"10.1007","volume":"345","author":[{"given":"Balendu Bhooshan","family":"Upadhyay","sequence":"first","affiliation":[]},{"given":"Subham","family":"Poddar","sequence":"additional","affiliation":[]},{"given":"Jen-Chih","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5437-5591","authenticated-orcid":false,"given":"Xiaopeng","family":"Zhao","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,5]]},"reference":[{"key":"6461_CR1","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s10957-018-1223-7","volume":"177","author":"S Al-Homidan","year":"2018","unstructured":"Al-Homidan, S., Hadjisavvas, N., & Shaalan, L. (2018). 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"},{"key":"6461_CR2","doi-asserted-by":"crossref","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. A., & Oliveira, P. R. (2016). A scalarization proximal point method for quasiconvex multiobjective minimization. Journal of Global Optimization, 64, 79\u201396.","journal-title":"Journal of Global Optimization"},{"issue":"1\u20132","key":"6461_CR3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s10107-011-0484-9","volume":"137","author":"H Attouch","year":"2013","unstructured":"Attouch, H., Bolte, J., & Svaiter, B. F. (2013). Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss\u2013Seidel methods. Mathematical Programming, 137(1\u20132), 91\u2013129.","journal-title":"Mathematical Programming"},{"key":"6461_CR4","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1287\/moor.24.3.645","volume":"24","author":"A Auslender","year":"1999","unstructured":"Auslender, A., Teboulle, M., & Ben-Tiba, S. (1999). Interior proximal and multiplier methods based on second order homogeneous kernels. Mathematics of Operations Research, 24, 645\u2013668.","journal-title":"Mathematics of Operations Research"},{"key":"6461_CR5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1023\/A:1022618915698","volume":"97","author":"D Aussel","year":"1998","unstructured":"Aussel, D. (1998). Subdifferential properties of quasiconvex and pseudoconvex function, unified approach. Journal of Optimization Theory and Applications, 97, 29\u201345.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"6461_CR6","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1137\/120866415","volume":"23","author":"JY Bello Cruz","year":"2013","unstructured":"Bello Cruz, J. Y. (2013). A subgradient method for vector optimization problems. SIAM Journal on Optimization, 23, 2169\u20132182.","journal-title":"SIAM Journal on Optimization"},{"key":"6461_CR7","doi-asserted-by":"crossref","unstructured":"Bento, G. C., Cruz Neto, J. X., Lopes, J. O., Mordukhovich, B. S., & Silva Filho P. R. (2024). A refined proximal algorithm for nonconvex multiobjective optimization in Hilbert spaces. arXiv preprint arXiv:2403.09922.","DOI":"10.1007\/s10898-024-01453-6"},{"issue":"2","key":"6461_CR8","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1137\/16M107534X","volume":"28","author":"GC Bento","year":"2018","unstructured":"Bento, G. C., Cruz Neto, J. X., L\u00f3pez, G., Soubeyran, A., & Souza, J. C. O. (2018). The proximal point method for locally Lipschitz functions in multiobjective optimization with application to the compromise problem. SIAM Journal on Optimization, 28(2), 1104\u20131120.","journal-title":"SIAM Journal on Optimization"},{"key":"6461_CR9","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1016\/j.ejor.2014.01.002","volume":"235","author":"GC Bento","year":"2014","unstructured":"Bento, G. C., Cruz Neto, J. X., Oliveira, P. R., & Soubeyran, A. (2014a). The self-regulation problem as an inexact steepest descent method for multicriteria optimization. European Journal of Operational Research, 235, 494\u2013502.","journal-title":"European Journal of Operational Research"},{"key":"6461_CR10","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s11228-014-0279-2","volume":"22","author":"GC Bento","year":"2014","unstructured":"Bento, G. C., Cruz, N. J. X., & Soubeyran, A. (2014b). A proximal point-type method for multicriteria optimization. Set-Valued and Variational Analysis, 22, 557\u2013573.","journal-title":"Set-Valued and Variational Analysis"},{"issue":"1","key":"6461_CR11","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10957-018-1330-5","volume":"179","author":"GC Bento","year":"2018","unstructured":"Bento, G. C., Neto, J. C., & Meireles, L. V. (2018). Proximal point method for locally Lipschitz functions in multiobjective optimization of Hadamard manifolds. Journal of Optimization Theory and Applications, 179(1), 37\u201352.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"6461_CR12","volume-title":"Constrained optimization and Lagrange multiplier methods","author":"DP Bertsekas","year":"2014","unstructured":"Bertsekas, D. P. (2014). Constrained optimization and Lagrange multiplier methods. New York: Academic Press."},{"issue":"4","key":"6461_CR13","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. (2005). Proximal methods in vector optimization. SIAM Journal on Optimization, 15(4), 953\u2013970.","journal-title":"SIAM Journal on Optimization"},{"issue":"2","key":"6461_CR14","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1287\/ijoc.10.2.209","volume":"10","author":"PS Bradley","year":"1998","unstructured":"Bradley, P. S., Mangasarian, O. L., & Street, W. N. (1998). Feature selection via mathematical programming. INFORMS Journal on Computing, 10(2), 209\u2013217.","journal-title":"INFORMS Journal on Computing"},{"issue":"3","key":"6461_CR15","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","volume":"7","author":"LM Bregman","year":"1967","unstructured":"Bregman, L. M. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics, 7(3), 200\u2013217.","journal-title":"USSR Computational Mathematics and Mathematical Physics"},{"key":"6461_CR16","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/BF02761171","volume":"29","author":"H Br\u00e9zis","year":"1978","unstructured":"Br\u00e9zis, H., & Lions, P. L. (1978). Produits infinis de r\u00e9solvantes. Israel Journal of Mathematics, 29, 329\u2013345.","journal-title":"Israel Journal of Mathematics"},{"key":"6461_CR17","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.ejor.2016.05.026","volume":"256","author":"AS Brito","year":"2017","unstructured":"Brito, A. S., Cruz Neto, J. X., Santos, P. S. M., & Souza, S. S. (2017). A relaxed projection method for solving multiobjective optimization problems. European Journal of Operational Research, 256, 17\u201323.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"6461_CR18","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1137\/S1052623495286302","volume":"8","author":"RS Burachik","year":"1998","unstructured":"Burachik, R. S., & Iusem, A. N. (1998). A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM Journal on Optimization, 8(1), 197\u2013216.","journal-title":"SIAM Journal on Optimization"},{"issue":"5","key":"6461_CR19","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1137\/S0363012998339745","volume":"39","author":"RS Burachik","year":"2001","unstructured":"Burachik, R. S., & Scheimberg, S. (2001). A proximal point algorithm for the variational inequality problem in Banach space. SIAM Journal on Control and Optimization, 39(5), 1633\u20131649.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"6461_CR20","volume-title":"Generalized convexity and optimization: Theory and applications","author":"A Cambini","year":"2009","unstructured":"Cambini, A., & Martein, L. (2009). Generalized convexity and optimization: Theory and applications. Berlin: Springer."},{"issue":"1","key":"6461_CR21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2006.09.070","volume":"183","author":"LC 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":"6461_CR22","first-page":"373","volume":"81","author":"Y Censor","year":"1998","unstructured":"Censor, Y., Iusem, A. N., & Zenios, S. A. (1998). An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Mathematical Programming, 81, 373\u2013400.","journal-title":"Mathematical Programming"},{"key":"6461_CR23","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/BF00934676","volume":"34","author":"Y Censor","year":"1981","unstructured":"Censor, Y., & Lent, A. (1981). An iterative row-action method for interval convex programming. Journal of Optimization Theory and Applications, 34, 321\u2013353.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"6461_CR24","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/BF00940051","volume":"73","author":"Y Censor","year":"1992","unstructured":"Censor, Y., & Zenios, S. A. (1992). The proximal minimization algorithm with D-function. Journal of Optimization Theory and Applications, 73, 451\u2013464.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"6","key":"6461_CR25","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1080\/00036811.2010.483428","volume":"90","author":"Z Chen","year":"2011","unstructured":"Chen, Z., Huang, X., & Yang, X. (2011). Generalized proximal point algorithms for multiobjective optimization problems. Applicable Analysis, 90(6), 935\u2013949.","journal-title":"Applicable Analysis"},{"issue":"3","key":"6461_CR26","doi-asserted-by":"crossref","first-page":"538","DOI":"10.1137\/0803026","volume":"3","author":"G Chen","year":"1993","unstructured":"Chen, G., & Teboulle, M. (1993). Convergence analysis of a proximal-like minimization algorithm using Bregman function. SIAM Journal on Optimization, 3(3), 538\u2013543.","journal-title":"SIAM Journal on Optimization"},{"key":"6461_CR27","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971309","volume-title":"Optimization and nonsmooth analysis","author":"FH Clarke","year":"1990","unstructured":"Clarke, F. H. (1990). Optimization and nonsmooth analysis. Philadelphia: SIAM."},{"issue":"3","key":"6461_CR28","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s10589-012-9494-7","volume":"54","author":"JX Da Cruz Neto","year":"2013","unstructured":"Da Cruz Neto, J. X., Da Silva, G. J. P., Ferreira, O. P., & Lopes, J. O. (2013). A subgradient method for multiobjective optimization. Computational Optimization and Applications, 54(3), 461\u2013472.","journal-title":"Computational Optimization and Applications"},{"key":"6461_CR29","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/BF00940283","volume":"51","author":"A De Pierro","year":"1986","unstructured":"De Pierro, A., & Iusem, A. N. (1986). A relaxed version of Bregman\u2019s method for convex programming. Journal of Optimization Theory and Applications, 51, 421\u2013440.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"10","key":"6461_CR30","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1016\/j.na.2005.03.104","volume":"64","author":"Y Dong","year":"2006","unstructured":"Dong, Y. (2006). A new relative error criterion for the proximal point algorithm. Nonlinear Analysis, 64(10), 2143\u20132148.","journal-title":"Nonlinear Analysis"},{"issue":"1","key":"6461_CR31","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1287\/moor.18.1.202","volume":"18","author":"J Eckstein","year":"1993","unstructured":"Eckstein, J. (1993). Nonlinear proximal point algorithms using Bregman function, with applications to convex programming. Mathematics of Operations Research, 18(1), 202\u2013226.","journal-title":"Mathematics of Operations Research"},{"issue":"1","key":"6461_CR32","first-page":"113","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":"2","key":"6461_CR33","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. (2009). Newton\u2019s method for multiobjective optimization. SIAM Journal on Optimization, 20(2), 602\u2013626.","journal-title":"SIAM Journal on Optimization"},{"key":"6461_CR34","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1080\/02331934.2010.522710","volume":"60","author":"EH Fukuda","year":"2011","unstructured":"Fukuda, E. H., & Gra\u00f1a Drummond, L. M. (2011). On the convergence of the projected gradient method for vector optimization. Optimization, 60, 1009\u20131021.","journal-title":"Optimization"},{"key":"6461_CR35","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. (2004). A projected gradient method for vector optimization problems. Computational Optimization and Applications, 28, 5\u201329.","journal-title":"Computational Optimization and Applications"},{"key":"6461_CR36","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.cam.2004.06.018","volume":"175","author":"LM Gra\u00f1a Drummond","year":"2005","unstructured":"Gra\u00f1a Drummond, L. M., & Svaiter, B. F. (2005). A steepest descent method for vector optimization. Journal of Computational and Applied Mathematics, 175, 395\u2013414.","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"6461_CR37","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1137\/0329022","volume":"29","author":"O G\u00fcler","year":"1991","unstructured":"G\u00fcler, O. (1991). On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization, 29, 403\u2013419.","journal-title":"SIAM Journal on Control and Optimization"},{"key":"6461_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-024-01419-8","author":"A Iusem","year":"2024","unstructured":"Iusem, A., Lara, F., Marcavillaca, R. T., & Yen, L. H. (2024). Two-step proximal point algorithms for pseudomonotone equlibrium problems with application in fractional programming. Journal of Global Optimization. https:\/\/doi.org\/10.1007\/s10898-024-01419-8","journal-title":"Journal of Global Optimization"},{"key":"6461_CR39","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-24828-6","volume-title":"Vector optimization: Theory, applications, and extensions","author":"J Jahn","year":"2004","unstructured":"Jahn, J. (2004). Vector optimization: Theory, applications, and extensions. Berlin: Springer."},{"issue":"2","key":"6461_CR40","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10898-022-01149-9","volume":"84","author":"A Kabgani","year":"2022","unstructured":"Kabgani, A., & Lara, F. (2022). Strong subdifferentials: Theory and applications in nonconvex optimization. Journal of Global Optimization, 84(2), 349\u2013368.","journal-title":"Journal of Global Optimization"},{"issue":"2","key":"6461_CR41","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1007\/s10957-015-0774-0","volume":"172","author":"PD Khanh","year":"2017","unstructured":"Khanh, P. D., Yao, J. C., & Yen, N. D. (2017). The Mordukhovich subdifferentials and directions of descent. Journal of Optimization Theory and Applications, 172(2), 518\u2013534.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"6461_CR42","doi-asserted-by":"crossref","first-page":"326","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, 326\u2013349.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"4","key":"6461_CR43","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s10898-009-9470-7","volume":"47","author":"N Langenberg","year":"2010","unstructured":"Langenberg, N. (2010). Pseudomonotone operators and the Bregman proximal point algorithm. Journal of Global Optimization, 47(4), 537\u2013555.","journal-title":"Journal of Global Optimization"},{"key":"6461_CR44","doi-asserted-by":"crossref","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"},{"issue":"3","key":"6461_CR45","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1080\/02331934.2022.2112580","volume":"73","author":"F Lara","year":"2024","unstructured":"Lara, F., & Marcavillaca, R. T. (2024). Bregman proximal point type algorithms for quasiconvex minimization. Optimization, 73(3), 497\u2013515.","journal-title":"Optimization"},{"key":"6461_CR46","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-50280-4","volume-title":"Theory of vector optimization","author":"DT Luc","year":"1989","unstructured":"Luc, D. T. (1989). Theory of vector optimization. Berlin: Springer."},{"issue":"2","key":"6461_CR47","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1137\/0322019","volume":"22","author":"FJ Luque","year":"1984","unstructured":"Luque, F. J. (1984). Asymptotic convergence analysis of the proximal point algorithm. SIAM Journal on Control and Optimization, 22(2), 277\u2013293.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"20","key":"6461_CR48","doi-asserted-by":"crossref","first-page":"5213","DOI":"10.1109\/TSP.2014.2340820","volume":"62","author":"M Malek-Mohammadi","year":"2014","unstructured":"Malek-Mohammadi, M., Babaie-Zadeh, M., & Skoglund, M. (2014). Iterative concave rank approximation for recovering low-rank matrices. IEEE Transactions on Signal Processing, 62(20), 5213\u20135226.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"6461_CR49","volume-title":"Nonlinear programming","author":"OL Mangasarian","year":"1969","unstructured":"Mangasarian, O. L. (1969). Nonlinear programming. New York: McGraw-Hill."},{"key":"6461_CR50","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet, B. (1970). Regularisation, d\u2019in\u00e9quations variationelles par approximations succesives. Revue fran\u00e7aise d\u2019informatique et de recherche op\u00e9rationnelle, 4, 154\u2013159.","journal-title":"Revue fran\u00e7aise d\u2019informatique et de recherche op\u00e9rationnelle"},{"key":"6461_CR51","volume-title":"Nonlinear multiobjective optimization","author":"K Miettinen","year":"1999","unstructured":"Miettinen, K. (1999). Nonlinear multiobjective optimization. Boston: Kluwer Academic Publishers."},{"issue":"6","key":"6461_CR52","doi-asserted-by":"crossref","first-page":"2424","DOI":"10.1109\/TSP.2009.2016889","volume":"57","author":"CJ Miosso","year":"2009","unstructured":"Miosso, C. J., von Borries, R., Argaez, M., Vel\u00e1zquez, L., Quintero, C., & Potes, C. (2009). Compressive sensing reconstruction with prior information by iteratively reweighted least-squares. IEEE Transactions on Signal Processing, 57(6), 2424\u20132431.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"6461_CR53","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s10898-011-9690-5","volume":"53","author":"SK Mishra","year":"2012","unstructured":"Mishra, S. K., Jaiswal, M., & Le Thi, H. A. (2012). Nonsmooth semi-infinite programming problem using limiting subdifferentials. Journal of Global Optimization, 53, 285\u2013296.","journal-title":"Journal of Global Optimization"},{"key":"6461_CR54","volume-title":"Variational analysis and generalized differentiation I: Basic theory","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B. S. (2006a). Variational analysis and generalized differentiation I: Basic theory. New York: Springer."},{"key":"6461_CR55","volume-title":"Variational analysis and generalized differentiation II: Applications","author":"BS Mordukhovich","year":"2006","unstructured":"Mordukhovich, B. S. (2006b). Variational analysis and generalized differentiation II: Applications. Berlin: Springer."},{"key":"6461_CR56","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s10898-007-9156-y","volume":"39","author":"S Pan","year":"2007","unstructured":"Pan, S., & Chen, J.-S. (2007). Entropy-like proximal algorithms based on a second-order homogeneous distance function for quasi-convex programming. Journal of Global Optimization, 39, 555\u2013575.","journal-title":"Journal of Global Optimization"},{"issue":"2","key":"6461_CR57","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1007\/s10479-020-03622-8","volume":"316","author":"EA Papa Quiroz","year":"2022","unstructured":"Papa Quiroz, E. A., & Cruzado, S. (2022). An inexact scalarization proximal point method for multiobjective quasiconvex minimization. Annals of Operations Research, 316(2), 1445\u20131470.","journal-title":"Annals of Operations Research"},{"key":"6461_CR58","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/j.ejor.2015.05.041","volume":"246","author":"EA Papa Quiroz","year":"2015","unstructured":"Papa Quiroz, E. A., Mallma Ramirez, L., & Oliveira, P. R. (2015). An inexact proximal method for quasiconvex minimization. European Journal of Operational Research, 246, 721\u2013729.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"6461_CR59","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1051\/ro\/2017078","volume":"52","author":"EA Papa Quiroz","year":"2018","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.","journal-title":"RAIRO Operations Research"},{"issue":"1","key":"6461_CR60","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.apm.2010.06.009","volume":"35","author":"L Pourkarimi","year":"2011","unstructured":"Pourkarimi, L., Yaghoobi, M. A., & Mashinchi, M. (2011). Efficient curve fitting: An application of multiobjective programming. Applied Mathematical Modelling, 35(1), 346\u2013365.","journal-title":"Applied Mathematical Modelling"},{"key":"6461_CR61","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.ejor.2014.01.022","volume":"235","author":"S Qu","year":"2014","unstructured":"Qu, S., Liu, C., Goh, M., Li, Y., & Ji, Y. (2014). Nonsmooth multiobjective programming with quasi-Newton methods. European Journal of Operational Research, 235, 503\u2013510.","journal-title":"European Journal of Operational Research"},{"issue":"4","key":"6461_CR62","doi-asserted-by":"crossref","first-page":"4369","DOI":"10.1007\/s40747-022-00963-8","volume":"9","author":"J Ren","year":"2023","unstructured":"Ren, J., Qiu, F., & Hu, H. (2023). Multiple sparse detection-based evolutionary algorithm for large-scale sparse multiobjective optimization problems. Complex & Intelligent Systems, 9(4), 4369\u20134388.","journal-title":"Complex & Intelligent Systems"},{"key":"6461_CR63","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1007\/BFb0120929","volume":"14","author":"S Robinson","year":"1981","unstructured":"Robinson, S. (1981). Some continuity properties of polyhedral multifunctions. Mathematical Programming Studies, 14, 206\u2013241.","journal-title":"Mathematical Programming Studies"},{"key":"6461_CR64","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press."},{"issue":"5","key":"6461_CR65","doi-asserted-by":"crossref","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":"6461_CR66","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. (1998). Variational analysis. Berlin: Springer."},{"issue":"5","key":"6461_CR67","doi-asserted-by":"crossref","first-page":"2809","DOI":"10.1109\/TWC.2014.040214.131770","volume":"13","author":"Y Shi","year":"2014","unstructured":"Shi, Y., Zhang, J., & Letaief, K. B. (2014). Group sparse beamforming for green cloud-RAN. IEEE Transactions on Wireless Communications, 13(5), 2809\u20132823.","journal-title":"IEEE Transactions on Wireless Communications"},{"key":"6461_CR68","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1016\/j.jmaa.2006.08.033","volume":"330","author":"M Soleimani-damaneh","year":"2007","unstructured":"Soleimani-damaneh, M. (2007). Characterization of nonsmooth quasiconvex and pseudoconvex functions. Journal of Mathematical Analysis and applications, 330, 1387\u20131392.","journal-title":"Journal of Mathematical Analysis and applications"},{"issue":"2","key":"6461_CR69","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1287\/moor.25.2.214.12222","volume":"25","author":"MV Solodov","year":"2000","unstructured":"Solodov, M. V., & Svaiter, B. F. (2000a). An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Mathematics of Operations Research, 25(2), 214\u2013230.","journal-title":"Mathematics of Operations Research"},{"key":"6461_CR70","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s101079900113","volume":"87","author":"MV Solodov","year":"2000","unstructured":"Solodov, M. V., & Svaiter, B. F. (2000b). Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming, 87, 189\u2013202.","journal-title":"Mathematical Programming"},{"issue":"2","key":"6461_CR71","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.ejor.2009.03.045","volume":"201","author":"S Souza","year":"2010","unstructured":"Souza, S., Oliviera, P., Cruz Neto, J., & Soubeyran, A. (2010). A proximal point algorithm with separable Bregman distances for quasiconvex optimization over the nonnegative orthant. European Journal of Operational Research, 201(2), 365\u2013376.","journal-title":"European Journal of Operational Research"},{"issue":"3","key":"6461_CR72","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1109\/TIT.2011.2178139","volume":"58","author":"W Stummer","year":"2012","unstructured":"Stummer, W., & Vajda, I. (2012). On Bregman distances and divergences of probability measures. IEEE Transactions on Information Theory, 58(3), 1277\u20131288.","journal-title":"IEEE Transactions on Information Theory"},{"issue":"6\u20137","key":"6461_CR73","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.orl.2014.06.009","volume":"42","author":"GJ Tang","year":"2014","unstructured":"Tang, G. J., & Huang, N. J. (2014). Rate of convergence for proximal point algorithms on Hadamard manifolds. Operation Research Letters, 42(6\u20137), 383\u2013387.","journal-title":"Operation Research Letters"},{"issue":"2","key":"6461_CR74","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1109\/TEVC.2019.2918140","volume":"24","author":"Y Tian","year":"2019","unstructured":"Tian, Y., Zhang, X., Wang, C., & Jin, Y. (2019). An evolutionary algorithm for large-scale sparse multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 24(2), 380\u2013393.","journal-title":"IEEE Transactions on Evolutionary Computation"},{"issue":"3","key":"6461_CR75","doi-asserted-by":"crossref","first-page":"523","DOI":"10.3390\/math10030523","volume":"10","author":"S Trean\u0163\u0103","year":"2022","unstructured":"Trean\u0163\u0103, S., Mishra, P., & Upadhyay, B. B. (2022). Minty variational principle for nonsmooth interval-valued vector optimization problems on Hadamard manifolds. Mathematics, 10(3), 523.","journal-title":"Mathematics"},{"key":"6461_CR76","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s41980-023-00791-7","volume":"49","author":"BB Upadhyay","year":"2023","unstructured":"Upadhyay, B. B., Ghosh, A., & Trean\u0163\u0103, S. (2023). Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems on Hadamard manifolds. Bulletin of the Iranian Mathematical Society, 49, 45.","journal-title":"Bulletin of the Iranian Mathematical Society"},{"issue":"1","key":"6461_CR77","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2023.127785","volume":"531","author":"BB Upadhyay","year":"2024","unstructured":"Upadhyay, B. B., Ghosh, A., & Trean\u0163\u0103, S. (2024). Optimality conditions and duality for nonsmooth multiobjective semi-infnite programming problems with vanishing constraints on Hadamard manifolds. Journal of Mathematical Analysis and Applications, 531(1), 127785.","journal-title":"Journal of Mathematical Analysis and Applications"},{"issue":"1","key":"6461_CR78","doi-asserted-by":"crossref","first-page":"282","DOI":"10.37398\/JSR.2020.640139","volume":"64","author":"BB Upadhyay","year":"2020","unstructured":"Upadhyay, B. B., & Mishra, P. (2020). On generalized Minty and Stampacchia vector variational-like inequalities and nonsmooth vector optimization problem involving higher order strong invexity. Journal of Scientific Research, 64(1), 282\u2013291.","journal-title":"Journal of Scientific Research"},{"issue":"4","key":"6461_CR79","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.3934\/jimo.2023138","volume":"20","author":"BB Upadhyay","year":"2024","unstructured":"Upadhyay, B. B., Pandey, R. K., & Liao, S. (2024a). Newton\u2019s method for interval-valued multiobjective optimization problem. Journal of Industrial and Management Optimization, 20(4), 1633\u20131661.","journal-title":"Journal of Industrial and Management Optimization"},{"key":"6461_CR80","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115550","volume":"438","author":"BB Upadhyay","year":"2024","unstructured":"Upadhyay, B. B., Pandey, R. K., Pan, J., & Zeng, S. (2024b). Quasi-Newton algorithms for solving interval-valued multiobjective optimization problems by using their certain equivalence. Journal of Computational and Applied Mathematics, 438, 115550.","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"6461_CR81","doi-asserted-by":"crossref","DOI":"10.1016\/j.fss.2024.109066","volume":"492","author":"BB Upadhyay","year":"2024","unstructured":"Upadhyay, B. B., Pandey, R. K., & Zeng, S. (2024c). A generalization of generalized Hukuhara Newton\u2019s method for interval-valued multiobjective optimization problems. Fuzzy Sets and Systems, 492, 109066.","journal-title":"Fuzzy Sets and Systems"},{"issue":"1","key":"6461_CR82","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1146\/annurev-statistics-022513-115643","volume":"1","author":"MJ Wainwright","year":"2014","unstructured":"Wainwright, M. J. (2014). Structured regularizers for high-dimensional problems: Statistical and computational issues. Annual Review of Statistics and Its Application, 1(1), 233\u2013253.","journal-title":"Annual Review of Statistics and Its Application"},{"key":"6461_CR83","first-page":"1439","volume":"3","author":"J Weston","year":"2003","unstructured":"Weston, J., Elisseeff, A., Sch\u00f6lkopf, B., & Tipping, M. (2003). Use of the zero-norm with linear models and kernel methods. Journal of Machine Learning Research, 3, 1439\u20131461.","journal-title":"Journal of Machine Learning Research"},{"issue":"7","key":"6461_CR84","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1109\/TSP.2009.2016892","volume":"57","author":"SJ Wright","year":"2009","unstructured":"Wright, S. J., Nowak, R. D., & Figueiredo, M. A. (2009). Sparse reconstruction by separable approximation. IEEE Transactions on Signal Processing, 57(7), 2479\u20132493.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"3","key":"6461_CR85","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1137\/20M1360748","volume":"32","author":"L Yang","year":"2022","unstructured":"Yang, L., & Toh, K. C. (2022). Bregman proximal point algorithm revisited: A new inexact version and its inertial variant. SIAM Journal on Optimization, 32(3), 1523\u20131554.","journal-title":"SIAM Journal on Optimization"},{"key":"6461_CR86","volume":"167","author":"C Yue","year":"2020","unstructured":"Yue, C., Liang, J., Qu, B., Han, Y., Zhu, Y., & Crisalle, O. D. (2020). A novel multiobjective optimization algorithm for sparse signal reconstruction. Signal Processing, 167, 107292.","journal-title":"Signal Processing"},{"issue":"5","key":"6461_CR87","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s40314-024-02828-x","volume":"43","author":"X Zhao","year":"2024","unstructured":"Zhao, X., Qi, M., Jolaoso, L. O., Shehu, Y., Yao, J. C., & Yao, Y. (2024). An inexact proximal point method for quasiconvex multiobjective optimization. Computational and Applied Mathematics, 43(5), 309.","journal-title":"Computational and Applied Mathematics"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-024-06461-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-024-06461-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-024-06461-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,27]],"date-time":"2025-01-27T12:25:26Z","timestamp":1737980726000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-024-06461-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,5]]},"references-count":87,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["6461"],"URL":"https:\/\/doi.org\/10.1007\/s10479-024-06461-z","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,5]]},"assertion":[{"value":"3 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2025","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 affirm that there are no actual or potential conflicts of interest associated with this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interests"}},{"value":"All the authors have read and endorsed the final manuscript.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}]}}