{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T01:41:15Z","timestamp":1773193275752,"version":"3.50.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2016,1,11]],"date-time":"2016-01-11T00:00:00Z","timestamp":1452470400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["458479\/2014-4"],"award-info":[{"award-number":["458479\/2014-4"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["471815\/2012-8"],"award-info":[{"award-number":["471815\/2012-8"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["303732\/2011-3"],"award-info":[{"award-number":["303732\/2011-3"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["312077\/2014-9"],"award-info":[{"award-number":["312077\/2014-9"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]},{"name":"CAPES-MES-CUBA","award":["226\/2012"],"award-info":[{"award-number":["226\/2012"]}]},{"DOI":"10.13039\/501100005285","name":"Funda\u00e7\u00e3o de Apoio a Pesquisa do Estado de Goi\u00e1s","doi-asserted-by":"publisher","award":["201210267000909 - 05\/2012"],"award-info":[{"award-number":["201210267000909 - 05\/2012"]}],"id":[{"id":"10.13039\/501100005285","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"CNPq","doi-asserted-by":"crossref","award":["305462\/2014-8"],"award-info":[{"award-number":["305462\/2014-8"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2016,3]]},"DOI":"10.1007\/s10957-015-0861-2","type":"journal-article","created":{"date-parts":[[2016,1,11]],"date-time":"2016-01-11T20:40:11Z","timestamp":1452544811000},"page":"743-755","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds"],"prefix":"10.1007","volume":"168","author":[{"given":"Glaydston","family":"de Carvalho Bento","sequence":"first","affiliation":[]},{"given":"Jo\u00e3o Xavier","family":"da Cruz Neto","sequence":"additional","affiliation":[]},{"given":"Paulo Roberto","family":"Oliveira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,1,11]]},"reference":[{"key":"861_CR1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/02331930290019413","volume":"51","author":"OP Ferreira","year":"2002","unstructured":"Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifold. Optimization 51, 257\u2013270 (2002)","journal-title":"Optimization"},{"issue":"1","key":"861_CR2","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10898-005-6741-9","volume":"35","author":"JX Cruz Neto","year":"2006","unstructured":"Cruz Neto, J.X., Ferreira, O.P., P\u00e9rez, L.R.L., N\u00e9meth, S.Z.: Convex- and monotone transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35(1), 53\u201369 (2006)","journal-title":"J. Global Optim."},{"issue":"3\u20134","key":"861_CR3","first-page":"417","volume":"9","author":"SZ N\u00e9meth","year":"1998","unstructured":"N\u00e9meth, S.Z.: Five kinds of monotone vector fields. Pure Math. Appl. 9(3\u20134), 417\u2013428 (1998)","journal-title":"Pure Math. Appl."},{"key":"861_CR4","first-page":"13","volume":"5","author":"SZ N\u00e9meth","year":"1999","unstructured":"N\u00e9meth, S.Z.: Geodesic monotone vector fields. Lobachevski. J. Math. 5, 13\u201328 (1999)","journal-title":"Lobachevski. J. Math."},{"issue":"3","key":"861_CR5","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1023\/A:1006624901670","volume":"84","author":"SZ N\u00e9meth","year":"1999","unstructured":"N\u00e9meth, S.Z.: Monotonicity of the complementary vector field of a nonexpansive map. Acta Math. Hungar. 84(3), 189\u2013197 (1999)","journal-title":"Acta Math. Hungar."},{"issue":"4","key":"861_CR6","doi-asserted-by":"crossref","first-page":"707","DOI":"10.5486\/PMD.2001.2408","volume":"58","author":"SZ N\u00e9meth","year":"2001","unstructured":"N\u00e9meth, S.Z.: Homeomorphisms and monotone vector fields. Publ. Math. Debrecen 58(4), 707\u2013716 (2001)","journal-title":"Publ. Math. Debrecen"},{"key":"861_CR7","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1112\/jlms\/jdn087","volume":"79","author":"C Li","year":"2009","unstructured":"Li, C., L\u00f3pez, G., Mart\u00edn-M\u00e1rquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. London Math. Soc. 79, 663\u2013683 (2009)","journal-title":"J. London Math. Soc."},{"issue":"1","key":"861_CR8","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s10898-003-3780-y","volume":"31","author":"OP Ferreira","year":"2005","unstructured":"Ferreira, O.P., P\u00e9rez, L.R.L., N\u00e9meth, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Global Optim. 31(1), 133\u2013151 (2005)","journal-title":"J. Global Optim."},{"key":"861_CR9","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1007\/s10957-010-9688-z","volume":"146","author":"JH Wang","year":"2010","unstructured":"Wang, J.H., L\u00f3pez, G., Mart\u00edn-M\u00e1rquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691\u2013708 (2010)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"861_CR10","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s11228-010-0169-1","volume":"19","author":"C Li","year":"2011","unstructured":"Li, C., L\u00f3pez, G., Mart\u00edn-M\u00e1rquez, V., Wang, J.H.: Resolvents of set-valued monotone vector fields in hadamard manifolds. Set-valued Var. Anal. 19(3), 361\u2013383 (2011)","journal-title":"Set-valued Var. Anal."},{"issue":"3","key":"861_CR11","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1007\/s10898-011-9773-3","volume":"54","author":"G Tang","year":"2012","unstructured":"Tang, G., Huang, N.: Korpelevich\u2019s method for variational inequality problems on Hadamard manifolds. J. Global Optim. 54(3), 493\u2013509 (2012)","journal-title":"J. Global Optim."},{"issue":"5","key":"861_CR12","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1080\/02331934.2013.840622","volume":"64","author":"G Tang","year":"2015","unstructured":"Tang, G., Wang, X., Liu, H.: A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence. Optimization 64(5), 1081\u20131096 (2015)","journal-title":"Optimization"},{"key":"861_CR13","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1080\/00207728108963798","volume":"12","author":"M Fukushima","year":"1981","unstructured":"Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain nonconvex minimization problems. Int. J. Syst. Sci. 12, 989\u20131000 (1981)","journal-title":"Int. J. Syst. Sci."},{"issue":"4","key":"861_CR14","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1023\/A:1008321423879","volume":"13","author":"A Kaplan","year":"1998","unstructured":"Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Global Optim. 13(4), 389\u2013406 (1998)","journal-title":"J. Global Optim."},{"issue":"2","key":"861_CR15","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/01630568208816109","volume":"4","author":"JE Spingarn","year":"1981","unstructured":"Spingarn, J.E.: Submonotone mappings and the proximal point algorithm. Numer. Funct. Anal. Optim. 4(2), 123\u2013150 (1981)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"861_CR16","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1287\/moor.27.1.170.331","volume":"27","author":"T Pennanen","year":"2002","unstructured":"Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170\u2013191 (2002)","journal-title":"Math. Oper. Res."},{"issue":"4","key":"861_CR17","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1137\/S1052623401399587","volume":"13","author":"AN Iusem","year":"2003","unstructured":"Iusem, A.N., Penannen, T., Svaiter, B.F.: Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13(4), 1080\u20131097 (2003)","journal-title":"SIAM J. Optim."},{"key":"861_CR18","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1137\/S0363012903427336","volume":"43","author":"PL Combettes","year":"2004","unstructured":"Combettes, P.L., Pennanen, T.: Proximal methods for cohypomonotone operators. SIAM J. Control Optim 43, 731\u2013742 (2004)","journal-title":"SIAM J. Control Optim"},{"issue":"1","key":"861_CR19","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/j.jmaa.2006.07.076","volume":"330","author":"R G\u00e1rciga-Otero","year":"2007","unstructured":"G\u00e1rciga-Otero, R., Iusem, A.N.: Proximal methods in reflexive Banach spaces without monotonicity. J. Math. Anal. Appl. 330(1), 433\u2013450 (2007)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"861_CR20","first-page":"49","volume":"16","author":"EA Papa Quiroz","year":"2009","unstructured":"Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16(1), 49\u201369 (2009)","journal-title":"J. Convex Anal."},{"key":"861_CR21","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1016\/j.na.2010.03.057","volume":"73","author":"GC Bento","year":"2010","unstructured":"Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Anal. 73, 564\u2013572 (2010)","journal-title":"Nonlinear Anal."},{"issue":"1","key":"861_CR22","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1090\/S0002-9947-1986-0857446-4","volume":"298","author":"T Yamaguchi","year":"1986","unstructured":"Yamaguchi, T.: Locally geodesically quasiconvex functions on complete riemannian manifolds. Trans. Am. Math. Soc. 298(1), 307\u2013330 (1986)","journal-title":"Trans. Am. Math. Soc."},{"key":"861_CR23","volume-title":"Optimization on Manifolds: Methods and Applications. Recent Advances in Optimization and its Applications in Engineering","author":"P-A Absil","year":"2010","unstructured":"Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization on Manifolds: Methods and Applications. Recent Advances in Optimization and its Applications in Engineering. Springer, New York (2010)"},{"key":"861_CR24","doi-asserted-by":"crossref","DOI":"10.1515\/9781400830244","volume-title":"Optimization Algorithms on Matrix Manifolds","author":"P-A Absil","year":"2008","unstructured":"Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)"},{"issue":"1\u20133","key":"861_CR25","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s10898-007-9212-7","volume":"40","author":"T Rapcs\u00e1k","year":"2008","unstructured":"Rapcs\u00e1k, T.: Sectional curvature in nonlinear optimization. J. Glob. Optim. 40(1\u20133), 375\u2013388 (2008)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"861_CR26","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1007\/s10957-011-9921-4","volume":"152","author":"GC Bento","year":"2012","unstructured":"Bento, G.C., Melo, J.G.: Subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152(3), 773\u2013785 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"861_CR27","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s10957-013-0307-7","volume":"159","author":"GC Bento","year":"2013","unstructured":"Bento, G.C., Cruz Neto, J.X.: A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 159(1), 125\u2013137 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"861_CR28","doi-asserted-by":"crossref","first-page":"3687","DOI":"10.1090\/S0002-9947-07-04075-5","volume":"359","author":"YS Ledyaev","year":"2007","unstructured":"Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687\u20133732 (2007)","journal-title":"Trans. Am. Math. Soc."},{"key":"861_CR29","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2201-7","volume-title":"Riemannian Geometry","author":"MP do Carmo","year":"1992","unstructured":"do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)"},{"issue":"1\u20132","key":"861_CR30","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10107-007-0133-5","volume":"116","author":"H Attouch","year":"2009","unstructured":"Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. Ser. B 116(1\u20132), 5\u201316 (2009)","journal-title":"Math. Program. Ser. B"},{"issue":"12","key":"861_CR31","doi-asserted-by":"crossref","first-page":"1383","DOI":"10.1080\/02331934.2011.564623","volume":"61","author":"FG Moreno","year":"2012","unstructured":"Moreno, F.G., Oliveira, P.R., Soubeyran, A.: A proximal algorithm with quasi distance. Application to habit\u2019s formation. Optimization 61(12), 1383\u20131403 (2012)","journal-title":"Optimization"},{"key":"861_CR32","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.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward\u2013backward splitting, and regularized Gauss\u2013Seidel methods. Math. Program. Ser. A 137, 91\u2013129 (2013)","journal-title":"Math. Program. Ser. A"},{"key":"861_CR33","unstructured":"Lojasiewicz, S.: Une proprit topologique des sous-ensembles analytiques rels. Les quations aux Drives Partielles, ditions du centre National de la Recherche Scientifique, pp. 87\u201389 (1963)"},{"key":"861_CR34","doi-asserted-by":"crossref","first-page":"769","DOI":"10.5802\/aif.1638","volume":"48","author":"K Kurdyka","year":"1998","unstructured":"Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48, 769\u2013783 (1998)","journal-title":"Ann. Inst. Fourier"},{"key":"861_CR35","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1215\/S0012-7094-96-08416-1","volume":"84","author":"L Dries van den","year":"1996","unstructured":"van den Dries, L., Miller, C.: Geometric categories and o-minimal structures. Duke Math. J. 84, 497\u2013540 (1996)","journal-title":"Duke Math. J."},{"issue":"4","key":"861_CR36","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1137\/050644641","volume":"17","author":"J Bolte","year":"2006","unstructured":"Bolte, J., Daniilidis, J.A., Lewis, A.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim 17(4), 1205\u20131223 (2006)","journal-title":"SIAM J. Optim"},{"issue":"2","key":"861_CR37","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1137\/060670080","volume":"18","author":"J Bolte","year":"2007","unstructured":"Bolte, J., Daniilidis, J.A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556\u2013572 (2007)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"861_CR38","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1287\/moor.1100.0449","volume":"35","author":"H Attouch","year":"2010","unstructured":"Attouch, H., Redont, P., Bolte, J., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka\u2013Lojasiewicz inequality. Math. Oper. Res. 35(2), 438\u2013457 (2010)","journal-title":"Math. Oper. Res."},{"key":"861_CR39","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2307\/2661354","volume":"152","author":"K Kurdyka","year":"2000","unstructured":"Kurdyka, K., Mostowski, T., Parusinski, A.: Proof of the gradient conjecture of R. Thom. Ann. Math. 152, 763\u2013792 (2000)","journal-title":"Thom. Ann. Math."},{"key":"861_CR40","unstructured":"Lageman, C.: Convergence of gradient-like dynamical systems and optimization algorithms, Ph.D. Thesis, Universitt Wrzburg (2007)"},{"key":"861_CR41","doi-asserted-by":"crossref","first-page":"3319","DOI":"10.1090\/S0002-9947-09-05048-X","volume":"362","author":"J Bolte","year":"2010","unstructured":"Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of lojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362, 3319\u20133363 (2010)","journal-title":"Trans. Am. Math. Soc."},{"key":"861_CR42","first-page":"395","volume":"20","author":"JX Cruz Neto","year":"2013","unstructured":"Cruz Neto, J.X., Oliveira, P.R., Soares J\u00fanior, P.A., Soubeyran, A.: Learning how to play Nash and alternating minimization method for structured nonconvex problems on Riemannian manifolds. J. Convex Anal. 20, 395\u2013438 (2013)","journal-title":"J. Convex Anal."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-015-0861-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-015-0861-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-015-0861-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T12:51:06Z","timestamp":1692190266000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-015-0861-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,11]]},"references-count":42,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,3]]}},"alternative-id":["861"],"URL":"https:\/\/doi.org\/10.1007\/s10957-015-0861-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,11]]}}}