{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T05:50:20Z","timestamp":1780552220270,"version":"3.54.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2015,2,20]],"date-time":"2015-02-20T00:00:00Z","timestamp":1424390400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2015,12]]},"DOI":"10.1007\/s10898-015-0282-7","type":"journal-article","created":{"date-parts":[[2015,2,19]],"date-time":"2015-02-19T09:14:08Z","timestamp":1424337248000},"page":"797-810","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["A proximal point algorithm for DC fuctions on Hadamard manifolds"],"prefix":"10.1007","volume":"63","author":[{"given":"J. C. O.","family":"Souza","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P. R.","family":"Oliveira","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2015,2,20]]},"reference":[{"key":"282_CR1","first-page":"154","volume":"4","author":"B Martinet","year":"1970","unstructured":"Martinet, B.: Regularisation d\u2019in\u00e9quations variationelles par approximations succesives. Rev. Franaise d\u2019Inform. Recherche Oper. 4, 154\u2013159 (1970)","journal-title":"Rev. Franaise d\u2019Inform. Recherche Oper."},{"key":"282_CR2","doi-asserted-by":"crossref","first-page":"273","DOI":"10.24033\/bsmf.1625","volume":"93","author":"JJ 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":"282_CR3","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877\u2013898 (1976)","journal-title":"SIAM J. Control Optim."},{"key":"282_CR4","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."},{"key":"282_CR5","doi-asserted-by":"crossref","first-page":"5924","DOI":"10.1016\/j.na.2012.06.005","volume":"75","author":"EA Papa Quiroz","year":"2012","unstructured":"Papa Quiroz, E.A., Oliveira, P.R.: Proximal point method for minimization quasiconvex locally Lipschitz functions on Hadamard manifolds. Nonlinear Anal. 75, 5924\u20135932 (2012)","journal-title":"Nonlinear Anal."},{"key":"282_CR6","doi-asserted-by":"crossref","unstructured":"Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257\u2013270 (2002)","DOI":"10.1080\/02331930290019413"},{"key":"282_CR7","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10898-005-6741-9","volume":"35","author":"JX Cruz Neto da","year":"2006","unstructured":"da Cruz Neto, J.X., Ferreira, O.P., Lucambio P\u00e9rez, L.R., N\u00e9meth, S.Z.: Convex-and monotone-transformable mathematical programming problems and a proximal-like point algorithm. J. Glob. Optim. 35, 53\u201369 (2006)","journal-title":"J. Glob. Optim."},{"key":"282_CR8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1023\/A:1022675100677","volume":"97","author":"OP Ferreira","year":"1998","unstructured":"Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93\u2013104 (1998)","journal-title":"J. Optim. Theory Appl."},{"key":"282_CR9","first-page":"89","volume":"3","author":"JX Cruz Neto da","year":"1998","unstructured":"da Cruz Neto, J.X., de Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balk. J. Geom. Appl. 3, 89\u2013100 (1998)","journal-title":"Balk. J. Geom. Appl."},{"key":"282_CR10","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1016\/j.matpur.2013.10.002","volume":"101","author":"A Krit\u00e1ly","year":"2014","unstructured":"Krit\u00e1ly, A.: Nash-type equilibria on Riemannian manifolds: a variational approach. J. Math. Pures Appl. 101, 660\u2013688 (2014)","journal-title":"J. Math. Pures Appl."},{"key":"282_CR11","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. Lond. Math. Soc. 79, 663\u2013683 (2009)","journal-title":"J. Lond. Math. Soc."},{"key":"282_CR12","doi-asserted-by":"crossref","first-page":"5695","DOI":"10.1016\/j.na.2009.04.048","volume":"71","author":"SL Li","year":"2009","unstructured":"Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695\u20135706 (2009)","journal-title":"Nonlinear Anal."},{"key":"282_CR13","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1016\/S0362-546X(02)00266-3","volume":"52","author":"SZ N\u00e9meth","year":"2003","unstructured":"N\u00e9meth, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491\u20131498 (2003)","journal-title":"Nonlinear Anal."},{"key":"282_CR14","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."},{"key":"282_CR15","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.jco.2007.12.003","volume":"24","author":"C Li","year":"2008","unstructured":"Li, C., Wang, J.H.: Newton\u2019s method for sections on Riemannian manifolds: generalized covariant $$\\alpha $$ \u03b1 -theory. J. Complex. 24, 423\u2013451 (2008)","journal-title":"J. Complex."},{"issue":"4","key":"282_CR16","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1137\/09075367X","volume":"21","author":"C Li","year":"2011","unstructured":"Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523\u20131560 (2011)","journal-title":"SIAM J. Optim."},{"key":"282_CR17","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s10957-011-9984-2","volume":"154","author":"GC Bento","year":"2012","unstructured":"Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Unconstrained steepest descent method for multicriteria optimization on Riemannian manifolds. J. Optim. Theory Appl. 154, 88\u2013107 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"4","key":"282_CR18","doi-asserted-by":"crossref","first-page":"2486","DOI":"10.1137\/110834962","volume":"50","author":"C Li","year":"2012","unstructured":"Li, C., Yao, J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J. Control Optim. 50(4), 2486\u20132514 (2012)","journal-title":"SIAM J. Control Optim."},{"key":"282_CR19","doi-asserted-by":"crossref","unstructured":"Li, C., L\u00f3pez, G., Wang, J.H., Yao, J.C.: Convergence analysis of inexact proximal point algorithms on Hadamard manifolds. J. Glob. Optim. (2014). doi: 10.1007\/s10898-014-0182-2","DOI":"10.1007\/s10898-014-0182-2"},{"key":"282_CR20","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1016\/j.orl.2013.04.003","volume":"41","author":"N Huang","year":"2013","unstructured":"Huang, N., Tang, G.: An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds. Oper. Res. Lett. 41, 586\u2013591 (2013)","journal-title":"Oper. Res. Lett."},{"key":"282_CR21","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10208-005-0179-9","volume":"7","author":"PA Absil","year":"2007","unstructured":"Absil, P.A., Baker, C.G.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7, 303\u2013330 (2007)","journal-title":"Found. Comput. Math."},{"key":"282_CR22","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1093\/imanum\/22.3.359","volume":"22","author":"RL Adler","year":"2002","unstructured":"Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton\u2019s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22, 359\u2013390 (2002)","journal-title":"IMA J. Numer. Anal."},{"key":"282_CR23","unstructured":"Lee, P.Y.: Geometric Optimization for Computer Vision. PhD thesis, Australian National University (2005)"},{"key":"282_CR24","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/0001-8708(84)90039-2","volume":"54","author":"RC Riddell","year":"1984","unstructured":"Riddell, R.C.: Minimax problems on Grassmann manifolds. Sums of eigenvalues. Adv. Math. 54, 107\u2013199 (1984)","journal-title":"Adv. Math."},{"key":"282_CR25","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.B.: From convex optimization to nonconvex optimization: necessary and sufficient conditions for global optimization. Nonsmooth optimization and related topics, pp. 219\u2013239. Springer, US (1989)","DOI":"10.1007\/978-1-4757-6019-4_13"},{"key":"282_CR26","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.B.: Generalized differentiabity, duality and optimization for problems dealing with diference of convex functions. In Convexity and duality in optimization. pp. 37\u201370. Springer, Berlin Hiedelberg (1985)","DOI":"10.1007\/978-3-642-45610-7_3"},{"key":"282_CR27","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.B., Tuy, H.: Essays on nonconvex optimization, Mathematical programming, vol. 41. North-Holland (1988)","DOI":"10.1007\/BF01580759"},{"issue":"1","key":"282_CR28","first-page":"1","volume":"20","author":"A Elhilali Alaoui","year":"1996","unstructured":"Elhilali Alaoui, A.: Caractrisation des fonctions D.C. (Characterization of D. C. functions). Ann. Sci. Math. Qu 20(1), 1\u201313 (1996)","journal-title":"Ann. Sci. Math. Qu"},{"key":"282_CR29","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/0022-247X(78)90243-3","volume":"66","author":"JF Toland","year":"1978","unstructured":"Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399\u2013415 (1978)","journal-title":"J. Math. Anal. Appl."},{"key":"282_CR30","first-page":"451","volume":"21","author":"W Sun","year":"2003","unstructured":"Sun, W., Sampaio, R.J.B., Candido, M.A.B.: Proximal point algorithm for minimization of DC functions. J. Comput. Math. 21, 451\u2013462 (2003)","journal-title":"J. Comput. Math."},{"key":"282_CR31","first-page":"475","volume":"24","author":"A Moudafi","year":"2006","unstructured":"Moudafi, A., Maing, P.-E.: On the convergence of an approximate proximal method for DC functions. J. Comput. Math. 24, 475\u2013480 (2006)","journal-title":"J. Comput. Math."},{"key":"282_CR32","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2201-7","volume-title":"Riemannian Geometry","author":"MP Carmo do","year":"1992","unstructured":"do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)"},{"key":"282_CR33","doi-asserted-by":"crossref","unstructured":"Sakai, T.: Riemannian geometry. Translations of mathematical monographs. Am. Math. Soc. Provid. 149 (1996)","DOI":"10.1090\/mmono\/149"},{"key":"282_CR34","doi-asserted-by":"crossref","unstructured":"Udriste, C.: Convex functions and optimization algorithms on Riemannian manifolds, Mathematics and its applications, vol. 297. Kluwer Academic, Dordrecht (1994)","DOI":"10.1007\/978-94-015-8390-9"},{"key":"282_CR35","unstructured":"Polyak, B.T.: Sharp Minima Institute of Control Sciences Lecture Notes, Moscow, USSR, 1979. Presented at the IIASA workshop on generalized Lagrangians and their applications, IIASA, Laxenburg, Austria (1979)"},{"key":"282_CR36","unstructured":"Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Ph.D. Thesis. University of Cambridge, UK (1988)"},{"key":"282_CR37","doi-asserted-by":"crossref","unstructured":"Bento G.C., Cruz Neto, J.X.: Finite termination of the proximal point method for convex functions on Hadamard manifolds. Optimization, 63, 1281\u20131288 (2014)","DOI":"10.1080\/02331934.2012.730050"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-015-0282-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-015-0282-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-015-0282-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,21]],"date-time":"2019-08-21T02:39:30Z","timestamp":1566355170000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-015-0282-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,20]]},"references-count":37,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,12]]}},"alternative-id":["282"],"URL":"https:\/\/doi.org\/10.1007\/s10898-015-0282-7","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2,20]]}}}