{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:06Z","timestamp":1772253066974,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,12]],"date-time":"2019-08-12T00:00:00Z","timestamp":1565568000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["MTM2015-66185-P"],"award-info":[{"award-number":["MTM2015-66185-P"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this paper is to show the existence and attainability of Karush\u2013Kuhn\u2013Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds.<\/jats:p>","DOI":"10.3390\/sym11081037","type":"journal-article","created":{"date-parts":[[2019,8,13]],"date-time":"2019-08-13T04:31:21Z","timestamp":1565670681000},"page":"1037","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0639-3776","authenticated-orcid":false,"given":"Gabriel","family":"Ruiz-Garz\u00f3n","sequence":"first","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de C\u00e1diz, 11405 C\u00e1diz, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1777-2395","authenticated-orcid":false,"given":"Rafaela","family":"Osuna-G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de Sevilla, 41012 Sevilla, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7951-4391","authenticated-orcid":false,"given":"Jaime","family":"Ruiz-Zapatero","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College of London, London WC1E 6BT, UK"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/BF01353421","article-title":"A generalization of Tychonoff\u2019s fixed point theorem","volume":"142","author":"Fan","year":"1961","journal-title":"Math. Ann."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/S0362-546X(02)00154-2","article-title":"New existence results for equilibrium problems","volume":"52","author":"Iusem","year":"2003","journal-title":"Nonlinear Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1080\/02331930290019413","article-title":"Proximal point algorithm on Riemannian manifolds","volume":"51","author":"Ferreira","year":"2002","journal-title":"Optimization"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1023\/A:1021221029301","article-title":"A Cartan-Hadamard Theorem for Banach-Finsler Manifolds","volume":"95","author":"Neeb","year":"2002","journal-title":"Geom. Dedicata"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.jmaa.2011.11.001","article-title":"Equilibrium problems in Hadamard manifolds","volume":"38","author":"Colao","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1016\/S0362-546X(02)00266-3","article-title":"Variational inequalities on Hadamard manifolds","volume":"52","year":"2003","journal-title":"Nonlinear Anal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Absil, P.A., Mahony, R., and Sepulchre, R. (2008). Optimization Algorithms on Matrix Manifolds, Princeton University Press.","DOI":"10.1515\/9781400830244"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.neucom.2004.11.035","article-title":"Learning algorithms utilizing quasigeodesic flows on the Stiefel manifold","volume":"67","author":"Nishimori","year":"2005","journal-title":"Neurocoputing"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Turaga, P., Veeraraghavan, A., and Chellapa, R. (2008, January 23\u201328). Statistical Analysis on Stiefel and Grasmann manifolds with applications in computer vision. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Anchorage, AK, USA.","DOI":"10.1109\/CVPR.2008.4587733"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5143","DOI":"10.1016\/j.neuroimage.2008.10.052","article-title":"The geometric median on Riemannian manifolds with application to robust atlas estimation","volume":"45","author":"Fletcher","year":"2009","journal-title":"NeuroImage"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/978-3-319-10605-2_17","article-title":"Canonical Correlation Analysis on Riemannian Manifolds and Its Applications","volume":"Volume 8690","author":"Fleet","year":"2014","journal-title":"Lecture Notes in Computer Science"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1016\/j.matpur.2013.10.002","article-title":"Nash-type equilibria on Riemannian manifolds: A variational approach","volume":"101","year":"2014","journal-title":"J. Math. Pures Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.1016\/j.na.2006.12.038","article-title":"Dini derivative and a characterization for Lipschiz and convex functions on Riemannian manifolds","volume":"68","author":"Ferreira","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","article-title":"On sufficiency of the Khun-Tucker conditions","volume":"80","author":"Hanson","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1016\/j.na.2008.02.085","article-title":"Invariant monotone vector fields on Riemannian manifolds","volume":"70","author":"Barani","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3884","DOI":"10.1016\/j.na.2011.02.023","article-title":"Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds","volume":"74","author":"Hosseini","year":"2011","journal-title":"Nonlinear Anal."},{"key":"ref_17","first-page":"1245","article-title":"Optimality and duality on Riemannian manifolds","volume":"22","year":"2018","journal-title":"Taiwanese J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Khan, M.A., and Ishan, A.A. (2019). Generalized Geodesic Convexity on Riemannian manifolds. Mathematics, 7.","DOI":"10.3390\/math7060547"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/S0022-247X(02)00535-8","article-title":"Generalized vector quasi-equilibrium problems with applications","volume":"277","author":"Ansari","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1016\/j.jmaa.2008.01.026","article-title":"Optimality conditions for vector equilibrium problems","volume":"342","author":"Gong","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3598","DOI":"10.1016\/j.na.2010.07.041","article-title":"Scalarization and optimality conditions for vector equilibrium problems","volume":"73","author":"Gong","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"842715","DOI":"10.1155\/2010\/842715","article-title":"Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems","volume":"2010","author":"Wei","year":"2010","journal-title":"J. Inequalities Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s11590-013-0695-5","article-title":"Optimality conditions for vector equilibrium problems with constraint in Banach spaces","volume":"8","author":"Feng","year":"2014","journal-title":"Optim. Lett."},{"key":"ref_24","unstructured":"Karush, W. (1939). Minima of Functions of Several Variables with Inequalities as Side Conditions. [Master\u2019s Thesis, Department of Mathematics, University of Chicago]."},{"key":"ref_25","unstructured":"Khun, H.W., and Tucker, A.W. (August, January 31). Nonlinear Programming. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Statistical Laboratory of the University of California, Berkeley, CA, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"333","DOI":"10.4310\/jdg\/1214436108","article-title":"Totally convex sets in complete Riemannian manifolds","volume":"16","author":"Bangert","year":"1981","journal-title":"J. Differ. Geom."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"833","DOI":"10.11650\/tjm.17.2013.1937","article-title":"Roughly Geodesic B-invex and Optimization problem on Hadamard Manifolds","volume":"17","author":"Zhou","year":"2013","journal-title":"Taiwanese J. Math."},{"key":"ref_28","unstructured":"Khajejpour, S., and Pouryayevali, M.R. (2018). Convexity of distance function to convex subsets of Riemannian manifolds. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1037\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:10:32Z","timestamp":1760188232000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/1037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,12]]},"references-count":28,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["sym11081037"],"URL":"https:\/\/doi.org\/10.3390\/sym11081037","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201907.0177.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,12]]}}}