{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T19:43:49Z","timestamp":1771530229095,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,12]],"date-time":"2020-05-12T00:00:00Z","timestamp":1589241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush\u2013Kuhn\u2013Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash\u2019s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces.<\/jats:p>","DOI":"10.3390\/sym12050804","type":"journal-article","created":{"date-parts":[[2020,5,12]],"date-time":"2020-05-12T10:53:55Z","timestamp":1589280835000},"page":"804","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0639-3776","authenticated-orcid":false,"given":"Gabriel","family":"Ruiz-Garz\u00f3n","sequence":"first","affiliation":[{"name":"Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de C\u00e1diz, 11406 C\u00e1diz, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1286-2823","authenticated-orcid":false,"given":"Antonio","family":"Rufi\u00e1n-Lizana","sequence":"additional","affiliation":[{"name":"Departamento de Estad\u00edstica e I.O., Universidad de Sevilla, 41012 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,12]]},"reference":[{"key":"ref_1","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. 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