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The relationships between them are studied. Also, we compare these concepts with the homologous ones given by considering the vector criterion. Finally, a Lagrange multiplier rule for Henig proper solutions of a set optimization problem with a cone constraint is obtained under convexity hypotheses. Illustrative examples are also given.<\/jats:p>","DOI":"10.1007\/s10957-022-02088-x","type":"journal-article","created":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T02:02:36Z","timestamp":1663380156000},"page":"878-902","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["New Notions of Proper Efficiency in Set Optimization with the Set Criterion"],"prefix":"10.1007","volume":"195","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6634-3482","authenticated-orcid":false,"given":"L.","family":"Huerga","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1513-8590","authenticated-orcid":false,"given":"B.","family":"Jim\u00e9nez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9033-3180","authenticated-orcid":false,"given":"V.","family":"Novo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"key":"2088_CR1","volume-title":"Applied Nonlinear Analysis","author":"JP Aubin","year":"1984","unstructured":"Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. 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