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The Karush\u2013Kuhn\u2013Tucker necessary optimality conditions are proved for such a nonsmooth extremum problem. Further, the exact <jats:inline-formula><jats:alternatives><jats:tex-math>$$l_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>l<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> penalty function method is used for solving the considered nonsmooth fuzzy optimization problem. Therefore, its associated fuzzy penalized optimization problem is constructed in this approach. Then, the exactness property of the exact <jats:inline-formula><jats:alternatives><jats:tex-math>$$l_{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>l<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> penalty function method is analyzed if it is used for solving the considered nonsmooth convex fuzzy optimization problem.<\/jats:p>","DOI":"10.1007\/s00500-022-07459-0","type":"journal-article","created":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T10:06:09Z","timestamp":1662545169000},"page":"11627-11643","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the exact $$l_{1}$$ penalty function method for convex nonsmooth optimization problems with fuzzy objective function"],"prefix":"10.1007","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2218-7668","authenticated-orcid":false,"given":"Tadeusz","family":"Antczak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,7]]},"reference":[{"key":"7459_CR1","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/0165-0114(92)90319-Y","volume":"49","author":"EE Ammar","year":"1992","unstructured":"Ammar EE (1992) On fuzzy convexity and parametric fuzzy optimization. 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