{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:09:13Z","timestamp":1760234953120,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T00:00:00Z","timestamp":1626134400000},"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>In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a \u03c3-porous set.<\/jats:p>","DOI":"10.3390\/sym13071253","type":"journal-article","created":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T04:26:06Z","timestamp":1626150366000},"page":"1253","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Well-Posedness and Porosity for Symmetric Optimization Problems"],"prefix":"10.3390","volume":"13","author":[{"given":"Alexander J.","family":"Zaslavski","sequence":"first","affiliation":[{"name":"Department of Mathematics, The Technion\u2014Israel Institute of Technology, Haifa 32000, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Zaslavski, A.J. (2020). 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Convex Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zaslavski, A.J. (2010). Optimization on metric and normed spaces. Springer Optimization and Its Applications, Springer.","DOI":"10.1007\/978-0-387-88621-3"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1137\/15M1035550","article-title":"Strongly convex functions, Moreau envelopes, and the generic nature of convex functions with strong minimizers","volume":"26","author":"Planiden","year":"2016","journal-title":"SIAM J. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1006\/jath.2000.3503","article-title":"On well posed generalized best approximation problems","volume":"107","author":"Li","year":"2000","journal-title":"J. Approx. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1016\/j.jmaa.2015.03.055","article-title":"Porosity results on fixed points for nonexpansive set-valued maps in hyperbolic spaces","volume":"428","author":"Peng","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Reich, S., and Zaslavski, A.J. (2014). Genericity in nonlinear analysis. Developments in Mathematics, Springer.","DOI":"10.1007\/978-1-4614-9533-8"},{"key":"ref_14","first-page":"791","article-title":"On the residuality of certain classes of convex functions","volume":"5","author":"Vanderwerff","year":"2020","journal-title":"Pure Appl. Funct. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.na.2013.03.008","article-title":"Most maximally monotone operators have a unique zero and a super-regular resolvent","volume":"87","author":"Wang","year":"2013","journal-title":"Nonlinear Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1253\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:29:37Z","timestamp":1760164177000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1253"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,13]]},"references-count":15,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071253"],"URL":"https:\/\/doi.org\/10.3390\/sym13071253","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,7,13]]}}}