{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:53:02Z","timestamp":1760233982782,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T00:00:00Z","timestamp":1615852800000},"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>Our main focus in this work is the classical variational inequality problem with Lipschitz continuous and pseudo-monotone mapping in real Hilbert spaces. An adaptive reflected subgradient-extragradient method is presented along with its weak convergence analysis. The novelty of the proposed method lies in the fact that only one projection onto the feasible set in each iteration is required, and there is no need to know\/approximate the Lipschitz constant of the cost function a priori. To illustrate and emphasize the potential applicability of the new scheme, several numerical experiments and comparisons in tomography reconstruction, Nash\u2013Cournot oligopolistic equilibrium, and more are presented.<\/jats:p>","DOI":"10.3390\/sym13030489","type":"journal-article","created":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T21:42:41Z","timestamp":1615930961000},"page":"489","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Projected-Reflected Subgradient-Extragradient Method and Its Real-World Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2150-553X","authenticated-orcid":false,"given":"Aviv","family":"Gibali","sequence":"first","affiliation":[{"name":"Department of Mathematics, ORT Braude College, Karmiel 2161002, Israel"},{"name":"The Center for Mathematics and Scientific Computation, U. Haifa, Mt. Carmel, Haifa 3498838, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9429-4975","authenticated-orcid":false,"given":"Olaniyi S.","family":"Iyiola","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physical Sciences, California University of Pennsylvania, California, PA 15419, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5920-250X","authenticated-orcid":false,"given":"Lanre","family":"Akinyemi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Prairie View A&amp;M University, Prairie View, TX 77446, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9224-7139","authenticated-orcid":false,"given":"Yekini","family":"Shehu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,16]]},"reference":[{"key":"ref_1","first-page":"138","article-title":"Sul problema elastostatico di Signorini con ambigue condizioni al contorno","volume":"34","author":"Fichera","year":"1963","journal-title":"Atti Accad. 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