{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:17:21Z","timestamp":1760235441962,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,15]],"date-time":"2021-08-15T00:00:00Z","timestamp":1628985600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the 2020 Shanghai Leading Talents Program of the Shanghai Municipal Human Resources and Social Security Bureau","award":["20LJ2006100"],"award-info":[{"award-number":["20LJ2006100"]}]},{"name":"the Innovation Program of Shanghai Municipal Education Commission","award":["15ZZ068"],"award-info":[{"award-number":["15ZZ068"]}]},{"name":"the Program for Outstanding Academic Leaders in Shanghai City","award":["15XD1503100"],"award-info":[{"award-number":["15XD1503100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.<\/jats:p>","DOI":"10.3390\/sym13081496","type":"journal-article","created":{"date-parts":[[2021,8,15]],"date-time":"2021-08-15T22:51:27Z","timestamp":1629067887000},"page":"1496","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds"],"prefix":"10.3390","volume":"13","author":[{"given":"Chun-Yan","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Lu-Chuan","family":"Ceng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Long","family":"He","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Hui-Ying","family":"Hu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Tu-Yan","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Dan-Qiong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]},{"given":"Hong-Ling","family":"Fan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Shanghai Normal University, Shanghai 200234, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,15]]},"reference":[{"key":"ref_1","first-page":"747","article-title":"The extragradient method for finding saddle points and other problems","volume":"12","author":"Korpelevich","year":"1976","journal-title":"Ekon. 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