{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:06:29Z","timestamp":1760709989197,"version":"3.41.2"},"reference-count":31,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2019,8,20]],"date-time":"2019-08-20T00:00:00Z","timestamp":1566259200000},"content-version":"vor","delay-in-days":231,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61801171","61871430","U1604155","61602155"],"award-info":[{"award-number":["61801171","61871430","U1604155","61602155"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004773","name":"Henan University","doi-asserted-by":"publisher","award":["19zx010"],"award-info":[{"award-number":["19zx010"]}],"id":[{"id":"10.13039\/501100004773","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We consider a distributed constrained optimization problem over graphs, where cost function of each agent is private. Moreover, we assume that the graphs are time\u2010varying and directed. In order to address such problem, a fully decentralized stochastic subgradient projection algorithm is proposed over time\u2010varying directed graphs. However, since the graphs are directed, the weight matrix may not be a doubly stochastic matrix. Therefore, we overcome this difficulty by using weight\u2010balancing technique. By choosing appropriate step\u2010sizes, we show that iterations of all agents asymptotically converge to some optimal solutions. Further, by our analysis, convergence rate of our proposed algorithm is <jats:italic>O<\/jats:italic>(ln\u200a\u0393\/\u0393) under local strong convexity, where \u0393 is the number of iterations. In addition, under local convexity, we prove that our proposed algorithm can converge with rate . In addition, we verify the theoretical results through simulations.<\/jats:p>","DOI":"10.1155\/2019\/8030792","type":"journal-article","created":{"date-parts":[[2019,8,20]],"date-time":"2019-08-20T23:32:34Z","timestamp":1566343954000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Distributed Stochastic Subgradient Projection Algorithms Based on Weight\u2010Balancing over Time\u2010Varying Directed Graphs"],"prefix":"10.1155","volume":"2019","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6411-7035","authenticated-orcid":false,"given":"Junlong","family":"Zhu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3460-5756","authenticated-orcid":false,"given":"Ping","family":"Xie","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2523-1089","authenticated-orcid":false,"given":"Mingchuan","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0932-8788","authenticated-orcid":false,"given":"Ruijuan","family":"Zheng","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5132-3817","authenticated-orcid":false,"given":"Ling","family":"Xing","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1572-5293","authenticated-orcid":false,"given":"Qingtao","family":"Wu","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2019,8,20]]},"reference":[{"key":"e_1_2_10_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21606-5"},{"key":"e_1_2_10_2_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139042918"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2009.2018648"},{"key":"e_1_2_10_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/090752651"},{"key":"e_1_2_10_5_2","unstructured":"JohanssonB. 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