{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:44:27Z","timestamp":1787341467375,"version":"build-2736575974"},"reference-count":59,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["389\/12"],"award-info":[{"award-number":["389\/12"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006260","name":"Technion-Israel Institute of Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006260","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In this article we consider a consistent convex feasibility problem in a real Hilbert space defined by a finite family of sets $C_i$. We are interested, in particular, in the case where for each $i$, $C_i={Fix} U_i=\\{z\\in \\mathcal H\\mid p_i(z)=0\\}$, $U_i\\colon\\mathcal H\\rightarrow \\mathcal H$ is a cutter and $p_i\\colon\\mathcal H\\rightarrow [0,\\infty)$ is a proximity function. Moreover, we make the following assumption: the computation of $p_i$ is at most as difficult as the evaluation of $U_i$ and this is at most as difficult as projecting onto $C_i$. We study a double-layer fixed point algorithm which applies two types of controls in every iteration step. The first one---the outer control---is assumed to be almost cyclic. The second one---the inner control---determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions. The convergence results presented in this manuscript depend on the conditions which, first, bind together the sets, the operators, and the proximity functions and, second, connect the inner and outer controls. In particular, weak regularity (demi-closedness principle), bounded regularity, and bounded linear regularity imply weak, strong, and linear convergence of our algorithm, respectively. The framework presented in this paper covers many known (subgradient) projection algorithms already existing in the literature, for example, those applied with (almost) cyclic, remotest-set, maximum displacement, most-violated constraint, and simultaneous controls. In addition, we provide several new examples, where the double-layer approach indeed accelerates the convergence speed as we demonstrate numerically.<\/jats:p>","DOI":"10.1137\/16m1087333","type":"journal-article","created":{"date-parts":[[2017,7,26]],"date-time":"2017-07-26T11:52:18Z","timestamp":1501069938000},"page":"1431-1458","source":"Crossref","is-referenced-by-count":12,"title":["Weak, Strong, and Linear Convergence of a Double-Layer Fixed Point Algorithm"],"prefix":"10.1137","volume":"27","author":[{"given":"Victor I.","family":"Kolobov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Simeon","family":"Reich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rafa\u0142","family":"Zalas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,7,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1954-037-2"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90375-3"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.01.087"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1950-0051437-7"},{"key":"atypb5","unstructured":"A. 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Combettes,\n                      Convex Analysis and Monotone Operator Theory in Hilbert Spaces\n                      , CMS Books Math.\/Ouvrages Math. SMC, Springer, New York, 2011.","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1111\/j.1475-3995.2008.00682.x"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2003.10.010"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2014.06.075"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/14096267X"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1080\/10556780310001604977"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/15M1045223"},{"key":"atypb17","first-page":"487","volume":"162","author":"Bre\u0300gman L. M.","year":"1965","journal-title":"Dokl. Akad. 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Cegielski,\n                      Iterative Methods for Fixed Point Problems in Hilbert Spaces\n                      , Lecture Notes in Math. 2057, Springer, Heidelberg, 2012.","DOI":"10.1007\/978-3-642-30901-4"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/15M1012438"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-014-0662-z"},{"key":"atypb23","first-page":"139","author":"Cegielski A.","year":"2016","journal-title":"RI"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2016.1147038"},{"key":"atypb25","first-page":"155","author":"Cegielski A.","year":"2011","journal-title":"New York"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1080\/01630563.2012.716807"},{"key":"atypb27","first-page":"399","volume":"15","author":"Cegielski A.","year":"2014","journal-title":"Fixed Point Theory"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2014.957701"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585107"},{"key":"atypb30","unstructured":"Y. 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