{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T08:14:40Z","timestamp":1776327280521,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T00:00:00Z","timestamp":1696464000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T00:00:00Z","timestamp":1696464000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DE200100063"],"award-info":[{"award-number":["DE200100063"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP230101749"],"award-info":[{"award-number":["DP230101749"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001782","name":"University of Melbourne","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001782","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.<\/jats:p>","DOI":"10.1007\/s11590-023-02064-y","type":"journal-article","created":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T07:01:46Z","timestamp":1696489306000},"page":"1541-1559","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Frugal and decentralised resolvent splittings defined by nonexpansive operators"],"prefix":"10.1007","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3654-6553","authenticated-orcid":false,"given":"Matthew K.","family":"Tam","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,5]]},"reference":[{"key":"2064_CR1","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s11075-022-01405-9","volume":"93","author":"FJ Arag\u00f3n-Artacho","year":"2022","unstructured":"Arag\u00f3n-Artacho, F.J., Bo\u0163, R.I., Torregrosa-Bel\u00e9n, D.: A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Numer. Algorithms 93, 103 (2022)","journal-title":"Numer. Algorithms"},{"issue":"2","key":"2064_CR2","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1007\/s10589-021-00291-6","volume":"80","author":"FJ Arag\u00f3n Artacho","year":"2021","unstructured":"Arag\u00f3n Artacho, F.J., Campoy, R., Tam, M.K.: Strengthened splitting methods for computing resolvents. Comput. Optim. Appl. 80(2), 549\u2013585 (2021)","journal-title":"Comput. Optim. Appl."},{"key":"2064_CR3","doi-asserted-by":"crossref","unstructured":"Arag\u00f3n-Artacho, F.J., Malitsky, Y., Tam, M.K., Torregrosa-Bel\u00e9n, D.: Distributed forward-backward methods for ring networks. Comput. Optim. Appl. pp. 1\u201326 (2022)","DOI":"10.1007\/s10589-022-00400-z"},{"key":"2064_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-48311-5","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. CMS Books in Mathematics, Springer International Publishing (2017)","edition":"2"},{"issue":"1","key":"2064_CR5","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/s10107-016-1086-3","volume":"164","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Moursi, W.M.: On the Douglas\u2013Rachford algorithm. Math. Program. 164(1), 263\u2013284 (2017)","journal-title":"Math. Program."},{"key":"2064_CR6","volume-title":"Parallel and Distributed Computation: Numerical Methods","author":"D Bertsekas","year":"2015","unstructured":"Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific (2015)"},{"issue":"2","key":"2064_CR7","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1137\/19M1308451","volume":"58","author":"MN B\u00fai","year":"2020","unstructured":"B\u00fai, M.N., Combettes, P.L.: The Douglas\u2013Rachford algorithm converges only weakly. SIAM J. Control Optim. 58(2), 1118\u20131120 (2020)","journal-title":"SIAM J. Control Optim."},{"issue":"1","key":"2064_CR8","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/s10589-022-00395-7","volume":"83","author":"R Campoy","year":"2022","unstructured":"Campoy, R.: A product space reformulation with reduced dimension for splitting algorithms. Comput. Optim. Appl. 83(1), 319\u2013348 (2022)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"2064_CR9","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","volume":"40","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120\u2013145 (2011)","journal-title":"J. Math. Imaging Vis."},{"issue":"92","key":"2064_CR10","first-page":"17","volume":"6","author":"FRK Chung","year":"1996","unstructured":"Chung, F.R.K.: Lectures on spectral graph theory. CBMS Lect. Fresno 6(92), 17\u201321 (1996)","journal-title":"CBMS Lect. Fresno"},{"issue":"2","key":"2064_CR11","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1007\/s10957-012-0245-9","volume":"158","author":"L Condat","year":"2013","unstructured":"Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158(2), 460\u2013479 (2013)","journal-title":"J. Optim. Theory Appl."},{"key":"2064_CR12","doi-asserted-by":"crossref","unstructured":"Condat, L., Kitahara, D., Contreras, A., Hirabayashi, A.: Proximal splitting algorithms for convex optimization: a tour of recent advances, with new twists. SIAM Rev p. to appear (2022)","DOI":"10.1137\/20M1379344"},{"issue":"1","key":"2064_CR13","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., Bertsekas, D.P.: On the Douglas\u2013Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1), 293\u2013318 (1992)","journal-title":"Math. Program."},{"key":"2064_CR14","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s10107-022-01906-4","volume":"201","author":"Y Malitsky","year":"2022","unstructured":"Malitsky, Y., Tam, M.K.: Resolvent splitting for sums of monotone operators with minimal lifting. Math. Program. 201, 231 (2022)","journal-title":"Math. Program."},{"key":"2064_CR15","unstructured":"Morin, M., Banert, S., Giselsson, P.: Frugal splitting operators: Representation, minimal lifting and convergence. arXiv preprint arXiv:2206.11177 (2022)"},{"key":"2064_CR16","first-page":"397","volume":"18","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Monotone operators associated with saddle-functions and minimax problems. J. Nonlinear Funct. Anal. 18, 397\u2013407 (1970)","journal-title":"J. Nonlinear Funct. Anal."},{"issue":"1","key":"2064_CR17","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s10107-019-01403-1","volume":"182","author":"EK Ryu","year":"2020","unstructured":"Ryu, E.K.: Uniqueness of DRS as the $$2$$ operator resolvent-splitting and impossibility of $$3$$ operator resolvent-splitting. Math. Program. 182(1), 233\u2013273 (2020)","journal-title":"Math. Program."},{"key":"2064_CR18","doi-asserted-by":"publisher","DOI":"10.1017\/9781009160865","volume-title":"Large-Scale Convex Optimization","author":"EK Ryu","year":"2022","unstructured":"Ryu, E.K., Yin, W.: Large-Scale Convex Optimization. Cambridge University Press (2022)"},{"issue":"2","key":"2064_CR19","doi-asserted-by":"publisher","first-page":"944","DOI":"10.1137\/14096668X","volume":"25","author":"W Shi","year":"2015","unstructured":"Shi, W., Ling, Q., Wu, G., Yin, W.: EXTRA: an exact first-order algorithm for decentralized consensus optimization. SIAM J. Optim. 25(2), 944\u2013966 (2015)","journal-title":"SIAM J. Optim."},{"issue":"22","key":"2064_CR20","doi-asserted-by":"publisher","first-page":"6013","DOI":"10.1109\/TSP.2015.2461520","volume":"63","author":"W Shi","year":"2015","unstructured":"Shi, W., Ling, Q., Wu, G., Yin, W.: A proximal gradient algorithm for decentralized composite optimization. IEEE Trans. Signal Process. 63(22), 6013\u20136023 (2015)","journal-title":"IEEE Trans. Signal Process."},{"issue":"1","key":"2064_CR21","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1137\/100788100","volume":"49","author":"BF Svaiter","year":"2011","unstructured":"Svaiter, B.F.: On weak convergence of the Douglas\u2013Rachford method. SIAM J. Control. Optim. 49(1), 280\u2013287 (2011)","journal-title":"SIAM J. Control. Optim."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-023-02064-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-023-02064-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-023-02064-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T04:17:41Z","timestamp":1724905061000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-023-02064-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,5]]},"references-count":21,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["2064"],"URL":"https:\/\/doi.org\/10.1007\/s11590-023-02064-y","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"value":"1862-4472","type":"print"},{"value":"1862-4480","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,5]]},"assertion":[{"value":"20 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}