{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T04:13:26Z","timestamp":1783656806955,"version":"3.55.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T00:00:00Z","timestamp":1657843200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T00:00:00Z","timestamp":1657843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-097960-B-C22"],"award-info":[{"award-number":["PGC2018-097960-B-C22"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003359","name":"Generalitat Valenciana","doi-asserted-by":"publisher","award":["AICO\/2021\/165"],"award-info":[{"award-number":["AICO\/2021\/165"]}],"id":[{"id":"10.13039\/501100003359","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Universitat de Valencia"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra\u2019s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.<\/jats:p>","DOI":"10.1007\/s10589-022-00395-7","type":"journal-article","created":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T15:44:00Z","timestamp":1657899840000},"page":"319-348","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A product space reformulation with reduced dimension for splitting algorithms"],"prefix":"10.1007","volume":"83","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1275-7831","authenticated-orcid":false,"given":"Rub\u00e9n","family":"Campoy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,7,15]]},"reference":[{"issue":"3","key":"395_CR1","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1007\/s00245-019-09599-6","volume":"80","author":"S Adly","year":"2019","unstructured":"Adly, S., Bourdin, L.: On a decomposition formula for the resolvent operator of the sum of two set-valued maps with monotonicity assumptions. Appl. Math. Opt. 80(3), 715\u2013732 (2019)","journal-title":"Appl. Math. Opt."},{"issue":"6","key":"395_CR2","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1016\/j.orl.2018.10.003","volume":"46","author":"S Alwadani","year":"2018","unstructured":"Alwadani, S., Bauschke, H.H., Moursi, W.M., Wang, X.: On the asymptotic behaviour of the Arag\u00f3n Artacho-Campoy algorithm. Oper. Res. Lett. 46(6), 585\u2013587 (2018)","journal-title":"Oper. Res. Lett."},{"issue":"1","key":"395_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10957-013-0488-0","volume":"163","author":"FJ Arag\u00f3n Artacho","year":"2014","unstructured":"Arag\u00f3n Artacho, F.J., Borwein, J.M., Tam, M.K.: Recent results on Douglas-Rachford methods for combinatorial optimization problem. J. Optim. Theory. Appl. 163(1), 1\u201330 (2014)","journal-title":"J. Optim. Theory. Appl."},{"issue":"1","key":"395_CR4","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s10589-017-9942-5","volume":"69","author":"FJ Arag\u00f3n Artacho","year":"2018","unstructured":"Arag\u00f3n Artacho, F.J., Campoy, R.: A new projection method for finding the closest point in the intersection of convex sets. Comput. Optim. Appl. 69(1), 99\u2013132 (2018)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"395_CR5","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1007\/s10957-019-01481-3","volume":"181","author":"FJ Arag\u00f3n Artacho","year":"2019","unstructured":"Arag\u00f3n Artacho, F.J., Campoy, R.: Computing the resolvent of the sum of maximally monotone operators with the averaged alternating modified reflections algorithm. J. Optim. Theory Appl. 181(3), 709\u2013726 (2019)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"395_CR6","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s00186-019-00691-9","volume":"91","author":"FJ Arag\u00f3n Artacho","year":"2020","unstructured":"Arag\u00f3n Artacho, F.J., Campoy, R., Tam, M.K.: The Douglas-Rachford algorithm for convex and nonconvex feasibility problems. Math. Methods Oper. Res. 91(2), 201\u2013240 (2020)","journal-title":"Math. Methods Oper. Res."},{"issue":"2","key":"395_CR7","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":"395_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-022-00631-6","author":"FJ Arag\u00f3n-Artacho","year":"2022","unstructured":"Arag\u00f3n-Artacho, F.J., Torregrosa-Bel\u00e9n, D.: A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes. Set-Valued Variat. Anal. (2022). https:\/\/doi.org\/10.1007\/s11228-022-00631-6","journal-title":"Set-Valued Variat. Anal."},{"key":"395_CR9","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. Springer, Berlin (2017)","edition":"2"},{"issue":"7","key":"395_CR10","doi-asserted-by":"publisher","first-page":"1334","DOI":"10.1364\/JOSAA.19.001334","volume":"19","author":"HH Bauschke","year":"2002","unstructured":"Bauschke, H.H., Combettes, P.L., Luke, D.R.: Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Am A 19(7), 1334\u20131345 (2002)","journal-title":"J. Opt. Soc. Am A"},{"issue":"1\u20132","key":"395_CR11","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-Rachford algorithm. Math. Program. 164(1\u20132), 263\u2013284 (2017)","journal-title":"Math. Program."},{"key":"395_CR12","unstructured":"Bauschke, H.H., Moursi, W.M.: On the Douglas-Rachford algorithm for solving possibly inconsistent optimization problems. ArXiv preprint Arxiv: arXiv:2106.11547 (2021)"},{"key":"395_CR13","doi-asserted-by":"crossref","unstructured":"Bauschke, H.H., Singh, S., Wang, X.: The splitting algorithms by Ryu, by Malitsky\u2013Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection. ArXiv preprint ArXiv: arXiv:2203.03832. (2022)","DOI":"10.1137\/22M1483165"},{"issue":"4","key":"395_CR14","doi-asserted-by":"publisher","first-page":"2541","DOI":"10.1137\/120901106","volume":"23","author":"RI Bot","year":"2013","unstructured":"Bot, R.I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23(4), 2541\u20132565 (2013)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"395_CR15","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s11228-020-00542-4","volume":"29","author":"V Cevher","year":"2021","unstructured":"Cevher, V., Vu, B.C.: A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators. Set-Valued Var. Anal. 29(1), 163\u2013174 (2021)","journal-title":"Set-Valued Var. Anal."},{"issue":"4","key":"395_CR16","first-page":"727","volume":"16","author":"PL Combettes","year":"2009","unstructured":"Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16(4), 727\u2013748 (2009)","journal-title":"J. Convex Anal."},{"key":"395_CR17","unstructured":"Condat, L., Kitahara, D., Contreras, A., Hirabayashi, A: Proximal splitting algorithms for convex optimization: a tour of recent advances, with new twists. ArXiv preprint Arxiv: arXiv:1912.00137 (2021)"},{"issue":"1","key":"395_CR18","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s10957-021-01878-z","volume":"190","author":"M Dao","year":"2021","unstructured":"Dao, M., Dizon, N., Hogan, J., Tam, M.K.: Constraint reduction reformulations for projection algorithms with applications to wavelet construction. J. Optim. Theory Appl. 190(1), 201\u2013233 (2021)","journal-title":"J. Optim. Theory Appl."},{"issue":"5","key":"395_CR19","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1007\/s11590-019-01432-x","volume":"14","author":"MN Dao","year":"2020","unstructured":"Dao, M.N., Phan, H.M.: Computing the resolvent of the sum of operators with application to best approximation problems. Optim. Lett. 14(5), 1193\u20131205 (2020)","journal-title":"Optim. Lett."},{"issue":"4","key":"395_CR20","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1007\/s11228-017-0421-z","volume":"25","author":"D Davis","year":"2017","unstructured":"Davis, D., Yin, W.: A three-operator splitting scheme and its optimization applications. Set-Valued Var. Anal. 25(4), 829\u2013858 (2017)","journal-title":"Set-Valued Var. Anal."},{"issue":"2","key":"395_CR21","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201\u2013213 (2002)","journal-title":"Math. Program."},{"issue":"2","key":"395_CR22","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","volume":"82","author":"J Douglas","year":"1956","unstructured":"Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421\u2013439 (1956)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"395_CR23","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1364\/JOSAA.20.000040","volume":"20","author":"V Elser","year":"2003","unstructured":"Elser, V.: Phase retrieval by iterated projections. J. Opt. Soc. Am. A 20(1), 40\u201355 (2003)","journal-title":"J. Opt. Soc. Am. A"},{"issue":"2","key":"395_CR24","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1073\/pnas.0606359104","volume":"104","author":"V Elser","year":"2007","unstructured":"Elser, V., Rankenburg, I., Thibault, P.: Searching with iterated maps. Proc. Natl. Acad. Sci. 104(2), 418\u2013423 (2007)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"395_CR25","doi-asserted-by":"crossref","unstructured":"Franklin, D.J., Hogan, J.A., Tam, M.K.: Higher-dimensional wavelets and the Douglas\u2013Rachford algorithm. In: 13th International Conference on Sampling Theory and Applications (SampTA), pp. 1\u20134. IEEE (2019)","DOI":"10.1109\/SampTA45681.2019.9030823"},{"issue":"1","key":"395_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10957-016-0889-y","volume":"169","author":"AF Izmailov","year":"2016","unstructured":"Izmailov, A.F., Solodov, M.V., Uskov, E.T.: Globalizing stabilized sequential quadratic programming method by smooth primal-dual exact penalty function. J. Optim. Theor. Appl. 169(1), 1\u201331 (2016)","journal-title":"J. Optim. Theor. Appl."},{"issue":"3","key":"395_CR27","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1007\/BF00968624","volume":"26","author":"AY Kruger","year":"1985","unstructured":"Kruger, A.Y.: Generalized differentials of nonsmooth functions, and necessary conditions for an extremum. Sib. Math. J. 26(3), 370\u2013379 (1985)","journal-title":"Sib. Math. J."},{"issue":"1","key":"395_CR28","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1017\/S1446181118000391","volume":"61","author":"BP Lamichhane","year":"2019","unstructured":"Lamichhane, B.P., Lindstrom, S.B., Sims, B.: Application of projection algorithms to differential equations: boundary value problems. ANZIAM J. 61(1), 23\u201346 (2019)","journal-title":"ANZIAM J."},{"issue":"6","key":"395_CR29","doi-asserted-by":"publisher","first-page":"964","DOI":"10.1137\/0716071","volume":"16","author":"PL Lions","year":"1979","unstructured":"Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964\u2013979 (1979)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"395_CR30","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1137\/18M1207260","volume":"30","author":"Y Malitsky","year":"2020","unstructured":"Malitsky, Y., Tam, M.K.: A forward-backward splitting method for monotone inclusions without cocoercivity. SIAM J. Optim. 30(2), 1451\u20131472 (2020)","journal-title":"SIAM J. Optim."},{"key":"395_CR31","doi-asserted-by":"crossref","unstructured":"Malitsky, Y., Tam, M.K.: Resolvent splitting for sums of monotone operators with minimal lifting. ArXiv preprint Arxiv: arXiv:2108.02897 (2021)","DOI":"10.1007\/s10107-022-01906-4"},{"issue":"2","key":"395_CR32","doi-asserted-by":"publisher","first-page":"87","DOI":"10.4169\/amer.math.monthly.119.02.087","volume":"119","author":"BS Mordukhovich","year":"2012","unstructured":"Mordukhovich, B.S., Nam, N.M., Salinas, J.: Solving a generalized Heron problem by means of convex analysis. Amer. Math. Monthly 119(2), 87\u201399 (2012)","journal-title":"Amer. Math. Monthly"},{"issue":"10","key":"395_CR33","doi-asserted-by":"publisher","first-page":"1915","DOI":"10.1080\/00036811.2011.604849","volume":"91","author":"BS Mordukhovich","year":"2012","unstructured":"Mordukhovich, B.S., Nam, N.M., Salinas, J.: Applications of variational analysis to a generalized Heron problem. Appl. Anal. 91(10), 1915\u20131942 (2012)","journal-title":"Appl. Anal."},{"key":"395_CR34","unstructured":"Pierra, G.: M\u00e9thodes de D\u00e9composition et Croisement D\u2019algorithmes Pour des Probl\u00e8mes D\u2019optimisation. Doctoral Dissertation. Institut National Polytechnique de Grenoble-INPG, Universit\u00e9 Joseph-Fourier-Grenoble I (1976)"},{"issue":"1","key":"395_CR35","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1007\/BF02612715","volume":"28","author":"G Pierra","year":"1984","unstructured":"Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28(1), 96\u2013115 (1984)","journal-title":"Math. Program."},{"issue":"3","key":"395_CR36","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1137\/120872802","volume":"6","author":"H Raguet","year":"2013","unstructured":"Raguet, H., Fadili, J., Peyr\u00e9, G.: A generalized forward-backward splitting. SIAM J. Imaging Sci. 6(3), 1199\u20131226 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"395_CR37","first-page":"125248","volume":"381","author":"J Rieger","year":"2020","unstructured":"Rieger, J., Tam, M.K.: Backward-forward-reflected-backward splitting for three operator monotone inclusions. Appl. Math. Comput. 381, 125248 (2020)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"395_CR38","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877\u2013898 (1976)","journal-title":"SIAM J. Control Optim."},{"issue":"1","key":"395_CR39","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."},{"issue":"2","key":"395_CR40","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1137\/S0363012998338806","volume":"38","author":"P Tseng","year":"2000","unstructured":"Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431\u2013446 (2000)","journal-title":"SIAM J. Control Optim."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-022-00395-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-022-00395-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-022-00395-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T16:36:19Z","timestamp":1700843779000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-022-00395-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,15]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["395"],"URL":"https:\/\/doi.org\/10.1007\/s10589-022-00395-7","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,15]]},"assertion":[{"value":"3 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 July 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author has no conflict of interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}