{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T23:20:18Z","timestamp":1776122418349,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,6,28]],"date-time":"2021-06-28T00:00:00Z","timestamp":1624838400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,6,28]],"date-time":"2021-06-28T00:00:00Z","timestamp":1624838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100006595","name":"Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii","doi-asserted-by":"publisher","award":["NO Grants 2014-2021, project ELO-Hyp, contract no. 24\/2020"],"award-info":[{"award-number":["NO Grants 2014-2021, project ELO-Hyp, contract no. 24\/2020"]}],"id":[{"id":"10.13039\/501100006595","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s10589-021-00294-3","type":"journal-article","created":{"date-parts":[[2021,6,28]],"date-time":"2021-06-28T22:04:03Z","timestamp":1624917843000},"page":"121-152","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Minibatch stochastic subgradient-based projection algorithms for feasibility problems with convex inequalities"],"prefix":"10.1007","volume":"80","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1102-2654","authenticated-orcid":false,"given":"Ion","family":"Necoara","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angelia","family":"Nedi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,28]]},"reference":[{"issue":"11","key":"294_CR1","doi-asserted-by":"publisher","first-page":"2545","DOI":"10.1109\/TAC.2009.2031207","volume":"54","author":"T Alamo","year":"2009","unstructured":"Alamo, T., Tempo, R., Camacho, E.F.: Randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems. IEEE Trans. Autom. Control 54(11), 2545\u20132559 (2009)","journal-title":"IEEE Trans. Autom. Control"},{"key":"294_CR2","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/S1570-579X(01)80003-3","volume-title":"Inherently Parallel Algorithms in Feasibility and Optimization and their Applications","author":"HH Bauschke","year":"2001","unstructured":"Bauschke, H.H.: Projection algorithms: Results and open problems. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, pp. 11\u201322. Elsevier, Amsterdam (2001)"},{"issue":"6","key":"294_CR3","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1364\/JOSAA.20.001025","volume":"20","author":"HH Bauschke","year":"2003","unstructured":"Bauschke, H.H., Combettes, P.L., Luke, D.R.: Hybrid projection-reflection method for phase retrieval. J. Opt. Soc. Am. 20(6), 1025\u20131034 (2003)","journal-title":"J. Opt. Soc. Am."},{"key":"294_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970777","volume-title":"Linear Matrix Inequalities in Systems and Control Theory","author":"S Boyd","year":"1994","unstructured":"Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. SIAM, Philadelphia (1994)"},{"key":"294_CR5","first-page":"620","volume":"7","author":"LM Bregman","year":"1967","unstructured":"Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. Zh. Vychisl. Mat. Mat. Fiz. 7, 620\u2013631 (1967)","journal-title":"Zh. Vychisl. Mat. Mat. Fiz."},{"issue":"6","key":"294_CR6","doi-asserted-by":"publisher","first-page":"1340","DOI":"10.1137\/0331063","volume":"31","author":"JV Burke","year":"1993","unstructured":"Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31(6), 1340\u20131359 (1993)","journal-title":"SIAM J. Control Optim."},{"issue":"9","key":"294_CR7","doi-asserted-by":"publisher","first-page":"3614","DOI":"10.1109\/TSP.2006.879312","volume":"54","author":"D Blatt","year":"2006","unstructured":"Blatt, D., Hero, A.: Energy based sensor network source localization via projection onto convex sets. IEEE Trans. Signal Process. 54(9), 3614\u20133619 (2006)","journal-title":"IEEE Trans. Signal Process."},{"issue":"11","key":"294_CR8","doi-asserted-by":"publisher","first-page":"1755","DOI":"10.1109\/9.964685","volume":"40","author":"G Calafiore","year":"2001","unstructured":"Calafiore, G., Polyak, B.T.: Stochastic algorithms for exact and approximate feasibility of robust LMI\u2019s. IEEE Trans. Autom. Control 40(11), 1755\u20131759 (2001)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"4","key":"294_CR9","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1109\/83.563316","volume":"6","author":"PL Combettes","year":"1997","unstructured":"Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6(4), 493\u2013506 (1997)","journal-title":"IEEE Trans. Image Process."},{"key":"294_CR10","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s002459900050","volume":"35","author":"PL Combettes","year":"1997","unstructured":"Combettes, P.L.: Hilbertian convex feasibility problem: convergence of projection methods. Appl. Math. Optim. 35, 311\u2013330 (1997)","journal-title":"Appl. Math. Optim."},{"key":"294_CR11","first-page":"96","volume-title":"Parametric Optimization and Approximation","author":"F Deutsch","year":"1983","unstructured":"Deutsch, F.: Rate of convergence of the method of alternating projections. In: Brosowski, B., Deutsch, F. (eds.) Parametric Optimization and Approximation, pp. 96\u2013107. Birkhauser, Basel (1983)"},{"key":"294_CR12","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.jat.2006.02.005","volume":"142","author":"F Deutsch","year":"2006","unstructured":"Deutsch, F., Hundal, H.: The rate of convergence for the cyclic projections algorithm I: Angles between convex sets. J. Approx. Theory 142, 36\u201355 (2006)","journal-title":"J. Approx. Theory"},{"key":"294_CR13","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)"},{"key":"294_CR14","unstructured":"Fercoq, O., Alacaoglu, A., Necoara, I., Cevher, V.: Almost surely constrained convex optimization. Presented at the (2019)"},{"issue":"6","key":"294_CR15","doi-asserted-by":"publisher","first-page":"1211","DOI":"10.1016\/0041-5553(67)90113-9","volume":"7","author":"LG Gubin","year":"1967","unstructured":"Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. Comput. Math. Math. Phys. 7(6), 1211\u20131228 (1967)","journal-title":"Comput. Math. Math. Phys."},{"key":"294_CR16","unstructured":"Guler, O., Hoffman, A., Rothblum, U.: Approximations to solutions to systems of linear inequalities. DIMACS technical report, DIMACS Center for Discrete Mathematics and Theoretical Computer Science (1992)"},{"key":"294_CR17","first-page":"96","volume":"23","author":"I Halperin","year":"1962","unstructured":"Halperin, I.: The product of projection operators. Acta Sci. Math. 23, 96\u201399 (1962)","journal-title":"Acta Sci. Math."},{"key":"294_CR18","unstructured":"Kundu, A., Bach, F., Bhattacharrya, C.: Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach. In: International Conference on Artificial Intelligence and Statistics, (2018)"},{"key":"294_CR19","first-page":"355","volume":"A35","author":"S Kaczmarz","year":"1937","unstructured":"Kaczmarz, S.: Angenaherte Auflosung von Systemen linearer Gleichungen. Bull. Acad. Sci. Pologne A35, 355\u2013357 (1937)","journal-title":"Bull. Acad. Sci. Pologne"},{"key":"294_CR20","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/978-1-4613-3341-8_3","volume-title":"Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications","author":"A Lewis","year":"1998","unstructured":"Lewis, A., Pang, J.: Error bounds for convex inequality systems. In: Crouzeix, J.P., Martinez-Legaz, J.E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications, pp. 75\u2013110. Springer, Berlin (1998)"},{"key":"294_CR21","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1109\/TCSVT.2005.848303","volume":"15","author":"A Liew","year":"2005","unstructured":"Liew, A., Yan, H., Law, N.: POCS-based blocking artifacts suppression using a smoothness constraint set with explicit region modeling. IEEE Trans. Circ. Syst. Video Tech. 15, 795\u2013800 (2005)","journal-title":"IEEE Trans. Circ. Syst. Video Tech."},{"key":"294_CR22","unstructured":"Lacoste-Julien, S., Schmidt, M., Bach, F.: A simpler approach to obtaining an $$O(1\/t)$$ convergence rate for the projected stochastic subgradient method. CoRR, abs\/1212.2002, (2012)"},{"key":"294_CR23","doi-asserted-by":"publisher","first-page":"393","DOI":"10.4153\/CJM-1954-038-x","volume":"6","author":"TS Motzkin","year":"1954","unstructured":"Motzkin, T.S., Shoenberg, I.: The relaxation method for linear inequalities. Can. J. Math. 6, 393\u2013404 (1954)","journal-title":"Can. J. Math."},{"issue":"4","key":"294_CR24","doi-asserted-by":"publisher","first-page":"2814","DOI":"10.1137\/18M1167061","volume":"29","author":"I Necoara","year":"2019","unstructured":"Necoara, I., Richtarik, P., Patrascu, A.: Randomized projection methods for convex feasibility problems. SIAM J. Optim. 29(4), 2814\u20132852 (2019)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"294_CR25","doi-asserted-by":"publisher","first-page":"1425","DOI":"10.1137\/19M1251643","volume":"40","author":"I Necoara","year":"2019","unstructured":"Necoara, I.: Faster randomized block Kaczmarz algorithms. SIAM J. Matrix Anal. Appl. 40(4), 1425\u20131452 (2019)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"294_CR26","doi-asserted-by":"crossref","unstructured":"Nedic, A.: Random projection algorithms for convex set intersection problems. In: Proceedings of IEEE Conference on Decision and Control, pp.\u00a07655\u20137660, (2010)","DOI":"10.1109\/CDC.2010.5717734"},{"issue":"2","key":"294_CR27","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s10107-011-0468-9","volume":"129","author":"A Nedic","year":"2011","unstructured":"Nedic, A.: Random algorithms for convex minimization problems. Math. Progr. 129(2), 225\u2013273 (2011)","journal-title":"Math. Progr."},{"issue":"4","key":"294_CR28","doi-asserted-by":"publisher","first-page":"1574","DOI":"10.1137\/070704277","volume":"19","author":"A Nemirovski","year":"2009","unstructured":"Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574\u20131609 (2009)","journal-title":"SIAM J. Optim."},{"key":"294_CR29","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970791","volume-title":"Interior-Point Polynomial Algorithms in Convex Programming","author":"Yu Nesterov","year":"1994","unstructured":"Nesterov, Yu., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)"},{"key":"294_CR30","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/0041-5553(69)90061-5","volume":"9","author":"BT Polyak","year":"1969","unstructured":"Polyak, B.T.: Minimization of unsmooth functionals. Comput. Math. Math. Phys. 9, 14\u201329 (1969)","journal-title":"Comput. Math. Math. Phys."},{"key":"294_CR31","volume-title":"Introduction to Optimization","author":"BT Polyak","year":"1987","unstructured":"Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)"},{"key":"294_CR32","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/S1570-579X(01)80024-0","volume-title":"Inherently Parallel Algorithms in Feasibility and Optimization and their Applications","author":"BT Polyak","year":"2001","unstructured":"Polyak, B.T.: Random algorithms for solving convex inequalities. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications, pp. 409\u2013422. Elsevier, Amsterdam (2001)"},{"issue":"198","key":"294_CR33","first-page":"1","volume":"18","author":"A Patrascu","year":"2018","unstructured":"Patrascu, A., Necoara, I.: Nonasymptotic convergence of stochastic proximal point algorithms for constrained convex optimization. J. Mach. Learn. Res. 18(198), 1\u201342 (2018)","journal-title":"J. Mach. Learn. Res."},{"key":"294_CR34","doi-asserted-by":"publisher","DOI":"10.1109\/tnnls.2019.2957003","author":"X Peng","year":"2019","unstructured":"Peng, X., Li, L., Wang, F.: Accelerating minibatch stochastic gradient descent using typicality sampling. IEEE Trans. Neural Networks Learn. Syst. (2019). https:\/\/doi.org\/10.1109\/tnnls.2019.2957003","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"294_CR35","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"294_CR36","unstructured":"Stark, H., Yang, Y.: Vector space projections: A numerical approach to signal and image processing. Wiley-Interscience, Neural Nets and Optics (1998)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-021-00294-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-021-00294-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-021-00294-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T17:21:52Z","timestamp":1627752112000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-021-00294-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,28]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["294"],"URL":"https:\/\/doi.org\/10.1007\/s10589-021-00294-3","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,28]]},"assertion":[{"value":"4 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2021","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 authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}