{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T00:31:20Z","timestamp":1760488280717,"version":"build-2065373602"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s40314-025-03388-4","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T08:33:45Z","timestamp":1755765225000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inertial algorithms for equilibrium problems with applications to compressed sensing and image reconstruction"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5469-1332","authenticated-orcid":false,"given":"Fridoun","family":"Moradlou","sequence":"first","affiliation":[]},{"given":"Maryam","family":"Safari","sequence":"additional","affiliation":[]},{"given":"Masoumeh","family":"Azghani","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,21]]},"reference":[{"key":"3388_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/9781108377447","volume-title":"Compressive imaging: structure, learning, sampling","author":"B Adcock","year":"2021","unstructured":"Adcock B, Hansen AC (2021) Compressive imaging: structure, learning, sampling. Cambridge University Press, Cambridge"},{"key":"3388_CR2","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/s00009-014-0462-6","volume":"13","author":"S Alizadeh","year":"2016","unstructured":"Alizadeh S, Moradlou F (2016) A strong convergence theorem for equilibrium problems and generalized hybrid mappings. Mediterr J Math 13:379\u2013390","journal-title":"Mediterr J Math"},{"key":"3388_CR3","first-page":"81","volume":"84","author":"E Altiparmak","year":"2022","unstructured":"Altiparmak E, Karahan I (2022a) A modified preconditioning algorithm for solving monotone inclusion problem and application to image restoration problem. Sci Bull Univ Politeh Bucharest A 84:81\u201392","journal-title":"Sci Bull Univ Politeh Bucharest A"},{"issue":"12","key":"3388_CR4","doi-asserted-by":"publisher","first-page":"2482","DOI":"10.1080\/00207160.2022.2068146","volume":"99","author":"E Altiparmak","year":"2022","unstructured":"Altiparmak E, Karahan I (2022b) A new preconditioning algorithm for finding a zero of the sum of two monotone operators and its application to image restoration problems. Int J Comput Math 99(12):2482\u20132498","journal-title":"Int J Comput Math"},{"key":"3388_CR5","doi-asserted-by":"crossref","unstructured":"Altiparmak E, Karahan I (2024) A modified inertial viscosity algorithm for an infinite family of nonexpansive mappings and its application to image restoration. J Ind Manag Optim 20:453\u2013477","DOI":"10.3934\/jimo.2023086"},{"key":"3388_CR6","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2024.2358412","author":"E Alt\u0131parmak","year":"2024","unstructured":"Alt\u0131parmak E, Jolaoso LO, Karahan I, Rehman HU (2024) Pre-conditioning CQ algorithm for solving the split feasibility problem and its application to image restoration problem. Optimization. https:\/\/doi.org\/10.1080\/02331934.2024.2358412","journal-title":"Optimization"},{"key":"3388_CR7","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1137\/S1052623403427859","volume":"14","author":"F Alvarez","year":"2004","unstructured":"Alvarez F (2004) Weak convergence of a relaxed an inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J Optim 14:773\u2013782","journal-title":"SIAM J Optim"},{"key":"3388_CR8","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1023\/A:1011253113155","volume":"9","author":"F Alvarez","year":"2001","unstructured":"Alvarez F, Attouch H (2001) An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Var Anal 9:3\u201311","journal-title":"Set-Valued Var Anal"},{"issue":"2","key":"3388_CR9","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1109\/TMI.2014.2352113","volume":"34","author":"M Azghani","year":"2014","unstructured":"Azghani M, Kosmas P, Marvasti F (2014) Microwave medical imaging based on sparsity and an iterative method with adaptive thresholding. IEEE Trans Med Imaging 34(2):357\u2013365","journal-title":"IEEE Trans Med Imaging"},{"key":"3388_CR10","series-title":"CMS books in mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex analysis and monotone operator theory in Hilbert spaces","author":"H Bauschke","year":"2011","unstructured":"Bauschke H, Combettes P (2011) Convex analysis and monotone operator theory in Hilbert spaces. CMS books in mathematics. Springer, Cham"},{"key":"3388_CR11","first-page":"123","volume":"63","author":"E Blum","year":"1994","unstructured":"Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Student 63:123\u2013145","journal-title":"Math Student"},{"key":"3388_CR12","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1007\/s10957-015-0730-z","volume":"171","author":"R Bot","year":"2016","unstructured":"Bot R, Csetnek E (2016a) An inertial Tseng\u2019s type proximal algorithm for nonsmooth and nonconvex optimization problems. J Optim Theory Appl 171:600\u2013616","journal-title":"J Optim Theory Appl"},{"key":"3388_CR13","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1007\/s11075-015-0007-5","volume":"71","author":"R Bot","year":"2016","unstructured":"Bot R, Csetnek E (2016b) An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer Algorithms 71:519\u2013540","journal-title":"Numer Algorithms"},{"key":"3388_CR14","first-page":"29","volume":"1","author":"R Bot","year":"2016","unstructured":"Bot R, Csetnek E (2016c) An inertial alternating direction method of multipliers. Minimax Theory Appl 1:29\u201349","journal-title":"Minimax Theory Appl"},{"key":"3388_CR15","first-page":"472","volume":"256","author":"R Bot","year":"2015","unstructured":"Bot R, Csetnek E, Hendrich C (2015) Inertial Douglas\u2013Rachford splitting for monotone inclusion problems. Appl Math Comput 256:472\u2013487","journal-title":"Appl Math Comput"},{"key":"3388_CR16","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s13675-015-0045-8","volume":"4","author":"R Bot","year":"2016","unstructured":"Bot R, Csetnek E, Laszlo S (2016) An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J Comput Optim 4:3\u201325","journal-title":"EURO J Comput Optim"},{"issue":"2","key":"3388_CR17","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","volume":"148","author":"Y Censor","year":"2011","unstructured":"Censor Y, Gibali A, Reich S (2011) The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl 148(2):318\u2013335","journal-title":"J Optim Theory Appl"},{"key":"3388_CR18","doi-asserted-by":"publisher","first-page":"2120","DOI":"10.1137\/140980910","volume":"25","author":"C Chen","year":"2015","unstructured":"Chen C, Ma S, Yang J (2015) A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J Optim 25:2120\u20132142","journal-title":"SIAM J Optim"},{"key":"3388_CR19","first-page":"117","volume":"6","author":"LP Combettes","year":"2005","unstructured":"Combettes LP, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117\u2013136","journal-title":"J Nonlinear Convex Anal"},{"key":"3388_CR20","doi-asserted-by":"publisher","first-page":"2217","DOI":"10.1080\/02331934.2016.1239266","volume":"65","author":"QL Dong","year":"2016","unstructured":"Dong QL, Lu YY, Yang J (2016) The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65:2217\u20132226","journal-title":"Optimization"},{"key":"3388_CR21","first-page":"103","volume-title":"A minimax inequality and applications","author":"K Fan","year":"1972","unstructured":"Fan K (1972) A minimax inequality and applications. Academic Press, Cambridge, pp 103\u2013113"},{"key":"3388_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4948-7","volume-title":"A mathematical introduction to compressive sensing","author":"S Foucart","year":"2013","unstructured":"Foucart S, Rauhut H (2013) A mathematical introduction to compressive sensing. Springer, New York"},{"key":"3388_CR23","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s11784-019-0726-7","volume":"21","author":"A Gibali","year":"2019","unstructured":"Gibali A, Van Hieu D (2019) A new inertial double-projection method for solving variational inequalities. J fixed point theory appl 21:97","journal-title":"J fixed point theory appl"},{"key":"3388_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-68407-9","volume-title":"Foundations of optimization","author":"O G\u00fcler","year":"2010","unstructured":"G\u00fcler O (2010) Foundations of optimization, vol 258. Springer Science & Business Media, Berlin"},{"key":"3388_CR25","first-page":"493","volume":"5","author":"PT Hoai","year":"2021","unstructured":"Hoai PT, Thuong NT, Vinh NT (2021) Golden ratio algorithms for solving equilibrium problems in Hilbert spaces. J Nonlinear Var Anal 5:493\u2013518","journal-title":"J Nonlinear Var Anal"},{"issue":"12","key":"3388_CR26","doi-asserted-by":"publisher","first-page":"2007","DOI":"10.3390\/sym12122007","volume":"12","author":"LO Jolaoso","year":"2020","unstructured":"Jolaoso LO, Aphane M, Khan SH (2020) Two Bregman projection methods for solving variational inequality problems in Hilbert spaces with applications to signal processing. Symmetry 12(12):2007","journal-title":"Symmetry"},{"key":"3388_CR27","doi-asserted-by":"publisher","first-page":"1391","DOI":"10.1080\/01630563.2017.1321017","volume":"38","author":"Z Jouymandi","year":"2017","unstructured":"Jouymandi Z, Moradlou F (2017) Extragradient methods for solving equilibrium problems, variational inequalities and fixed point problems. Numer Funct Anal Optim 38:1391\u20131409","journal-title":"Numer Funct Anal Optim"},{"key":"3388_CR28","doi-asserted-by":"publisher","first-page":"523","DOI":"10.24193\/fpt-ro.2019.2.34","volume":"20","author":"Z Jouymandi","year":"2019","unstructured":"Jouymandi Z, Moradlou F (2019) Extragradient and linesearch algorithms for solving equilibrium problems, variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory 20:523\u2013540","journal-title":"Fixed Point Theory"},{"key":"3388_CR29","unstructured":"Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. In: Pure and Applied Mathematics, New York"},{"key":"3388_CR30","unstructured":"Korpelevi\u010d GM (1976) An extragradient method for finding saddle points and for other problems. E\u2018konom i Mat Metody 12:747\u2013756"},{"key":"3388_CR31","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s10957-013-0494-2","volume":"163","author":"R Kraikaew","year":"2014","unstructured":"Kraikaew R, Saejung S (2014) Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl 163:399\u2013412","journal-title":"J Optim Theory Appl"},{"key":"3388_CR32","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1007\/s10957-016-0972-4","volume":"171","author":"PE Maing\u00e9","year":"2016","unstructured":"Maing\u00e9 PE, Gobinddass ML (2016) Convergence of one-step projected gradient methods for variational inequalities. J Optim Theory Appl 171:146\u2013168","journal-title":"J Optim Theory Appl"},{"key":"3388_CR33","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1137\/14097238X","volume":"25","author":"Y Malitsky","year":"2015","unstructured":"Malitsky Y (2015) Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim 25:502\u2013520","journal-title":"SIAM J Optim"},{"key":"3388_CR34","doi-asserted-by":"crossref","unstructured":"Malitsky Y (2020) Golden ratio algorithms for variational inequalities. Math Program 184:383\u2013410","DOI":"10.1007\/s10107-019-01416-w"},{"key":"3388_CR35","first-page":"899","volume":"16","author":"A Maugeri","year":"2009","unstructured":"Maugeri A, Raciti F (2009) On existence theorems for monotone and nonmonotone variational inequalities. J Convex Anal 16:899\u2013911","journal-title":"J Convex Anal"},{"key":"3388_CR36","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1016\/0362-546X(92)90159-C","volume":"18","author":"L Muu","year":"1992","unstructured":"Muu L, Oettli W (1992) Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal 18:1159\u20131166","journal-title":"Nonlinear Anal"},{"key":"3388_CR37","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1137\/S0097539792240406","volume":"24","author":"BK Natarajan","year":"1995","unstructured":"Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24:227\u2013234","journal-title":"SIAM J Comput"},{"key":"3388_CR38","unstructured":"Nesterov Y E (1983) A method for solving the convex programming problem with convergence rate $${O}(1\/\\kappa ^2)$$. Dokl Akad Nauk SSSR 269:543\u2013547"},{"key":"3388_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0041-5553(64)90137-5","volume":"4","author":"BT Polyak","year":"1964","unstructured":"Polyak BT (1964) Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys 4:1\u201317","journal-title":"USSR Comput Math Math Phys"},{"issue":"4","key":"3388_CR40","first-page":"1","volume":"56","author":"Y Shehu","year":"2019","unstructured":"Shehu Y, Cholamjiak P (2019) Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo 56(4):1\u201321","journal-title":"Calcolo"},{"issue":"1","key":"3388_CR41","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","volume":"58","author":"R Tibshirani","year":"1996","unstructured":"Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B Stat Methodol 58(1):267\u2013288","journal-title":"J R Stat Soc Ser B Stat Methodol"},{"key":"3388_CR42","doi-asserted-by":"publisher","first-page":"4655","DOI":"10.1109\/TIT.2007.909108","volume":"53","author":"JA Tropp","year":"2007","unstructured":"Tropp JA, Gilbert AC (2007) Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans Inform Theor 53:4655\u20134666","journal-title":"IEEE Trans Inform Theor"},{"key":"3388_CR43","doi-asserted-by":"publisher","first-page":"101","DOI":"10.3390\/axioms9030101","volume":"9","author":"N Wairojjana","year":"2020","unstructured":"Wairojjana N, Rehman H, Sen MD, Pakkatanang N (2020) A general inertial projection-type algorithm for solving equilibrium problem in Hilbert spaces with applications in fixed-point problems. Axioms 9:101","journal-title":"Axioms"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03388-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03388-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03388-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T11:16:32Z","timestamp":1760440592000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03388-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,21]]},"references-count":43,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["3388"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03388-4","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,8,21]]},"assertion":[{"value":"15 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"421"}}