[{"abstract":[{"lang":"eng","text":"This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis."}],"type":"journal_article","file":[{"file_name":"2020_OptimizationEngineering_Shehu.pdf","access_level":"open_access","creator":"dernst","file_size":2137860,"content_type":"application/pdf","file_id":"8197","relation":"main_file","date_updated":"2020-08-03T15:24:39Z","date_created":"2020-08-03T15:24:39Z","success":1}],"oa_version":"Published Version","_id":"8196","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","intvolume":" 22","title":"New strong convergence method for the sum of two maximal monotone operators","ddc":["510"],"status":"public","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","day":"25","scopus_import":"1","date_published":"2021-02-25T00:00:00Z","citation":{"ama":"Shehu Y, Dong Q-L, Liu L-L, Yao J-C. New strong convergence method for the sum of two maximal monotone operators. Optimization and Engineering. 2021;22:2627-2653. doi:10.1007/s11081-020-09544-5","ista":"Shehu Y, Dong Q-L, Liu L-L, Yao J-C. 2021. New strong convergence method for the sum of two maximal monotone operators. Optimization and Engineering. 22, 2627–2653.","apa":"Shehu, Y., Dong, Q.-L., Liu, L.-L., & Yao, J.-C. (2021). New strong convergence method for the sum of two maximal monotone operators. Optimization and Engineering. Springer Nature. https://doi.org/10.1007/s11081-020-09544-5","ieee":"Y. Shehu, Q.-L. Dong, L.-L. Liu, and J.-C. Yao, “New strong convergence method for the sum of two maximal monotone operators,” Optimization and Engineering, vol. 22. Springer Nature, pp. 2627–2653, 2021.","mla":"Shehu, Yekini, et al. “New Strong Convergence Method for the Sum of Two Maximal Monotone Operators.” Optimization and Engineering, vol. 22, Springer Nature, 2021, pp. 2627–53, doi:10.1007/s11081-020-09544-5.","short":"Y. Shehu, Q.-L. Dong, L.-L. Liu, J.-C. Yao, Optimization and Engineering 22 (2021) 2627–2653.","chicago":"Shehu, Yekini, Qiao-Li Dong, Lu-Lu Liu, and Jen-Chih Yao. “New Strong Convergence Method for the Sum of Two Maximal Monotone Operators.” Optimization and Engineering. Springer Nature, 2021. https://doi.org/10.1007/s11081-020-09544-5."},"publication":"Optimization and Engineering","page":"2627-2653","article_type":"original","ec_funded":1,"file_date_updated":"2020-08-03T15:24:39Z","author":[{"orcid":"0000-0001-9224-7139","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","last_name":"Shehu","first_name":"Yekini","full_name":"Shehu, Yekini"},{"last_name":"Dong","first_name":"Qiao-Li","full_name":"Dong, Qiao-Li"},{"first_name":"Lu-Lu","last_name":"Liu","full_name":"Liu, Lu-Lu"},{"first_name":"Jen-Chih","last_name":"Yao","full_name":"Yao, Jen-Chih"}],"volume":22,"date_updated":"2024-03-07T14:39:29Z","date_created":"2020-08-03T14:29:57Z","year":"2021","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). The project of Yekini Shehu has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Program (FP7—2007–2013) (Grant Agreement No. 616160). The authors are grateful to the anonymous referees and the handling Editor for their comments and suggestions which have improved the earlier version of the manuscript greatly.","department":[{"_id":"VlKo"}],"publisher":"Springer Nature","publication_status":"published","publication_identifier":{"issn":["1389-4420"],"eissn":["1573-2924"]},"month":"02","doi":"10.1007/s11081-020-09544-5","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000559345400001"]},"oa":1,"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160","call_identifier":"FP7","name":"Discrete Optimization in Computer Vision: Theory and Practice"}],"isi":1,"quality_controlled":"1"},{"language":[{"iso":"eng"}],"doi":"10.1007/s11590-020-01603-1","project":[{"grant_number":"616160","_id":"25FBA906-B435-11E9-9278-68D0E5697425","name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"quality_controlled":"1","isi":1,"external_id":{"isi":["000537342300001"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"publication_identifier":{"eissn":["1862-4480"],"issn":["1862-4472"]},"month":"09","volume":15,"date_created":"2020-06-04T11:28:33Z","date_updated":"2024-03-07T15:00:43Z","author":[{"id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-9224-7139","first_name":"Yekini","last_name":"Shehu","full_name":"Shehu, Yekini"},{"full_name":"Gibali, Aviv","last_name":"Gibali","first_name":"Aviv"}],"publisher":"Springer Nature","department":[{"_id":"VlKo"}],"publication_status":"published","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). The authors are grateful to the referees for their insightful comments which have improved the earlier version of the manuscript greatly. The first author has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Program (FP7-2007-2013) (Grant agreement No. 616160).","year":"2021","ec_funded":1,"file_date_updated":"2024-03-07T14:58:51Z","date_published":"2021-09-01T00:00:00Z","page":"2109-2126","article_type":"original","citation":{"mla":"Shehu, Yekini, and Aviv Gibali. “New Inertial Relaxed Method for Solving Split Feasibilities.” Optimization Letters, vol. 15, Springer Nature, 2021, pp. 2109–26, doi:10.1007/s11590-020-01603-1.","short":"Y. Shehu, A. Gibali, Optimization Letters 15 (2021) 2109–2126.","chicago":"Shehu, Yekini, and Aviv Gibali. “New Inertial Relaxed Method for Solving Split Feasibilities.” Optimization Letters. Springer Nature, 2021. https://doi.org/10.1007/s11590-020-01603-1.","ama":"Shehu Y, Gibali A. New inertial relaxed method for solving split feasibilities. Optimization Letters. 2021;15:2109-2126. doi:10.1007/s11590-020-01603-1","ista":"Shehu Y, Gibali A. 2021. New inertial relaxed method for solving split feasibilities. Optimization Letters. 15, 2109–2126.","ieee":"Y. Shehu and A. Gibali, “New inertial relaxed method for solving split feasibilities,” Optimization Letters, vol. 15. Springer Nature, pp. 2109–2126, 2021.","apa":"Shehu, Y., & Gibali, A. (2021). New inertial relaxed method for solving split feasibilities. Optimization Letters. 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We give convergence of the sequence generated by our method under some suitable assumptions. Some numerical implementations from sparse signal and image deblurring are reported to show the efficiency of our method."}],"type":"journal_article"},{"project":[{"name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7","_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160"}],"quality_controlled":"1","isi":1,"external_id":{"isi":["000528979000015"]},"oa":1,"language":[{"iso":"eng"}],"doi":"10.1007/s11075-019-00758-y","publication_identifier":{"issn":["1017-1398"],"eissn":["1572-9265"]},"month":"05","publisher":"Springer Nature","department":[{"_id":"VlKo"}],"publication_status":"published","year":"2020","acknowledgement":"The research of this author is supported by the ERC grant at the IST.","volume":84,"date_updated":"2023-08-17T13:51:18Z","date_created":"2019-06-27T20:09:33Z","author":[{"first_name":"Yekini","last_name":"Shehu","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-9224-7139","full_name":"Shehu, Yekini"},{"full_name":"Li, Xiao-Huan","last_name":"Li","first_name":"Xiao-Huan"},{"full_name":"Dong, Qiao-Li","first_name":"Qiao-Li","last_name":"Dong"}],"ec_funded":1,"file_date_updated":"2020-07-14T12:47:34Z","page":"365-388","article_type":"original","citation":{"chicago":"Shehu, Yekini, Xiao-Huan Li, and Qiao-Li Dong. “An Efficient Projection-Type Method for Monotone Variational Inequalities in Hilbert Spaces.” Numerical Algorithms. Springer Nature, 2020. https://doi.org/10.1007/s11075-019-00758-y.","mla":"Shehu, Yekini, et al. “An Efficient Projection-Type Method for Monotone Variational Inequalities in Hilbert Spaces.” Numerical Algorithms, vol. 84, Springer Nature, 2020, pp. 365–88, doi:10.1007/s11075-019-00758-y.","short":"Y. Shehu, X.-H. Li, Q.-L. Dong, Numerical Algorithms 84 (2020) 365–388.","ista":"Shehu Y, Li X-H, Dong Q-L. 2020. An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numerical Algorithms. 84, 365–388.","apa":"Shehu, Y., Li, X.-H., & Dong, Q.-L. (2020). An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numerical Algorithms. Springer Nature. https://doi.org/10.1007/s11075-019-00758-y","ieee":"Y. Shehu, X.-H. Li, and Q.-L. Dong, “An efficient projection-type method for monotone variational inequalities in Hilbert spaces,” Numerical Algorithms, vol. 84. Springer Nature, pp. 365–388, 2020.","ama":"Shehu Y, Li X-H, Dong Q-L. An efficient projection-type method for monotone variational inequalities in Hilbert spaces. Numerical Algorithms. 2020;84:365-388. doi:10.1007/s11075-019-00758-y"},"publication":"Numerical Algorithms","date_published":"2020-05-01T00:00:00Z","scopus_import":"1","has_accepted_license":"1","article_processing_charge":"No","day":"01","intvolume":" 84","status":"public","title":"An efficient projection-type method for monotone variational inequalities in Hilbert spaces","ddc":["000"],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"6593","file":[{"creator":"kschuh","content_type":"application/pdf","file_size":359654,"access_level":"open_access","file_name":"ExtragradientMethodPaper.pdf","checksum":"bb1a1eb3ebb2df380863d0db594673ba","date_updated":"2020-07-14T12:47:34Z","date_created":"2019-10-01T13:14:10Z","file_id":"6927","relation":"main_file"}],"oa_version":"Submitted Version","type":"journal_article","abstract":[{"text":"We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising.","lang":"eng"}]},{"has_accepted_license":"1","article_processing_charge":"No","day":"01","scopus_import":"1","date_published":"2020-11-01T00:00:00Z","citation":{"ama":"Shehu Y, Iyiola OS. Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics. 2020;157:315-337. doi:10.1016/j.apnum.2020.06.009","ieee":"Y. Shehu and O. S. Iyiola, “Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence,” Applied Numerical Mathematics, vol. 157. Elsevier, pp. 315–337, 2020.","apa":"Shehu, Y., & Iyiola, O. S. (2020). Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics. Elsevier. https://doi.org/10.1016/j.apnum.2020.06.009","ista":"Shehu Y, Iyiola OS. 2020. Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics. 157, 315–337.","short":"Y. Shehu, O.S. Iyiola, Applied Numerical Mathematics 157 (2020) 315–337.","mla":"Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” Applied Numerical Mathematics, vol. 157, Elsevier, 2020, pp. 315–37, doi:10.1016/j.apnum.2020.06.009.","chicago":"Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” Applied Numerical Mathematics. Elsevier, 2020. https://doi.org/10.1016/j.apnum.2020.06.009."},"publication":"Applied Numerical Mathematics","page":"315-337","article_type":"original","abstract":[{"lang":"eng","text":"The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejér monotonicity of the iterates with respect to the solution, which is being enjoyed by their corresponding non-inertial projection methods for variational inequalities. This lack of Fejér monotonicity makes projection methods with vanilla inertial extrapolation step for variational inequalities not to converge faster than their corresponding non-inertial projection methods at times. Also, it has recently been proved that the projection methods with vanilla inertial extrapolation step may provide convergence rates that are worse than the classical projected gradient methods for strongly convex functions. In this paper, we introduce projection methods with alternated inertial extrapolation step for solving variational inequalities. We show that the sequence of iterates generated by our methods converges weakly to a solution of the variational inequality under some appropriate conditions. The Fejér monotonicity of even subsequence is recovered in these methods and linear rate of convergence is obtained. The numerical implementations of our methods compared with some other inertial projection methods show that our method is more efficient and outperforms some of these inertial projection methods."}],"type":"journal_article","file":[{"checksum":"87d81324a62c82baa925c009dfcb0200","date_created":"2020-07-02T09:08:59Z","date_updated":"2020-07-14T12:48:09Z","relation":"main_file","file_id":"8078","file_size":2874203,"content_type":"application/pdf","creator":"dernst","access_level":"open_access","file_name":"2020_AppliedNumericalMath_Shehu.pdf"}],"oa_version":"Submitted Version","_id":"8077","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","intvolume":" 157","title":"Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence","ddc":["510"],"status":"public","publication_identifier":{"issn":["0168-9274"]},"month":"11","doi":"10.1016/j.apnum.2020.06.009","language":[{"iso":"eng"}],"external_id":{"isi":["000564648400018"]},"oa":1,"project":[{"name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7","grant_number":"616160","_id":"25FBA906-B435-11E9-9278-68D0E5697425"}],"isi":1,"quality_controlled":"1","ec_funded":1,"file_date_updated":"2020-07-14T12:48:09Z","author":[{"last_name":"Shehu","first_name":"Yekini","orcid":"0000-0001-9224-7139","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","full_name":"Shehu, Yekini"},{"full_name":"Iyiola, Olaniyi S.","first_name":"Olaniyi S.","last_name":"Iyiola"}],"volume":157,"date_updated":"2023-08-22T07:50:43Z","date_created":"2020-07-02T09:02:33Z","acknowledgement":"The authors are grateful to the two anonymous referees for their insightful comments and suggestions which have improved the earlier version of the manuscript greatly. The first author has received funding from the European Research Council (ERC) under the European Union Seventh Framework Programme (FP7 - 2007-2013) (Grant agreement No. 616160).","year":"2020","department":[{"_id":"VlKo"}],"publisher":"Elsevier","publication_status":"published"},{"abstract":[{"lang":"eng","text":"In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature."}],"type":"journal_article","oa_version":"Submitted Version","file":[{"relation":"main_file","file_id":"8647","embargo":"2021-03-15","checksum":"9f6dc6c6bf2b48cb3a2091a9ed5feaf2","date_updated":"2021-03-16T23:30:04Z","date_created":"2020-10-12T10:40:27Z","access_level":"open_access","file_name":"2020_JourOptimizationTheoryApplic_Shehu.pdf","content_type":"application/pdf","file_size":332641,"creator":"dernst"}],"_id":"7161","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":" 184","status":"public","title":"Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces","ddc":["518","510","515"],"has_accepted_license":"1","article_processing_charge":"No","day":"01","scopus_import":"1","date_published":"2020-03-01T00:00:00Z","citation":{"short":"Y. Shehu, A. Gibali, S. Sagratella, Journal of Optimization Theory and Applications 184 (2020) 877–894.","mla":"Shehu, Yekini, et al. “Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces.” Journal of Optimization Theory and Applications, vol. 184, Springer Nature, 2020, pp. 877–894, doi:10.1007/s10957-019-01616-6.","chicago":"Shehu, Yekini, Aviv Gibali, and Simone Sagratella. “Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces.” Journal of Optimization Theory and Applications. Springer Nature, 2020. https://doi.org/10.1007/s10957-019-01616-6.","ama":"Shehu Y, Gibali A, Sagratella S. Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces. Journal of Optimization Theory and Applications. 2020;184:877–894. doi:10.1007/s10957-019-01616-6","ieee":"Y. Shehu, A. Gibali, and S. Sagratella, “Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 184. Springer Nature, pp. 877–894, 2020.","apa":"Shehu, Y., Gibali, A., & Sagratella, S. (2020). Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces. Journal of Optimization Theory and Applications. Springer Nature. https://doi.org/10.1007/s10957-019-01616-6","ista":"Shehu Y, Gibali A, Sagratella S. 2020. Inertial projection-type methods for solving quasi-variational inequalities in real Hilbert spaces. Journal of Optimization Theory and Applications. 184, 877–894."},"publication":"Journal of Optimization Theory and Applications","page":"877–894","article_type":"original","ec_funded":1,"file_date_updated":"2021-03-16T23:30:04Z","author":[{"full_name":"Shehu, Yekini","orcid":"0000-0001-9224-7139","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","last_name":"Shehu","first_name":"Yekini"},{"full_name":"Gibali, Aviv","first_name":"Aviv","last_name":"Gibali"},{"last_name":"Sagratella","first_name":"Simone","full_name":"Sagratella, Simone"}],"volume":184,"date_created":"2019-12-09T21:33:44Z","date_updated":"2023-09-06T11:27:15Z","year":"2020","acknowledgement":"We are grateful to the anonymous referees and editor whose insightful comments helped to considerably improve an earlier version of this paper. The research of the first author is supported by an ERC Grant from the Institute of Science and Technology (IST).","department":[{"_id":"VlKo"}],"publisher":"Springer Nature","publication_status":"published","publication_identifier":{"eissn":["1573-2878"],"issn":["0022-3239"]},"month":"03","doi":"10.1007/s10957-019-01616-6","language":[{"iso":"eng"}],"oa":1,"external_id":{"isi":["000511805200009"]},"project":[{"call_identifier":"FP7","name":"Discrete Optimization in Computer Vision: Theory and Practice","_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160"}],"quality_controlled":"1","isi":1}]