Convergence Results of Forward-Backward Algorithms for Sum of Monotone Operators in Banach Spaces
It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two m...
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| Vydáno v: | Resultate der Mathematik Ročník 74; číslo 4 |
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01.12.2019
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| Abstract | It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper. |
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| AbstractList | It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper. |
| ArticleNumber | 138 |
| Author | Shehu, Yekini |
| Author_xml | – sequence: 1 givenname: Yekini orcidid: 0000-0001-9224-7139 surname: Shehu fullname: Shehu, Yekini email: yekini.shehu@unn.edu.ng organization: Institute of Science and Technology (IST), Department of Mathematics, Zhejiang Normal University |
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| Keywords | 65K15 47J20 forward-backward algorithm 90C25 2-uniformly convex Banach space Inclusion problem weak convergence 47H05 47J25 strong convergence |
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| References_xml | – reference: IusemANSvaiterBFSplitting methods for finding zeroes of sums of maximal monotone operators in Banach spacesJ. Nonlinear Convex Anal.201415237939731843291291.47055 – reference: TakahashiSTakahashiWToyodaMStrong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spacesJ. Optim. Theory Appl.20101471274127205901208.4707110.1007/s10957-010-9713-2 – reference: WangYXuH-KStrong convergence for the proximal-gradient methodJ. Nonlinear Convex Anal.201415358159331838241295.46058 – reference: CombettesPWajsVRSignal recovery by proximal forward-backward splittingMultiscale Model. Simul.2005441168120022038491179.9403110.1137/050626090 – reference: GülerOOn the convergence of the proximal point algorithm for convex minimizationSIAM J. 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