An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems
In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm d...
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| Published in: | Journal of inequalities and applications Vol. 2021; no. 1; pp. 1 - 19 |
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| Language: | English |
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| Abstract | In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238,
2016
] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362,
2019
] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature. |
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| AbstractList | In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature. In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature. Abstract In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature. In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016 ] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019 ] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature. |
| ArticleNumber | 42 |
| Author | Jailoka, Pachara Suantai, Suthep Hanjing, Adisak |
| Author_xml | – sequence: 1 givenname: Suthep surname: Suantai fullname: Suantai, Suthep organization: Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University – sequence: 2 givenname: Pachara surname: Jailoka fullname: Jailoka, Pachara organization: Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University – sequence: 3 givenname: Adisak surname: Hanjing fullname: Hanjing, Adisak email: adisak_h@cmu.ac.th organization: Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University |
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| Keywords | Strong convergence 65K05 90C30 Inertial techniques 90C25 Forward-backward splitting Convex minimization problems Linesearch Viscosity approximation |
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B, Methodol.19965826728813792420850.62538 Aremu, K.O., Izuchukwu, C., Grace, O.N., Mewomo, O.T.: Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. J. Ind. Manag. Optim. 13(5) (2020). https://doi.org/10.3934/jimo.2020063 NesterovY.A method for solving the convex programming problem with convergence rate O(1/k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(1/k^{2})$\end{document}Dokl. Akad. Nauk SSSR1983269543547701288 OkekeC.C.IzuchukwuC.A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spacesOptim. 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Appl.20002414655173833210.1006/jmaa.1999.6615 ThungK.RaveendranP.A survey of image quality measuresProceedings of the International Conference for Technical Postgraduates (TECHPOS)2009Los AlamitosIEEE Comput. Soc.14 SaejungS.YotkaewP.Approximation of zeros of inverse strongly monotone operators in Banach spacesNonlinear Anal.201275724750284745310.1016/j.na.2011.09.005 WangZ.BovikA.C.SheikhH.R.SimoncelliE.P.Image quality assessment: from error visibility to structural similarityIEEE Trans. Image Process.20041360061210.1109/TIP.2003.819861 LinL.J.TakahashiW.A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applicationsPositivity201216429453297430810.1007/s11117-012-0161-0 TakahashiW.Introduction to Nonlinear and Convex Analysis2009YokohamaYokohama Publishers1183.46001 B.T. Polyak (2571_CR24) 1964; 4 Z. Wang (2571_CR33) 2004; 13 Y. Nesterov (2571_CR22) 1983; 269 S. Suantai (2571_CR28) 2020; 8 J.Y.B. Cruz (2571_CR9) 2016; 31 J.C. 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| References_xml | – reference: Hale, E., Yin, W., Zhang, Y.: A fixed-point continuation method for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{1}$\end{document}-regularized minimization with applications to compressed sensing, Rice University: Department of Computational and Applied Mathematics (2007) – reference: TakahashiW.Introduction to Nonlinear and Convex Analysis2009YokohamaYokohama Publishers1183.46001 – reference: RockafellarR.T.On the maximal monotonicity of subdifferential mappingsPac. J. Math.19703320921626282710.2140/pjm.1970.33.209 – reference: SaejungS.YotkaewP.Approximation of zeros of inverse strongly monotone operators in Banach spacesNonlinear Anal.201275724750284745310.1016/j.na.2011.09.005 – reference: LinL.J.TakahashiW.A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applicationsPositivity201216429453297430810.1007/s11117-012-0161-0 – reference: MartinetB.Régularisation d’inéquations variationnelles par approximations successivesRev. Fr. Inform. Rech. Oper.197041541580215.21103 – reference: OkekeC.C.IzuchukwuC.A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spacesOptim. Methods Softw.201934611681183401440110.1080/10556788.2018.1472259 – reference: WangZ.BovikA.C.SheikhH.R.SimoncelliE.P.Image quality assessment: from error visibility to structural similarityIEEE Trans. Image Process.20041360061210.1109/TIP.2003.819861 – reference: IzuchukwuC.GraceO.N.MewomoO.T.An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusionsOptimization202010.1080/02331934.2020.1808648 – reference: XuH.K.Viscosity approximation methods for nonexpansive mappingsJ. Math. Anal. Appl.2004298279291208654610.1016/j.jmaa.2004.04.059 – reference: DaubechiesI.DefriseM.MolC.D.An iterative thresholding algorithm for linear inverse problems with a sparsity constraintCommun. Pure Appl. Math.20045714131457207770410.1002/cpa.20042 – reference: CruzJ.Y.B.NghiaT.T.A.On the convergence of the forward-backward splitting method with linesearchesOptim. Methods Softw.20163112091238354706310.1080/10556788.2016.1214959 – reference: BauschkeH.H.CombettesP.L.Convex Analysis and Monotone Operator Theory in Hilbert Spaces2011New YorkSpringer10.1007/978-1-4419-9467-7 – reference: CombettesP.L.PesquetJ.C.A Douglas-Rachford splitting approach to nonsmooth convex variational signal recoveryIEEE J. Sel. Top. Signal Process.2007156457410.1109/JSTSP.2007.910264 – reference: KankamK.PholasaN.CholamjiakP.On convergence and complexity of the modified forward-backward method involving new linesearches for convex minimizationMath. Methods Appl. Sci.20194213521362392815610.1002/mma.5420 – reference: ThungK.RaveendranP.A survey of image quality measuresProceedings of the International Conference for Technical Postgraduates (TECHPOS)2009Los AlamitosIEEE Comput. Soc.14 – reference: LionsP.L.MercierB.Splitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. 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| Snippet | In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting... Abstract In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward... |
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| SubjectTerms | Algorithms Analysis Applications of Mathematics Approximation Convex minimization problems Forward-backward splitting Hilbert space Inertial techniques Linesearch Mathematical analysis Mathematics Mathematics and Statistics Optimization Signal reconstruction Splitting Strong convergence Viscosity Viscosity approximation |
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| Title | An accelerated viscosity forward-backward splitting algorithm with the linesearch process for convex minimization problems |
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