Novel algorithms based on forward-backward splitting technique: effective methods for regression and classification
In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The pract...
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| Published in | Journal of global optimization Vol. 90; no. 4; pp. 869 - 890 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
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Springer US
01.12.2024
Springer Nature B.V |
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| Abstract | In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios. |
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| AbstractList | In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence analysis, emphasizing the new algorithms and contrasting them with existing ones. Our findings are validated through a numerical example. The practical utility of these algorithms in real-world applications, including machine learning for tasks such as classification, regression, and image deblurring reveal that these algorithms consistently approach optimal solutions with fewer iterations, highlighting their efficiency in real-world scenarios. |
| Author | Atalan, Yunus Milovanović, Gradimir V. Hacıoğlu, Emirhan Gürsoy, Faik Ertürk, Müzeyyen |
| Author_xml | – sequence: 1 givenname: Yunus surname: Atalan fullname: Atalan, Yunus organization: Department of Mathematics, Aksaray University – sequence: 2 givenname: Emirhan orcidid: 0000-0003-0195-1935 surname: Hacıoğlu fullname: Hacıoğlu, Emirhan email: emirhanhacioglu@hotmail.com organization: Department of Mathematics, Trakya University – sequence: 3 givenname: Müzeyyen surname: Ertürk fullname: Ertürk, Müzeyyen organization: Department of Mathematics, Adiyaman University – sequence: 4 givenname: Faik surname: Gürsoy fullname: Gürsoy, Faik organization: Department of Mathematics, Adiyaman University – sequence: 5 givenname: Gradimir V. surname: Milovanović fullname: Milovanović, Gradimir V. organization: Serbian Academy of Sciences and Arts, Faculty of Science and Mathematics, University of Niš |
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| Cites_doi | 10.1007/s10589-017-9909-6 10.1007/BF01109805 10.1016/S0377-0427(00)00360-5 10.37193/CMI.2016.01.04 10.1137/050626090 10.24193/fpt-ro.2017.2.50 10.1007/s40065-018-0236-2 10.1137/16M1073741 10.1137/0314056 10.1007/s10957-015-0746-4 10.1006/jmaa.2000.7042 10.22436/jnsa.009.05.72 10.1016/0022-247X(76)90152-9 10.1137/080716542 10.1080/02331934.2017.1411485 10.1007/s11075-019-00706-w 10.1090/S0002-9939-1953-0054846-3 10.1016/S0252-9602(12)60127-1 10.1090/S0002-9939-1991-1086345-8 10.1002/zamm.19670470202 10.1007/978-3-319-48311-5 10.1007/s11075-015-0030-6 10.1080/02331934.2019.1702040 10.2298/FIL1610829G 10.1007/978-0-387-70914-7 10.1109/ICIP.2008.4711847 10.3390/sym10110563 10.1016/j.acha.2012.08.004 10.1007/978-1-4613-0299-5_1 10.1007/s10796-006-8779-8 10.1090/S0002-9939-1974-0336469-5 10.1080/02331934.2012.733883 10.1137/0716071 10.1186/s13660-019-2097-4 |
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| Keywords | cocoercive mappings Iterative algorithm , Nonexpansive mappings Relaxed Variational inequalities |
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| References | WengXFixed point iteration for local strictly pseudocontractive mappingProc. Am. Math. Soc.1991113372773110.1090/S0002-9939-1991-1086345-8 GürsoyFErtürkMAbbasMA Picard-type iterative algorithm for general variational inequalities and nonexpansive mappingsNumer. Algorithms2020833867883406435610.1007/s11075-019-00706-w ZongCTangYChoYJConvergence analysis of an inexact three-operator splitting algorithmSymmetry2018101156310.3390/sym10110563 BrowderFEConvergence theorems for sequences of nonlinear operators in Banach spacesMath. Z.1967100320122521514110.1007/BF01109805 Osei-BrysonKMGilesKSplitting methods for decision tree induction: An exploration of the relative performance of two entropy-based familiesInf. Syst. Front.20068319520910.1007/s10796-006-8779-8 Milovanović, G.V.: Numerical Analysis, Part I, Naučna Knjiga, Beograd, 1991 (Serbian) CombettesPLVuBCVariable metric forward-backward splitting with applications to monotone inclusions in dualityOptimization201463912891318322584510.1080/02331934.2012.733883 MannWRMean value methods in iterationProc. Am. Math. Soc.195345065105484610.1090/S0002-9939-1953-0054846-3 NoorMANew approximation schemes for general variational inequalitiesJ. Math. Anal. Appl.2000251217229179040610.1006/jmaa.2000.7042 SalzoSThe variable metric forward-backward splitting algorithm under mild differentiability assumptionsSIAM J. Optim.201727421532181370789910.1137/16M1073741 Bioucas-Dias, J.M., Figueiredo, M.A.: An iterative algorithm for linear inverse problems with compound regularizers. Proceedings – International Conference on Image Processing, ICIP. (pp. 685–688) (2008). https://doi.org/10.1109/ICIP.2008.4711847 DunnJCConvexity, monotonicity, and gradient processes in Hilbert spaceJ. Math. Anal. Appl.19765314515838817610.1016/0022-247X(76)90152-9 PholasaNCholamjiakPChoYJModified forward-backward splitting methods for accretive operators in Banach spacesJ. Nonlinear Sci. Appl.20169527662778349114410.22436/jnsa.009.05.72 SahuDRApplications of the S-iteration process to constrained minimization problems and split feasibility problemsFixed Point Theory20111211872042797080 VoroninSWoerdemanHJA new iterative firm-thresholding algorithm for inverse problems with sparsity constraintsAppl. Comput. Harmon. Anal.2013351151164305375110.1016/j.acha.2012.08.004 BauschkeHHCombettesPLConvex Analysis and Monotone Operator Theory in Hilbert Spaces20172BerlinSpringer10.1007/978-3-319-48311-5 LionsPLMercierBSplitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. Anal.19791696497955131910.1137/0716071 BerindeVPicard iteration converges faster than Mann iteration for a class of quasi-contractive operatorsFixed Point Theory Appl.20042971052086709 Chambolle, A., Dossal, C.: On the convergence of the iterates of the “fast iterative shrinkage/thresholding algorithm.” J. Optim. Theory Appl. 166(3), 968–982 (2015) CholamjiakPA generalized forward–backward splitting method for solving quasi inclusion problems in Banach spacesNumer. Algorithms2016714915932347974710.1007/s11075-015-0030-6 DadashiVPostolacheMForward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operatorsArab. J. Math.2020918999406289510.1007/s40065-018-0236-2 SahuDRYaoJCVermaMShuklaKKConvergence rate analysis of proximal gradient methods with applications to composite minimization problemsOptimization202170175100419578210.1080/02331934.2019.1702040 ChoSYKangSMApproximation of common solutions of variational inequalities via strict pseudocontractionsActa Math. Sci.20123216071618292744810.1016/S0252-9602(12)60127-1 Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer, New York (2011) CombettesPLWajsRSignal recovery by proximal forward-backward splittingMultiscale Model. Simul.2005411681200220384910.1137/050626090 LatafatPPatrinosPAsymmetric forward-backward-adjoint splitting for solving monotone inclusions involving three operatorsComput. Optim. Appl.20176815793368290810.1007/s10589-017-9909-6 AgarwalRPO’ReganDSahuDRIterative construction of fixed points of nearly asymptotically nonexpansive mappingsJ. Nonlinear Convex Anal.20078161792314666 Cui, F., Tang, Y., Zhu, C.: Convergence analysis of a variable metric forward-backward splitting algorithm with applications. J. Inequal. Appl. 2019(1), 141, 27 pp. (2019) KarakayaVAtalanYDoganKBouzaraNEHSome fixed point results for a new three steps iteration process in Banach spacesFixed Point Theory2017182625640370116510.24193/fpt-ro.2017.2.50 BrezinskiCConvergence acceleration during the 20th centuryJ. Comput. Appl. Math.20001221–2121179464910.1016/S0377-0427(00)00360-5 AtalanYOn a new fixed point iterative algorithm for general variational inequalitiesJ. Nonlinear Convex Anal.20192011237123864055652 GürsoyFSahuDRAnsariQHS-iteration process for variational inclusions and its rate of convergenceJ. Nonlinear Convex Anal.201617175317673562626 BeckATeboulleMA fast iterative shrinkage-thresholding algorithm for linear inverse problemsSIAM J. Imaging Sci.200921183202248652710.1137/080716542 WangYWangFStrong convergence of the forward-backward splitting method with multiple parameters in Hilbert spacesOptimization2018674493505376000310.1080/02331934.2017.1411485 PicardEMémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successivesJ. de Math. Pur. et App.189046145210 BerindeVOn a notion of rapidity of convergence used in the study of fixed point iterative methodsCreat. Math. Inform.2016252940355867110.37193/CMI.2016.01.04 IshikawaSFixed points by a new iteration methodProc. Am. Math. Soc.19744414715033646910.1090/S0002-9939-1974-0336469-5 Ansari, Q.H.: Vector equilibrium problems and vector variational inequalities. In Vector Variational Inequalities and Vector Equilibria, Springer, Boston, MA (2000) RockafellarRTMonotone operators and the proximal point algorithmSIAM J. Control. Optim.19761487789841048310.1137/0314056 GürsoyFA Picard-S iterative method for approximating fixed point of weak-contraction mappingsFilomat2016301028292845358340810.2298/FIL1610829G OstrowskiAThe round-off stability of iterationsZAMM-J. Appl. Math. Mech.1967472778121673110.1002/zamm.19670470202 KinderlehrerDStampacchiaGAn Introduction to Variational Inequalities and Their Applications1980New YorkAcademic Press F Gürsoy (1425_CR28) 2016; 30 PL Lions (1425_CR8) 1979; 16 V Berinde (1425_CR35) 2016; 25 RT Rockafellar (1425_CR9) 1976; 14 1425_CR3 N Pholasa (1425_CR14) 2016; 9 RP Agarwal (1425_CR27) 2007; 8 1425_CR36 1425_CR38 1425_CR15 MA Noor (1425_CR26) 2000; 251 C Zong (1425_CR18) 2018; 10 C Brezinski (1425_CR33) 2000; 122 1425_CR39 Y Atalan (1425_CR41) 2019; 20 F Gürsoy (1425_CR42) 2020; 83 D Kinderlehrer (1425_CR1) 1980 SY Cho (1425_CR2) 2012; 32 JC Dunn (1425_CR10) 1976; 53 V Dadashi (1425_CR17) 2020; 9 HH Bauschke (1425_CR30) 2017 Y Wang (1425_CR19) 2018; 67 P Latafat (1425_CR13) 2017; 68 E Picard (1425_CR23) 1890; 4 V Karakaya (1425_CR29) 2017; 18 FE Browder (1425_CR21) 1967; 100 DR Sahu (1425_CR6) 2021; 70 WR Mann (1425_CR24) 1953; 4 S Salzo (1425_CR12) 2017; 27 DR Sahu (1425_CR22) 2011; 12 A Beck (1425_CR37) 2009; 2 PL Combettes (1425_CR11) 2014; 63 V Berinde (1425_CR34) 2004; 2 A Ostrowski (1425_CR32) 1967; 47 F Gürsoy (1425_CR4) 2016; 17 X Weng (1425_CR31) 1991; 113 PL Combettes (1425_CR5) 2005; 4 KM Osei-Bryson (1425_CR7) 2006; 8 P Cholamjiak (1425_CR16) 2016; 71 1425_CR20 S Ishikawa (1425_CR25) 1974; 44 S Voronin (1425_CR40) 2013; 35 |
| References_xml | – reference: KinderlehrerDStampacchiaGAn Introduction to Variational Inequalities and Their Applications1980New YorkAcademic Press – reference: LionsPLMercierBSplitting algorithms for the sum of two nonlinear operatorsSIAM J. Numer. Anal.19791696497955131910.1137/0716071 – reference: CholamjiakPA generalized forward–backward splitting method for solving quasi inclusion problems in Banach spacesNumer. Algorithms2016714915932347974710.1007/s11075-015-0030-6 – reference: CombettesPLWajsRSignal recovery by proximal forward-backward splittingMultiscale Model. Simul.2005411681200220384910.1137/050626090 – reference: PholasaNCholamjiakPChoYJModified forward-backward splitting methods for accretive operators in Banach spacesJ. Nonlinear Sci. Appl.20169527662778349114410.22436/jnsa.009.05.72 – reference: OstrowskiAThe round-off stability of iterationsZAMM-J. Appl. Math. Mech.1967472778121673110.1002/zamm.19670470202 – reference: LatafatPPatrinosPAsymmetric forward-backward-adjoint splitting for solving monotone inclusions involving three operatorsComput. Optim. Appl.20176815793368290810.1007/s10589-017-9909-6 – reference: MannWRMean value methods in iterationProc. Am. Math. Soc.195345065105484610.1090/S0002-9939-1953-0054846-3 – reference: IshikawaSFixed points by a new iteration methodProc. Am. Math. Soc.19744414715033646910.1090/S0002-9939-1974-0336469-5 – reference: Chambolle, A., Dossal, C.: On the convergence of the iterates of the “fast iterative shrinkage/thresholding algorithm.” J. Optim. Theory Appl. 166(3), 968–982 (2015) – reference: Ansari, Q.H.: Vector equilibrium problems and vector variational inequalities. In Vector Variational Inequalities and Vector Equilibria, Springer, Boston, MA (2000) – reference: GürsoyFErtürkMAbbasMA Picard-type iterative algorithm for general variational inequalities and nonexpansive mappingsNumer. Algorithms2020833867883406435610.1007/s11075-019-00706-w – reference: BrowderFEConvergence theorems for sequences of nonlinear operators in Banach spacesMath. Z.1967100320122521514110.1007/BF01109805 – reference: BeckATeboulleMA fast iterative shrinkage-thresholding algorithm for linear inverse problemsSIAM J. Imaging Sci.200921183202248652710.1137/080716542 – reference: WangYWangFStrong convergence of the forward-backward splitting method with multiple parameters in Hilbert spacesOptimization2018674493505376000310.1080/02331934.2017.1411485 – reference: NoorMANew approximation schemes for general variational inequalitiesJ. Math. Anal. Appl.2000251217229179040610.1006/jmaa.2000.7042 – reference: CombettesPLVuBCVariable metric forward-backward splitting with applications to monotone inclusions in dualityOptimization201463912891318322584510.1080/02331934.2012.733883 – reference: SalzoSThe variable metric forward-backward splitting algorithm under mild differentiability assumptionsSIAM J. Optim.201727421532181370789910.1137/16M1073741 – reference: RockafellarRTMonotone operators and the proximal point algorithmSIAM J. Control. Optim.19761487789841048310.1137/0314056 – reference: SahuDRApplications of the S-iteration process to constrained minimization problems and split feasibility problemsFixed Point Theory20111211872042797080 – reference: ZongCTangYChoYJConvergence analysis of an inexact three-operator splitting algorithmSymmetry2018101156310.3390/sym10110563 – reference: Milovanović, G.V.: Numerical Analysis, Part I, Naučna Knjiga, Beograd, 1991 (Serbian) – reference: Bioucas-Dias, J.M., Figueiredo, M.A.: An iterative algorithm for linear inverse problems with compound regularizers. Proceedings – International Conference on Image Processing, ICIP. (pp. 685–688) (2008). https://doi.org/10.1109/ICIP.2008.4711847 – reference: BauschkeHHCombettesPLConvex Analysis and Monotone Operator Theory in Hilbert Spaces20172BerlinSpringer10.1007/978-3-319-48311-5 – reference: DadashiVPostolacheMForward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operatorsArab. J. Math.2020918999406289510.1007/s40065-018-0236-2 – reference: KarakayaVAtalanYDoganKBouzaraNEHSome fixed point results for a new three steps iteration process in Banach spacesFixed Point Theory2017182625640370116510.24193/fpt-ro.2017.2.50 – reference: Cui, F., Tang, Y., Zhu, C.: Convergence analysis of a variable metric forward-backward splitting algorithm with applications. J. Inequal. Appl. 2019(1), 141, 27 pp. (2019) – reference: AtalanYOn a new fixed point iterative algorithm for general variational inequalitiesJ. Nonlinear Convex Anal.20192011237123864055652 – reference: DunnJCConvexity, monotonicity, and gradient processes in Hilbert spaceJ. Math. Anal. Appl.19765314515838817610.1016/0022-247X(76)90152-9 – reference: Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. 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| Snippet | In this paper, we introduce two novel forward-backward splitting algorithms (FBSAs) for nonsmooth convex minimization. We provide a thorough convergence... |
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| SubjectTerms | Algorithms Classification Computer Science Hilbert space Machine learning Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Real Functions Splitting |
| Title | Novel algorithms based on forward-backward splitting technique: effective methods for regression and classification |
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