Inertial methods for fixed point problems and zero point problems of the sum of two monotone mappings

References

• Combettes PL, Wajs V. Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model Simul. 2005;4:1168–1200. doi: 10.1137/050626090
   Web of Science ®Google Scholar
• Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math. 2004;57:1413–1457. doi: 10.1002/cpa.20042
   Web of Science ®Google Scholar
• Duchi J, Singer Y. Efficient online and batch learning using forward backward splitting. J Mach Learn Res. 2009;10:2899–2934.
   Web of Science ®Google Scholar
• Raguet H, Fadili J, Peyré G. A generalized forward-backward splitting. SIAM J Imaging Sci. 2013;6:1199–1226. doi: 10.1137/120872802
   Web of Science ®Google Scholar
• Bruck R. On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J Math Anal Appl. 1977;61:159–164. doi: 10.1016/0022-247X(77)90152-4
   Web of Science ®Google Scholar
• Dong YD, Fischer A. A family of operator splitting methods revisited. Nonlinear Anal. 2010;72:4307–4315. doi: 10.1016/j.na.2010.02.010
   Web of Science ®Google Scholar
• Gibali A, Thong DV. Tseng type methods for solving inclusion problems and its applications. Calcolo. 2018;55:49. doi:10.1007/s10092-018-0292-1.
   Web of Science ®Google Scholar
• Goldstein AA. Convex programming in Hilbert spaces. Bull Am Math Soc. 1964;70:709–710. doi: 10.1090/S0002-9904-1964-11178-2
   Web of Science ®Google Scholar
• Huang YY, Dong YD. New properties of forward-backward splitting and a practical proximal-descent algorithm. Appl Math Comput. 2014;237:60–68.
   Web of Science ®Google Scholar
• Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal. 1979;16:964–979. doi: 10.1137/0716071
   Web of Science ®Google Scholar
• Levitin ES, Polyak BT. Constrained minimization methods USSR. Comput Math Math Phys. 1966;6:1–50. doi: 10.1016/0041-5553(66)90114-5
   Google Scholar
• Passty GB. Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J Math Anal Appl. 1979;72:383–390. doi: 10.1016/0022-247X(79)90234-8
   Web of Science ®Google Scholar
• Tseng P. A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim. 2000;38:431–446. doi: 10.1137/S0363012998338806
   Web of Science ®Google Scholar
• Moreau JJ. Proximité et dualité dans un espace Hilbertien. Bull Soc Math France. 1965;93:273–299. doi: 10.24033/bsmf.1625
   Google Scholar
• Bruck RE, Reich S. Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J Math. 1977;3:459–470.
   Google Scholar
• Rockafellar RT. Monotone operators and the proximal point algorithm. SIAM J Control Optim. 1976;14:877–898. doi: 10.1137/0314056
   Web of Science ®Google Scholar
• Takahashi W. Proximal point algorithms and four resolvents of nonlinear operators of monotone type in Banach spaces. Taiwan J Math. 2008;12:1883–1910. doi: 10.11650/twjm/1500405125
   Web of Science ®Google Scholar
• Wang F, Cui H. On the contraction-proximal point algorithms with multiparameters. J Global Optim. 2012;54:485–491. doi: 10.1007/s10898-011-9772-4
   Web of Science ®Google Scholar
• Wang CY, Xiu NH. Convergence of gradient projection methods for generalized convex minimization. Comput Optim Appl. 2000;16:111–120. doi: 10.1023/A:1008714607737
   Web of Science ®Google Scholar
• Xu HK. Averaged mappings and the gradient-projection algorithm. J Optim Theory Appl. 2011;150:360–378. doi: 10.1007/s10957-011-9837-z
   Web of Science ®Google Scholar
• Cruz JYB, Nghia TTA. On the convergence of the forward-backward splitting method with linesearches. Optim Method Softw. 2016;31:1209–1238. doi: 10.1080/10556788.2016.1214959
   Web of Science ®Google Scholar
• Dong QL, Jiang D, Cholamjiak P, et al. A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions. J Fixed Point Theory Appl. 2017;19:3097–3118. doi: 10.1007/s11784-017-0472-7
   Web of Science ®Google Scholar
• López G, Márquez MV, Wang F, et al. Forward-backward splitting methods for accretive operators in Banach spaces. Abstr Appl Anal. 2010;2012. Article ID 109236.
   Google Scholar
• Lorenz DA, Pock T. An inertial forward-backward algorithm for monotone inclusions. J Math Imaging Vis. 2015;51:311–325. doi: 10.1007/s10851-014-0523-2
   Web of Science ®Google Scholar
• Moudafi A, Oliny M. Convergence of a splitting inertial proximal method for monotone operators. J Comput Appl Math. 2003;155:447–454. doi: 10.1016/S0377-0427(02)00906-8
   Web of Science ®Google Scholar
• Attouch H, Goudon X, Redont P. The heavy ball with friction. I. The continuous dynamical system. Commun Contemp Math. 2000;2:1–34.
   Web of Science ®Google Scholar
• Attouch H, Czarnecki MO. Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J Diff Eq. 2002;179:278–310. doi: 10.1006/jdeq.2001.4034
   Web of Science ®Google Scholar
• Alvarez F, Attouch H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 2001;9:3–11. doi: 10.1023/A:1011253113155
   Google Scholar
• Attouch H, Peypouquet J. The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1k2. SIAM J Optim. 2016;26:1824–1834. doi: 10.1137/15M1046095
   Web of Science ®Google Scholar
• Maingé PE. Regularized and inertial algorithms for common fixed points of nonlinear operators. J Math Anal Appl. 2008;34:876–887. doi: 10.1016/j.jmaa.2008.03.028
   Web of Science ®Google Scholar
• Polyak BT. Some methods of speeding up the convergence of iterarive methods. Zh Vychisl Mat Mat Fiz. 1964;4:1–17.
   Google Scholar
• Alvarez F. Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J Optim. 2004;14:773–782. doi: 10.1137/S1052623403427859
   Web of Science ®Google Scholar
• Bot RI, Csetnek ER. An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer Algorithms. 2016;71:519–540. doi: 10.1007/s11075-015-0007-5
   Web of Science ®Google Scholar
• Bot RI, Csetnek ER, Hendrich C. Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl Math Comput. 2015;256:472–487.
   Web of Science ®Google Scholar
• Bot RI, Csetnek ER. An inertial alternating direction method of multipliers. Minimax Theory Appl. 2016;1:29–49.
   Google Scholar
• Bot RI, Csetnek ER. A hybrid proximal-extragradient algorithm with inertial effects. Numer Funct Anal Optim. 2015;36:951–963. doi: 10.1080/01630563.2015.1042113
   Web of Science ®Google Scholar
• Chen C, Ma S, Yang J. A general inertial proximal point algorithm for mixed variational inequality problem. SIAM J Optim. 2015;25:2120–2142. doi: 10.1137/140980910
   Web of Science ®Google Scholar
• Maingé PE. Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set Valued Anal. 2007;15:67–79. doi: 10.1007/s11228-006-0027-3
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Modified subgradient extragradient method for inequality variational problems. Numer Alg. 2018;79:597–610. doi: 10.1007/s11075-017-0452-4
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization. 2018;67:83–102. doi: 10.1080/02331934.2017.1377199
   Web of Science ®Google Scholar
• Thong DV, Hieu DV. An inertial method for solving split common fixed point problems. J Fixed Point Theory Appl. 2017;19:3029–3051. doi: 10.1007/s11784-017-0464-7
   Web of Science ®Google Scholar
• Liang JW. Convergence rates of first-order operator splitting methods. optimization and control [math.OC]. Normandie Universite; GREYC CNRS UMR 6072, 2016. English.
   Google Scholar
• Diaz JB, Metcalf FT. Subsequential limit points of successive approximations. Trans Amer Math Soc. 1969;135:459–485.
   Web of Science ®Google Scholar
• Chidume CE. Geometric properties of Banach spaces and nonlinear iterations. Springer Verlag Series: Lecture Notes in Mathematics vol. 1965, XVII, p. 326 (2009). ISBN 978-1-84882-189-7.
   Google Scholar
• Dotson WG. Fixed points of quasi-nonexpansive mappings. J Aust Math Soc. 1972;13:167–170. doi: 10.1017/S144678870001123X
   Google Scholar
• Dotson WG. An iterative process for nonlinear monotonic nonexpansive operators in Hilbert spaces. Math Comput. 1978;32:223–225. doi: 10.1090/S0025-5718-1978-0470779-8
   Web of Science ®Google Scholar
• Thong DV. Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J Fixed Point Theory Appl. 2017;19:1481–1499. doi: 10.1007/s11784-016-0323-y
   Web of Science ®Google Scholar
• Falset JG, Liorens FE, Marino G, et al. On strong convegence of Halpern's method for quasi-nonexpansive mappings in Hilbert spaces. Math Modell Anal. 2016;21:63–82. doi: 10.3846/13926292.2016.1132787
   Web of Science ®Google Scholar
• Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. New York: Springer; 2011.
   Google Scholar
• Cegielski A. Iterative methods for fixed point problems in Hilbert spaces. Berlin: Springer; 2012.
   Google Scholar
• Mann WR. Mean value metho ds in iteration. Proc Amer Math Soc. 1953;4:506–510. doi: 10.1090/S0002-9939-1953-0054846-3
   Web of Science ®Google Scholar
• Moudafi A. Viscosity approximation methods for fixed-points problems. J Math Anal Appl. 2000;241:46–55. doi: 10.1006/jmaa.1999.6615
   Web of Science ®Google Scholar
• Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker; 1984.
   Google Scholar
• Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 2003;118:417–428. doi: 10.1023/A:1025407607560
   Web of Science ®Google Scholar
• Takahashi W. Nonlinear functional analysis-fixed point theory and its applications. Yokohama: Yokohama Publishers; 2000.
   Google Scholar
• Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc. 1967;73:591–597. doi: 10.1090/S0002-9904-1967-11761-0
   Web of Science ®Google Scholar
• Xu HK. Iterative algorithms for nonlinear operators. J Lond Math Soc. 2002;66:240–256. doi: 10.1112/S0024610702003332
   Web of Science ®Google Scholar
• Maingé PE. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim. 2008;47:1499–1515. doi: 10.1137/060675319
   Web of Science ®Google Scholar
• Suantai S, Pholasa N, Cholamjiak P. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J Ind Manag Optim. 2018;14:1595–1615.
   Web of Science ®Google Scholar
• Cholamjiak W, Cholamjiak P, Suantai S. An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J Fixed Point Theory Appl. 2018;20:42. doi:10.1007/s11784-018-0526-5.
   Web of Science ®Google Scholar
• Rockafellar RT. On the maximal monotonicity of subdifferential mappings. Pac J Math. 1970;33:209–216. doi: 10.2140/pjm.1970.33.209
   Web of Science ®Google Scholar
• Censor Y, Elfving T. A multiprojection algorithm using Bregman projection in a product space. Numerical Algorithms. 1994;8:221–239. doi: 10.1007/BF02142692
   Google Scholar