References
- Douglas J, Rachford HH. On the numerical solution of heat conduction problems in two and three space variables. Trans Am Math Soc. 1956;82:421–439. doi: https://doi.org/10.1090/S0002-9947-1956-0084194-4
- Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal. 1979;16:964–979. doi: https://doi.org/10.1137/0716071
- Vũ BC. A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv Comput Math. 2013;38:667–681. doi: https://doi.org/10.1007/s10444-011-9254-8
- Liang J. Convergence Rates of First-Order Operator Splitting Methods. Optimization and Control [math.OC]. Normandie Université; GREYC CNRS UMR 6072, 2016. English.
- Davis D, Yin W. A three-operator splitting scheme and its optimization applications. Set-Valued Var Anal. 2017;25:829–858. doi: https://doi.org/10.1007/s11228-017-0421-z
- He B, Yuan X. On the convergence rate of Douglas–Rachford operator splitting method. Math Program Ser A. 2015;153:715–722. doi: https://doi.org/10.1007/s10107-014-0805-x
- Krasnosel'ski MA. Two remarks on the method of successive approximations. Usp Mat Nauk. 1955;10:123–127.
- Mann WR. Mean value methods in iteration. Proc Amer Math Soc. 1953;4:506–510. doi: https://doi.org/10.1090/S0002-9939-1953-0054846-3
- Dong QL, Yuan BH, Cho YJ, et al. Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim Lett. 2018;12:87–102. doi: https://doi.org/10.1007/s11590-016-1102-9
- He S, Yang C, Duan P. Realization of the hybrid method for Mann iterations. Appl Math Comput. 2010;217:4239–4247.
- Iutzeler F, Hendrickx MJ. A generic linear rate acceleration of optimization algorithms via relaxation and inertia. Optim Method Softw. 2019;34:383–405. doi: https://doi.org/10.1080/10556788.2017.1396601
- Kim TH, Xu HK. Strong convergence of modified Mann iterations. Nonlinear Anal. 2005;61:51–60. doi: https://doi.org/10.1016/j.na.2004.11.011
- Kim TH, Xu HK. Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 2006;64:1140–1152. doi: https://doi.org/10.1016/j.na.2005.05.059
- Liang J, Fadili F, Peyré G. Convergence rates with inexact non-expansive operators. Math Program Ser A. 2016;159:403–434. doi: https://doi.org/10.1007/s10107-015-0964-4
- Lions PL. Approximation de points fixes de contractions, C.R. Acad Sci Series A-B Paris. 1977;284:1357–1359.
- Reich S. Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl. 1979;67:274–276. doi: https://doi.org/10.1016/0022-247X(79)90024-6
- Stathopoulos G, Jones CN. An inertial parallel and asynchronous forward–backward iteration for distributed convex optimization. J Optim Theory Appl. 2019;182:1088–1119. doi: https://doi.org/10.1007/s10957-019-01542-7
- Wittmann R. Approximation of fixed points of nonexpansive mappings. Arch Math. 1992;58:486–491. doi: https://doi.org/10.1007/BF01190119
- Xu HK. Iterative algorithms for nonlinear operators. J Lond Math Soc. 2002;66:240–256. doi: https://doi.org/10.1112/S0024610702003332
- Xu HK. The parameter selection problem for Mann's fixed point algorithm. Taiwanese J of Math. 2008;12(8):1911–1920. doi: https://doi.org/10.11650/twjm/1500405126
- Bauschke HH, Combettes PL. Convex analysis and monotone operator theory in Hilbert spaces. 2nd ed, Berlin: Springer; 2017.
- Cominetti R, Soto JA, Vaisman J. On the rate of convergence of Krasnosel'skii–Mann iterations and their connection with sums of Bernoullis. Isr J Math. 2014;199:757–772. doi: https://doi.org/10.1007/s11856-013-0045-4
- Bravo M, Cominetti R. Sharp convergence rates for averaged nonexpansive maps. Israel J Math. 2018;227:163–188. doi: https://doi.org/10.1007/s11856-018-1723-z
- Bravo M, Cominetti R, Pavez-Signé M. Rates of convergence for inexact Krasnosel'skii-Mann iterations in Banach spaces. Math Program Ser. A. 2019;175:241–262. doi: https://doi.org/10.1007/s10107-018-1240-1
- Matsushita SY. On the convergence rates of the Krasnosel'skii–Mann iteration. Bull Aust Math Soc. 2017;96:162–170. doi: https://doi.org/10.1017/S000497271600109X
- Mainge PE. Convergence theorems for inertial KM-type algorithms. J Comput Appl Math. 2008;219:223–236. doi: https://doi.org/10.1016/j.cam.2007.07.021
- Bot RI, Csetnek ER, Hendrich C. Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl Math Comput. 2015;256:472–487.
- Dong QL, Cho YJ, Rassias TM. General inertial Mann algorithms and their convergence analysis for nonexpansive mappings. In: Rassias TM, editor. Applications of nonlinear analysis. Berlin: Springer; 2018. p. 175–191.
- Dong QL, Huang J, Li XH, et al. MiKM: multi-step inertial Krasnosel'ski–Mann algorithm and its applications. J Global Optim. 2019;73(4):801–824. doi: https://doi.org/10.1007/s10898-018-0727-x
- Combettes PL. Inconsistent signal feasibility problems: least-squares solutions in a product space. IEEE T Signal Proces. 1994;42(11):2955–2966. doi: https://doi.org/10.1109/78.330356
- Combettes PL, Yamada I. Compositions and convex combinations of averaged nonexpansive operators. J Math Anal Appl. 2015;425:55–70. doi: https://doi.org/10.1016/j.jmaa.2014.11.044
- Goebel K, Kirk WA. Topics in metric fixed point theory. Vol. 28. Cambridge: Cambridge University Press; 1990.
- Combettes PL. Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization. 2004;53:475–504. doi: https://doi.org/10.1080/02331930412331327157
- Ogura N, Yamada I. Non-strictly convex minimization over the fixed point set of an asymptotically shrinking nonexpansive mapping. Numer Funct Anal Optim. 2002;23:113–137. doi: https://doi.org/10.1081/NFA-120003674
- López Acedo G, Xu HK. Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2007;67:2258–2271. doi: https://doi.org/10.1016/j.na.2006.08.036
- Byrne CL. Iterative optimization in inverse problems. Boca Raton: CRC Press; 2014.
- Combettes PL, Condat L, Pesquet JC. A forward–backward view of some primal–dual optimization methods in image recovery. Proceedings of the IEEE International Conference on Image Processing; IEEE; 2014. p. 4141–4145.
- Demanet L, Zhang X. Eventual linear convergence of the Douglas–Rachford iteration for basis pursuit. Math Comput. 2016;85:209–238. doi: https://doi.org/10.1090/mcom/2965
- Eckstein J. Splitting methods for monotone operators with applications to parallel optimization. Cambridge, MA: Doctoral dissertation, Department of Civil Engineering, Massachusetts Institute of Technology; 1989. (Available as Report LIDS-TH-1877, Laboratory for Information and Decision Sciences, MIT).
- Eckstein J, Bertsekas DP. On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program. 1992;55:293–318. doi: https://doi.org/10.1007/BF01581204
- Aubin JP, Frankowska H. Set-valued analysis. Boston: Birkhäuser; 1990.
- Minty GJ. Monotone (nonlinear) operators in Hilbert space. Duke Math J. 1962;29:341–346. doi: https://doi.org/10.1215/S0012-7094-62-02933-2
- Bauschke HH, Bello Cruz JY, Nghia TTA, et al. Optimal rates of linear convergence of relaxed alternating projections and generalized Douglas–Rachford methods for two subspaces. Numer Algorithms. 2016;73:33–76. doi: https://doi.org/10.1007/s11075-015-0085-4
- Dong QL, Cho YJ, Zhong LL, et al. Inertial projection and contraction algorithms for variational inequalities. J Global Optim. 2018;70:687–704. doi: https://doi.org/10.1007/s10898-017-0506-0
- Xu HK. Averaged mappings and the gradient-projection algorithm. J Optim Theory Appl. 2011;150:360–378. doi: https://doi.org/10.1007/s10957-011-9837-z