×

Inertial Krasnosel’skiǐ-Mann type hybrid algorithms for solving hierarchical fixed point problems. (English) Zbl 1475.47089

Summary: In this paper, we suggest two inertial Krasnosel’skiǐ-Mann type hybrid algorithms to approximate a solution of a hierarchical fixed point problem for nonexpansive mappings in Hilbert space. We prove strong convergence theorems for these algorithms and the conditions of the convergence are very weak comparing other algorithms for the hierarchical fixed point problems. Further, we derive some consequences from the main results. Finally, we present two academic numerical examples for comparing these two algorithms with the algorithm in [Q.-L. Dong et al., J. Fixed Point Theory Appl. 19, No. 4, 3097–3118 (2017; Zbl 1482.47118)], which illustrate the advantage of the proposed algorithms. The methods and results presented in this paper generalize and unify previously known corresponding methods and results of this area.

MSC:

47J26 Fixed-point iterations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1482.47118
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38(4), 1102-1119 (2000) · Zbl 0954.34053 · doi:10.1137/S0363012998335802
[2] Alvarez, F., Attouch, H.: An inertial proximalmethod for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3-11 (2001) · Zbl 0991.65056 · doi:10.1023/A:1011253113155
[3] Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward – backward algorithm for convex minimization. SIAM J. Optim. 24(4), 232-256 (2014) · Zbl 1295.90044 · doi:10.1137/130910294
[4] Beck, A., Teboulle, M.: A fast iterative shrinkage-thersholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183-202 (2009) · Zbl 1175.94009 · doi:10.1137/080716542
[5] Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472-487 (2015) · Zbl 1338.65145
[6] Bot, R.I., Csetnek, E.R.: An inertial forward – backward – forward primal – dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithm 71, 519-540 (2016) · Zbl 1338.47076 · doi:10.1007/s11075-015-0007-5
[7] Brézis, H.: Mathematical Studies 5. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. American Elsevier, North Holland (1973) · Zbl 0252.47055
[8] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[9] Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: application to hierarchical minimization. SIAM J. Optim. 15, 555-572 (2005) · Zbl 1079.90098 · doi:10.1137/S105262340343467X
[10] Chan, R.H., Ma, S., Yang, J.F.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8(4), 2239-2267 (2015) · Zbl 1328.65134 · doi:10.1137/15100463X
[11] Chen, P., Huang, J., Zhang, X.: A primal – dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Probl. 29, 025011, 33 (2013) · Zbl 1279.65075
[12] Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, ThM: Inertial projection and contraction algorithms for variational inequalities. J. Global Optim. 70, 687-704 (2018) · Zbl 1390.90568 · doi:10.1007/s10898-017-0506-0
[13] Dong, Q., Jiang, J., Cholamjiak, P., Shehu, Y.: A strong convergence result involving an inertial forward – backward algorithm for monotone inclusions. J. Fixed Point Theory A 19(4), 3097-3118 (2017) · Zbl 1482.47118 · doi:10.1007/s11784-017-0472-7
[14] Dong, Q.L., Lu, Y.Y.: A new hybrid algorithm for a nonexpansive mapping. Fixed Point Theory Appl. 2015(37), 7 (2015) · Zbl 1317.90322
[15] Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, ThM: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett. 12(1), 87-102 (2018) · Zbl 1462.65058 · doi:10.1007/s11590-016-1102-9
[16] Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293-318 (1992) · Zbl 0765.90073 · doi:10.1007/BF01581204
[17] Goebel, K., Kirk, W.A.: Cambridge Studies in Advanced Mathematics. Topics in metric fixed point theory, 28th edn. Cambridge University Press, Cambridge (1990)
[18] Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001
[19] He, S., Yang, C., Duan, P.: Realization of the hybrid method for Mann iterations. Appl. Math. Comput. 217, 4239-4247 (2010) · Zbl 1207.65065
[20] Iiduka, H.: Iterative algorithm for triple-hierarchical constrained noncnvex optimization problem and its application to network bandwith allocation. SIAM. J. Optim. 22, 862-878 (2012) · Zbl 1267.90139 · doi:10.1137/110849456
[21] Iiduka, H.: Fixed point optimization algorithms for distributed optimization in networkd systems. SIAM. J. Optim. 23, 1-26 (2013) · Zbl 1266.49067 · doi:10.1137/120866877
[22] Kazmi, K.R., Ali, R., Furkan, M.: Krasnoselski-Mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem. Numer. Algorithms 77, 289-308 (2018) · Zbl 1481.65082 · doi:10.1007/s11075-017-0316-y
[23] Kazmi, K.R., Ali, R., Furkan, M.: Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings. Numer. Algorithms 79, 499-527 (2018) · Zbl 06945626 · doi:10.1007/s11075-017-0448-0
[24] Kazmi, K.R., Rizvi, S.H., Ali, R.: A hybrid iterative method without extrapolating step for solving mixed equilibrium problem. Creative Math. Inf. 24(2), 163-170 (2015) · Zbl 1389.49026
[25] Kim, T.H., Xu, H.K.: Strong convergence of modified Mann iterations. Nonlinear Anal. 61, 51-60 (2005) · Zbl 1091.47055 · doi:10.1016/j.na.2004.11.011
[26] Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747-756 (1976) · Zbl 0342.90044
[27] Lorenz, D., Pock, T.: An inertial forward – backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311-325 (2015) · Zbl 1327.47063 · doi:10.1007/s10851-014-0523-2
[28] Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) · Zbl 0898.90006 · doi:10.1017/CBO9780511983658
[29] Maingé, P.E.: Convergence theorem for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223-236 (2008) · Zbl 1156.65054 · doi:10.1016/j.cam.2007.07.021
[30] Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876-887 (2008) · Zbl 1146.47042 · doi:10.1016/j.jmaa.2008.03.028
[31] Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolating step for solving variational inequality problems. J. Global Optim. 61, 193-202 (2015) · Zbl 1366.47018 · doi:10.1007/s10898-014-0150-x
[32] Matinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400-2411 (2006) · Zbl 1105.47060 · doi:10.1016/j.na.2005.08.018
[33] Micchelli, C.A., Shen, L., Xu, Y.: Proximity algorithms for image models: denoising. Inverse Probl. 27(4), 045009 (2011) · Zbl 1216.94015 · doi:10.1088/0266-5611/27/4/045009
[34] Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447-454 (2003) · Zbl 1027.65077 · doi:10.1016/S0377-0427(02)00906-8
[35] Moudafi, A., Maige, P.E.: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory Appl. 2006(95453), 10 (2006) · Zbl 1143.47305
[36] Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635-1640 (2007) · Zbl 1128.47060 · doi:10.1088/0266-5611/23/4/015
[37] Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230-1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[38] Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroup. J. Math. Anal. Appl. 279, 372-379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[39] Popov, L.D.: A modification of the Arrow-Hurwicz method for searching for saddle points. Mat. Zametki 28(5), 777-784 (1980) · Zbl 0456.90068
[40] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274-276 (1979) · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6
[41] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[42] Shehu, Y., Dong, Q.L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68, 385-409 (2019) · Zbl 1431.49009 · doi:10.1080/02331934.2018.1522636
[43] Shehu, Y., Iyiola, O.S.: Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J. Fixed Point Theory Appl. 19, 2483-2510 (2017) · Zbl 1493.47100 · doi:10.1007/s11784-017-0435-z
[44] Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619-655 (2004) · Zbl 1095.47049 · doi:10.1081/NFA-200045815
[45] Yang, Q., Zhao, J.: Generalized KM theorem and their applications. Inverse Probl. 22, 833-844 (2006) · Zbl 1117.65081 · doi:10.1088/0266-5611/22/3/006
[46] Yao, Y., Liou, Y.C.: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 24, 501-508 (2008) · Zbl 1154.47055 · doi:10.1088/0266-5611/24/1/015015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.