Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
maximal operator; Bregman distance; reflexive Banach space; weak convergence; strong convergence
Summary:
The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature.
References:
[1] Agarwal, R. P., O'Regan, D., Sahu, D. R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Topological Fixed Point Theory and Its Applications 6. Springer, New York (2009). DOI 10.1007/978-0-387-75818-3 | MR 2508013 | Zbl 1176.47037
[2] Alber, Y. I.: Generalized projection operators in Banach spaces: Properties and applications. Funct. Differ. Equ. 1 (1993), 1-21. MR 1297965 | Zbl 0882.47046
[3] Alber, Y. I.: Metric and generalized projection operators in Banach spaces: Properties and applications. Theory and Applications of Nonlinear Operator of Accretive and Monotone Type Lecture Notes in Pure and Applied Mathematics 178. Marcel Dekker, New York (1996), 15-50. MR 1386667 | Zbl 0883.47083
[4] Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. Springer Monographs in Mathematics. Springer, Dordrecht (2012). DOI 10.1007/978-94-007-2247-7 | MR 3025420 | Zbl 1244.49001
[5] Bauschke, H. H., Borwein, J. M., Combettes, P. L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3 (2001), 615-647. DOI 10.1142/S0219199701000524 | MR 1869107 | Zbl 1032.49025
[6] Bauschke, H. H., Wang, X., Yao, L.: General resolvents for monotone operators: Characterization and extension. Biomedical Mathematics: Promising Directions in Imagine, Therapy Planning and Inverse Problem (Huangguoshu 2008) Medical Physics Publishing, Madison (2010), 57-74 Y. Censor, M. Jiang, G. Wang.
[7] Beck, A.: First-Order Methods in Optimization. MOS-SIAM Series on Optimization 25. SIAM, Philadelphia (2017). DOI 10.1137/1.9781611974997 | MR 3719240 | Zbl 1384.65033
[8] Bertsekas, D. P., Tsitsiklis, J. N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997). MR 3587745 | Zbl 0743.65107
[9] Bonnans, J. F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000). DOI 10.1007/978-1-4612-1394-9 | MR 1756264 | Zbl 0966.49001
[10] Borwein, J. M., Reich, S., Sabach, S.: A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept. J. Nonlinear Convex Anal. 12 (2011), 161-184. MR 2816416 | Zbl 1221.26019
[11] Bregman, L. M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. U.S.S.R. Comput. Math. Math. Phys. 7 (1967), 200-217 translation from Zh. Vychisl. Mat. Mat. Fiz. 7 1967 620-631. DOI 10.1016/0041-5553(67)90040-7 | MR 0215617 | Zbl 0186.23807
[12] Butnariu, D., Resmerita, E.: Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006 (2006), Article ID 84919, 39 pages. DOI 10.1155/AAA/2006/84919 | MR 2211675 | Zbl 1130.47046
[13] Chang, S.-S., Wen, C.-F., Yao, J.-C.: Generalized viscosity implicit rules for solving quasi-inclusion problems of accretive operators in Banach spaces. Optimization 66 (2017), 1105-1117. DOI 10.1080/02331934.2017.1325888 | MR 3656611 | Zbl 06774306
[14] Chang, S.-S., Wen, C.-F., Yao, J.-C.: Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization 67 (2018), 183-1196. DOI 10.1080/02331934.2018.1470176 | MR 3820582 | Zbl 1402.90119
[15] Chang, S.-S., Wen, C.-F., Yao, J.-C.: A generalized forward-backward splitting method for solving a system of quasi variational inclusions in Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113 (2019), 729-747. DOI 10.1007/s13398-018-0511-2 | MR 3942363 | Zbl 07086844
[16] Chang, S.-S., Wen, C.-F., Yao, J.-C.: Zero point problem of accretive operators in Banach spaces. Bull. Malays. Math. Sci. Soc. (2) 42 (2019), 105-118. DOI 10.1007/s40840-017-0470-3 | MR 3894618 | Zbl 07009736
[17] Chen, G. H.-G., Rockafellar, R. T.: Convergence rates in forward-backward splitting. SIAM J. Optim. 7 (1997), 421-444. DOI 10.1137/S1052623495290179 | MR 1443627 | Zbl 0876.49009
[18] Cholamjiak, P., Pholasa, N., Suantai, S., Sunthrayuth, P.: The generalized viscosity explicit rules for solving variational inclusion problems in Banach spaces. (to appear) in Optimization. DOI 10.1080/02331934.2020.1789131 | MR 4343296
[19] Combettes, P. L., Wajs, V. R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4 (2005), 1168-1200. DOI 10.1137/050626090 | MR 2203849 | Zbl 1179.94031
[20] Dunn, J. C.: Convexity, monotonicity, and gradient processes in Hilbert space. J. Math. Anal. Appl. 53 (1976), 145-158. DOI 10.1016/0022-247X(76)90152-9 | MR 0388176 | Zbl 0321.49025
[21] Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29 (1991), 403-419. DOI 10.1137/0329022 | MR 1092735 | Zbl 0737.90047
[22] He, S., Yang, C.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. 2013 (2013), Article ID 942315, 8 pages. DOI 10.1155/2013/942315 | MR 3055865 | Zbl 1273.47099
[23] Iyer, R., Bilmes, J. A.: Submodular-Bregman and the Lovász-Bregman divergences with applications. Advances in Neural Information Processing Systems 25 (NIPS 2012) MIT Press, Cambridge (2012), 9 pages.
[24] Kimura, Y., Nakajo, K.: Strong convergence for a modified forward-backward splitting method in Banach spaces. J. Nonlinear Var. Anal. 3 (2019), 5-18. DOI 10.23952/jnva.3.2019.1.02 | MR 4055639 | Zbl 07133169
[25] Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (1979), 964-979. DOI 10.1137/0716071 | MR 0551319 | Zbl 0426.65050
[26] López, G., Martín-Márquez, V., Wang, F., Xu, H.-K.: Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 109236, pages 25 pages. DOI 10.1155/2012/109236 | MR 2955015 | Zbl 1252.47043
[27] Maingé, P. E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325 (2007), 469-479. DOI 10.1016/j.jmaa.2005.12.066 | MR 2273538 | Zbl 1111.47058
[28] Martín-Márquez, V., Reich, S., Sabach, S.: Iterative Methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst., Ser. S 6 (2013), 1043-1063. DOI 10.3934/dcdss.2013.6.1043 | MR 3009055 | Zbl 1266.26023
[29] Naraghirad, E.: Halpern's iteration for Bregman relatively nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 34 (2013), 1129-1155. DOI 10.1080/01630563.2013.767269 | MR 3175612 | Zbl 1301.47088
[30] Naraghirad, E., Yao, J.-C.: Bregman weak relatively nonexpansive mappings in Banach space. Fixed Point Theory Appl. 2013 (2013), Article ID 141, 43 pages. DOI 10.1186/1687-1812-2013-141 | MR 3072832 | Zbl 1423.47046
[31] Nielsen, F., Boltz, S.: The Burbea-Rao and Bhattacharyya centroids. IEEE Trans. Inf. Theory 57 (2011), 5455-5466. DOI 10.1109/TIT.2011.2159046 | MR 2849367 | Zbl 1365.94159
[32] Nielsen, F., Nock, R.: Generalizing skew Jensen divergences and Bregman divergences with comparative convexity. IEEE Signal Process. Lett. 24 (2017), 1123-1127. DOI 10.1109/LSP.2017.2712195
[33] Ogbuisi, F. U., Izuchukwu, C.: Approximating a zero of sum of two monotone operators which solves a fixed point problem in reflexive Banach spaces. Numer. Funct. Anal. Optim. 41 (2020), 322-343. DOI 10.1080/01630563.2019.1628050 | MR 4041318 | Zbl 07150136
[34] Pathak, H. K.: An Introduction to Nonlinear Analysis and Fixed Point Theory. Springer, Singapore (2018). DOI 10.1007/978-981-10-8866-7 | MR 3728339 | Zbl 1447.47002
[35] Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type Lecture Notes in Pure and Applied Mathematics 178. Marcel Dekker, New York (1996), 313-318. MR 1386686 | Zbl 0943.47040
[36] Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31 (2010), 22-44. DOI 10.1080/01630560903499852 | MR 2677243 | Zbl 1200.47085
[37] Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal., Theory Methods Appl., Ser. A 73 (2010), 122-135. DOI 10.1016/j.na.2010.03.005 | MR 2645837 | Zbl 1226.47089
[38] Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. Fixed-Point Algorithms for Inverse Problems in Science and Engineering Springer Optimization and Its Applications 49. Springer, New York (2011), 301-316. DOI 10.1007/978-1-4419-9569-8_15 | MR 2858843 | Zbl 1245.47015
[39] Rockafellar, R. T.: Convex Analysis. Princeton University Press, Princeton (1970). DOI 10.1515/9781400873173 | MR 0274683 | Zbl 0193.18401
[40] Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33 (1970), 209-216. DOI 10.2140/pjm.1970.33.209 | MR 0262827 | Zbl 0199.47101
[41] Rockafellar, R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14 (1976), 877-898. DOI 10.1137/0314056 | MR 0410483 | Zbl 0358.90053
[42] Sabach, S.: Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces. SIAM J. Optim. 21 (2011), 1289-1308. DOI 10.1137/100799873 | MR 2854584 | Zbl 1237.47073
[43] Shehu, Y.: Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces. Result. Math. 74 (2019), Article ID 138, 24 pages. DOI 10.1007/s00025-019-1061-4 | MR 3978049 | Zbl 07099993
[44] Shehu, Y., Cai, G.: Strong convergence result of forward-backward splitting methods for accretive operators in Banach spaces with applications. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 112 (2018), 71-87. DOI 10.1007/s13398-016-0366-3 | MR 3742991 | Zbl 06836237
[45] Suantai, S., Cholamjiak, P., Sunthrayuth, P.: Iterative methods with perturbations for the sum of two accretive operators in $q$-uniformly smooth Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113 (2019), 203-223. DOI 10.1007/s13398-017-0465-9 | MR 3902940 | Zbl 1450.47026
[46] Sunthrayuth, P., Cholamjiak, P.: Iterative methods for solving quasi-variational inclusion and fixed point problem in $q$-uniformly smooth Banach spaces. Numer. Algorithms 78 (2018), 1019-1044. DOI 10.1007/s11075-017-0411-0 | MR 3827320 | Zbl 06916361
[47] Takahashi, W., Wong, N.-C., Yao, J.-C.: Two generalized strong convergence theorems of Halpern's type in Hilbert spaces and applications. Taiwanese J. Math. 16 (2012), 1151-1172. DOI 10.11650/twjm/1500406684 | MR 2917261 | Zbl 06062770
[48] Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38 (2000), 431-446. DOI 10.1137/S0363012998338806 | MR 1741147 | Zbl 0997.90062
[49] Wang, Y., Wang, F.: Strong convergence of the forward-backward splitting method with multiple parameters in Hilbert spaces. Optimization 67 (2018), 493-505. DOI 10.1080/02331934.2017.1411485 | MR 3760003 | Zbl 06865949
[50] Xu, H.-K.: Inequalities in Banach spaces with applications. Nonlinear Anal., Theory Methods Appl. 16 (1991), 1127-1138. DOI 10.1016/0362-546X(91)90200-K | MR 1111623 | Zbl 0757.46033
[51] Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002). DOI 10.1142/5021 | MR 1921556 | Zbl 1023.46003
[52] Zegeye, H., Shahzad, N.: An algorithm for finding a common point of the solution set of a variational inequality and the fixed point set of a Bregman relatively nonexpansive mapping. Appl. Math. Comput. 248 (2014), 225-234. DOI 10.1016/j.amc.2014.08.071 | MR 3276677 | Zbl 1338.47115
[53] Zhu, J.-H., Chang, S.-S.: Halpern-Mann's iterations for Bregman strongly nonexpansive mappings in reflexive Banach spaces with applications. J. Inequal. Appl. 2013 (2013), Article ID 146, 14 pages. DOI 10.1186/1029-242X-2013-146 | MR 3055828 | Zbl 1279.47091
Partner of
EuDML logo