A Modified Inertial Shrinking Projection Method for Solving Inclusion Problems and Split Equilibrium Problems in Hilbert Spaces

Authors

  • Watcharaporn Cholamjiak School of Science, University of Phayao, Phayao 56000
  • Suhel Ahmad Khan Department of Mathematics, BITS-Pilani, Dubai Campus, P.O. Box 345055, Dubai
  • Suthep Suantai Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200

DOI:

https://doi.org/10.26713/cma.v10i2.1074

Keywords:

Inertial method, Inclusion problem, SP-iteration, Forward-backward algorithm, Split equilibrium problem

Abstract

In this paper, we propose a modified inertial forward-backward splitting method for solving the split equilibrium problem and the inclusion problem. Then we establish the weak convergence theorem of the proposed method. Using the shrinking projection method, we obtain strong convergence theorem. Moreover, we provide some numerical experiments to show the efficiency and the comparison.

Downloads

Download data is not yet available.

Author Biographies

Watcharaporn Cholamjiak, School of Science, University of Phayao, Phayao 56000

Mathematics

Suhel Ahmad Khan, Department of Mathematics, BITS-Pilani, Dubai Campus, P.O. Box 345055, Dubai

Mathematics

Suthep Suantai, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200

Mathematics

References

F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces, SIAM. J. Optim. 14(3) (2004), 773 – 782, DOI: 10.1137/S1052623403427859.

F. Alvarez and H. Attouch, An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal. 9 (2001), 3 – 11, DOI: 10.1023/A:1011253113155.

J. B. Baillon and G. Haddad, Quelques proprietes des operateurs angle-bornes et cycliquement monotones, Isr. J. Math. 26 (1977), 137 – 150, DOI: 10.1007/BF03007664.

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics, Springer, New York (2011), DOI: 10.1007/978-3-319-48311-5.

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejérmonotone methods in Hilbert spaces, Math. Oper. Research 26 (2001), 248 – 264, DOI: 10.1287/moor.26.2.248.10558.

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123 – 145.

F. E. Browder, Nonexpansive nonlinear operators in a Banach space, in Proceedings of the National Academy of Sciences of the United States of America 54 (1965), 1041 – 1044, DOI: 10.1073/pnas.54.4.1041.

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems 18 (2002), 441 – 453, DOI: 10.1088/0266-5611/18/2/310.

C. Byrne, A unified treatment of some iterative algorithm in signal processing and image reconstruction, Inverse Problems 20 (2004), 103 – 120, DOI: 10.1088/0266-5611/20/1/006.

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algor. 71 (2016), 915 – 932, DOI: 10.1007/s11784-018-0526-5.

L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186 – 201, DOI: 10.1016/j.cam.2007.02.022.

P. Cholamjiak, A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Numer. Algor. 8 (1994), 221 – 239, DOI: 10.1007/s11075-015-0030-6.

P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005), 1168 – 1200, DOI: 10.1137/050626090.

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117 – 136.

G. Crombez, A geometrical look at iteration methods for operators with fixed point, Numer. Funct. Anal. Optim. 26 (2005), 157 – 175, DOI: 10.1081/NFA-200063882.

G. Crombez, A hierarchical presentation of operators with fixed points on Hilbert spaces, Numer. Funct. Anal. Optim. 27 (2006), 259 – 277, DOI: 10.1080/01630560600569957.

Y. Dang, J. Sun and H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim. 13(3) (2017), 1383 – 1394, DOI: 10.3934/jimo.2016078.

J. Deepho, J. Martinez-Moreno and P. Kumam, A viscosity of Cesí ro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems, Journal of Nonlinear Sciences and Applications 9(4) (2016), 1475 – 1496, DOI: 10.22436/jnsa.009.04.07.

Q. L. Dong, H. B. Yuan, Y. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optimization Letters 12(1) (2018), 87 – 102, DOI: 10.1007/s11590-016-1102-9.

J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Trans. Amer. Math. Soc. 82 (1956), 421 – 439, DOI: 10.2307/1993056.

S. He and C. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal. 2013 (2013), Article ID 942315, DOI: 10.1155/2013/942315.

S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147 – 150, DOI: 10.1090/S0002-9939-1974-0336469-5.

K. R. Kazmi and S. H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egyptian Math. Soc. 21 (2013), 44 – 51, DOI: 10.1016/j.joems.2012.10.009.

P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 964 – 979, DOI: 10.1137/0716071.

G. López, V. Martí­n-Márquez, F. Wang and H. K. Xu, Forward-Backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal. 2012 (2012), Article ID 109236, DOI: 10.1155/2012/109236.

D. Lorenz and T. Pock, An Inertial Forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision 51 (2015), 311 – 325, DOI: 10.1007/s10851-014-0523-2.

W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506 – 510.

C. Martinez-Yanes and H.-K. Xu, Strong convergence of CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2400 – 2411, DOI: 10.1016/j.na.2005.08.018.

A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math. 155 (2003), 447 – 454, DOI: 10.1016/S0377-0427(02)00906-8.

K. Nakajo and W. Takahashi, Strongly convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372 – 379, DOI: 10.1016/S0022-247X(02)00458-4.

Y. Nesterov, A method for solving the convex programming problem with convergence rate O(1/k2), Dokl. Akad. Nauk SSSR 269 (1983), 543 – 547.

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217 – 229, DOI: 10.1006/jmaa.2000.7042.

G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (1979), 383 – 390, DOI: 10.1016/0022-247X(79)90234-8.

D. H. Peaceman and H. H. Rachford, The numerical solution of parabolic and elliptic differentials, J. Soc. Indust. Appl. Math. 3 (1955), 28 – 41, DOI: 10.1137/0103003.

J. W. Peng, Y. C. Liou and J. C. Yao, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions, Fixed Point Theory Appl. 2009 (2009), Article ID 794178, 21 p., DOI: 10.1155/2009/794178.

W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval, J. Com. Appl. Math. 235 (2011), 3006 – 3014, DOI: 10.1016/j.cam.2010.12.022.

B. T. Polyak, Introduction to Optimization, Optimization Software, New York (1987).

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz. 4 (1964), 1 – 17, DOI: 10.1016/0041-5553(64)90137-5.

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control. Optim. 14 (1976), 877 – 898, DOI: 10.1137/0314056.

R. T. Rockafellar, On the maximality of subdifferential mappings, Pac. J. Math. 33 (1970), 209 – 216, DOI: 10.2140/pjm.1970.33.209.

K. Sitthithakerngkiet, J. Deepho, J. Martí­nez-Moreno and P. Kumam, An iterative approximation scheme for solving a split generalized equilibrium, variational inequalities and fixed point problems, International Journal of Computer Mathematics 94(12) (2017), 2373 – 2395, DOI: 10.1080/00207160.2017.1283409.

S. Suantai,Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005), 506 – 517, DOI: 10.1016/j.jmaa.2005.03.002.

S. Suantai and P. Cholamjiak, Algorithms for solving generalized equilibrium problems and fixed points of nonexpansive semigroups in Hilbert spaces, Optimization 63 (2014), 799 – 815, DOI: 10.1080/02331934.2012.684355.

S. Suantai, P. Cholamjiak, Y. J. Cho and W. Cholamjiak, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory and Appl. 2016 (2016), 35, DOI: 10.1186/s13663-016-0509-4.

A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in Nonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka (eds.), Yokohama Publishers, Yokohama (2005), DOI: 10.1016/j.na.2003.07.023.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).

W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), 276 – 286, DOI: 10.1016/j.jmaa.2007.09.062.

P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control. Optim. 38 (2000), 431 – 446, DOI: 10.1137/S0363012998338806.

S. Wang, X. Gong, A. A. Abdou and Y. J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl. 2016 (2016), 4, DOI: 10.1186/s13663-015-0475-2.

U. Witthayarat, A. A. Abdou and Y. J. Cho, Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings

Downloads

Published

30-06-2019
CITATION

How to Cite

Cholamjiak, W., Khan, S. A., & Suantai, S. (2019). A Modified Inertial Shrinking Projection Method for Solving Inclusion Problems and Split Equilibrium Problems in Hilbert Spaces. Communications in Mathematics and Applications, 10(2), 191–213. https://doi.org/10.26713/cma.v10i2.1074

Issue

Section

Research Article