Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem

Peiyuan Wang; Jianjun Zhou; Risheng Wang; Jie Chen

doi:doi:10.11648/j.dmath.20170203.14

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Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem

Peiyuan Wang, Jianjun Zhou, Risheng Wang, Jie Chen

Published in International Journal of Discrete Mathematics (Volume 2, Issue 3)

Received: 6 February 2017 Accepted: 1 March 2017 Published: 24 March 2017

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Abstract

In this paper, we present several explicit and hybrid strong convergence algorithms for solving the multiple-sets split feasibility problem (MSSFP). Firstly, we modify the existing successive, parallel and cyclic algorithms with the hybrid steepest descent method; then two new hybrid formulas based on the Mann type method are presented; Two general hybrid algorithms which can cover the former ones are further proposed. Strong convergence properties are investigated, and numerical experiments shows the compromise is promising.

Published in	International Journal of Discrete Mathematics (Volume 2, Issue 3)
DOI	10.11648/j.dmath.20170203.14
Page(s)	80-87
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Variational Inequalities, Multiple-Sets Split Feasibility Problem, Hybrid Steepest Descent Method, Lipschitz Continuous, Inverse Strongly Monotone

References

[1]	Y. Yamad, “The hybrid steepest descent method for variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications”, North-Holland, Amsterdam, 2001, pp. 473-504.
[2]	H. K. Xu and T. H. Kim, “Convergence of hybrid steepest descent methods for variational inequalities”, J. Optim. Theory Appl. 2003, vol. 119, pp. 185-201.
[3]	L. C. Zeng, N. C. Wong and J. C. Yao, “Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities”, J. Optim. Theory Appl. 2007, vol. 132, pp. 51-69.
[4]	X. Liu and Y. Cui, “The common minimal-norm fixed point of a finite family of nonexpansive mappings”, Nonlinear Anal. 2010, vol. 73, pp. 76-83.
[5]	S. Iemoto and W. “Takahashi, Strong convergence theorems by a hybrid steepest descent method for countable nonexpansive mappings in Hilbert spaces”, Sciential Mathematical Japonicae Online, 2008, pp. 557-570.
[6]	D. Buong and L. T. Duong, “An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces”, J. Optim. Theory Appl. 2011, vol. 151, pp. 513-524.
[7]	H. Y. Zhou and P. Y. Wang, “A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces”, J. Optim. Theory Appl. 2013, DOI: 10. 1007/s10957-013-0470-x.
[8]	J. K. Kim and D. Buong, “A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings”, Journal of Inequalities and Applications, 2013: 419 (2013), DOI: 10.1186/1029-242X-2013-419.
[9]	Y. Censor, T. Elfving and N. Kopf et al., “The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems”, 2005, vol. 21, pp. 2071–2084.
[10]	H. K. Xu, “A variable Krasnosel’ski–Mann algorithm and the multiple-set split feasibility problem”, Inverse Problems, 2006, vol. 22, pp. 2021–2034.
[11]	Y. Censor, “A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem”, J. Math. Anal. Appl. 2007, vol. 327, pp. 1244–1256.
[12]	S. Chang, Y. Cho and J. K. Kim et al., “Multiple-set split feasibility problems for asymptotically strict pseudocontractions,” Abstract and Applied Analysis, 2012, DOI: 10.1155/2012/491760.
[13]	H. K. Xu, “Iterative methods for the split feasibility problem”, http://www. doc88. com/p-035415145998. html.
[14]	Y. Guo, Y. Yu and R. Chen, “Strong convergence theorem of the CQ algorithm for the multiple-set split feasibility problem”, 2011 International Conference on Future Computer Sciences and Application, 2011, pp. 61-64.
[15]	H. He, S. Liu and M. A. Noor, “Some Krasnonsel’ski-Mann Algorithms and the Multiple-Set Split Feasibility Problem”, Fixed Point Theory and Applications, 2010, DOI: 10.1155/2010/513956.
[16]	B. Deng, T. Chen and Q. Dong, “Viscosity iteration methods for a split feasibility problem and a mixed equilibrium problem in a Hilbert space”, Fixed Point Theory and Applications, 2012: 226.
[17]	F. E. Browder, “Fixed point theorems for noncompact mappings in Hilbert space”, Proc, Natl Acad. Sci. USA, 1965, vol. 53, pp. 1272–1276.
[18]	T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces”, Fixed Point Theory Appl. 2005, vol. 1, pp. 103–123.
[19]	H. K. Xu, “Iterative algorithms for nonlinear operators”, J. Lond. Math. Soc. 2002, vol. 2, pp. 240–256.
[20]	W. Zhang, D. Han and Z. Li, “A self-adaptive projection method for solving the multiple-sets split feasibility problem”, Inverse Problems, 2009, vol. 25, DOI: 10.1088/0266-5611/25/11/115001.
[21]	Y. Dang, Y. Gao, “A New simultanseou subgradint projection for solving a mutiple-sets split feasibility problem,” Applications of Mathematics, 2014, vol. 59, pp. 37–51.
[22]	Y. Dang, Y. Gao, “Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem”, Appl. Math. J. Chinese Univ. 2014, vol. 29 (3), pp. 283-294.

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APA Style

Peiyuan Wang, Jianjun Zhou, Risheng Wang, Jie Chen. (2017). Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. International Journal of Discrete Mathematics, 2(3), 80-87. https://doi.org/10.11648/j.dmath.20170203.14

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ACS Style

Peiyuan Wang; Jianjun Zhou; Risheng Wang; Jie Chen. Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. Int. J. Discrete Math. 2017, 2(3), 80-87. doi: 10.11648/j.dmath.20170203.14

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AMA Style

Peiyuan Wang, Jianjun Zhou, Risheng Wang, Jie Chen. Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. Int J Discrete Math. 2017;2(3):80-87. doi: 10.11648/j.dmath.20170203.14

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@article{10.11648/j.dmath.20170203.14,
  author = {Peiyuan Wang and Jianjun Zhou and Risheng Wang and Jie Chen},
  title = {Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {3},
  pages = {80-87},
  doi = {10.11648/j.dmath.20170203.14},
  url = {https://doi.org/10.11648/j.dmath.20170203.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.14},
  abstract = {In this paper, we present several explicit and hybrid strong convergence algorithms for solving the multiple-sets split feasibility problem (MSSFP). Firstly, we modify the existing successive, parallel and cyclic algorithms with the hybrid steepest descent method; then two new hybrid formulas based on the Mann type method are presented; Two general hybrid algorithms which can cover the former ones are further proposed. Strong convergence properties are investigated, and numerical experiments shows the compromise is promising.},
 year = {2017}
}

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TY  - JOUR
T1  - Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem
AU  - Peiyuan Wang
AU  - Jianjun Zhou
AU  - Risheng Wang
AU  - Jie Chen
Y1  - 2017/03/24
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170203.14
DO  - 10.11648/j.dmath.20170203.14
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 80
EP  - 87
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170203.14
AB  - In this paper, we present several explicit and hybrid strong convergence algorithms for solving the multiple-sets split feasibility problem (MSSFP). Firstly, we modify the existing successive, parallel and cyclic algorithms with the hybrid steepest descent method; then two new hybrid formulas based on the Mann type method are presented; Two general hybrid algorithms which can cover the former ones are further proposed. Strong convergence properties are investigated, and numerical experiments shows the compromise is promising.
VL  - 2
IS  - 3
ER  -

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Author Information

Peiyuan Wang

Postdoctoral Workstation, China Marine Development and Reserch Center (CMDRC), Beijing, China
Jianjun Zhou

China Marine Development and Reserch Center (CMDRC), Beijing, China
Risheng Wang

China Marine Development and Reserch Center (CMDRC), Beijing, China
Jie Chen

China Marine Development and Reserch Center (CMDRC), Beijing, China

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APA Style

Peiyuan Wang, Jianjun Zhou, Risheng Wang, Jie Chen. (2017). Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. International Journal of Discrete Mathematics, 2(3), 80-87. https://doi.org/10.11648/j.dmath.20170203.14

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ACS Style

Peiyuan Wang; Jianjun Zhou; Risheng Wang; Jie Chen. Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. Int. J. Discrete Math. 2017, 2(3), 80-87. doi: 10.11648/j.dmath.20170203.14

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AMA Style

Peiyuan Wang, Jianjun Zhou, Risheng Wang, Jie Chen. Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem. Int J Discrete Math. 2017;2(3):80-87. doi: 10.11648/j.dmath.20170203.14

Copy | Download

@article{10.11648/j.dmath.20170203.14,
  author = {Peiyuan Wang and Jianjun Zhou and Risheng Wang and Jie Chen},
  title = {Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {3},
  pages = {80-87},
  doi = {10.11648/j.dmath.20170203.14},
  url = {https://doi.org/10.11648/j.dmath.20170203.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.14},
  abstract = {In this paper, we present several explicit and hybrid strong convergence algorithms for solving the multiple-sets split feasibility problem (MSSFP). Firstly, we modify the existing successive, parallel and cyclic algorithms with the hybrid steepest descent method; then two new hybrid formulas based on the Mann type method are presented; Two general hybrid algorithms which can cover the former ones are further proposed. Strong convergence properties are investigated, and numerical experiments shows the compromise is promising.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Some Explicit and Hybrid Strong Convergence Algorithms for Solving the Multiple-Sets Split Feasibility Problem
AU  - Peiyuan Wang
AU  - Jianjun Zhou
AU  - Risheng Wang
AU  - Jie Chen
Y1  - 2017/03/24
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170203.14
DO  - 10.11648/j.dmath.20170203.14
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 80
EP  - 87
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170203.14
AB  - In this paper, we present several explicit and hybrid strong convergence algorithms for solving the multiple-sets split feasibility problem (MSSFP). Firstly, we modify the existing successive, parallel and cyclic algorithms with the hybrid steepest descent method; then two new hybrid formulas based on the Mann type method are presented; Two general hybrid algorithms which can cover the former ones are further proposed. Strong convergence properties are investigated, and numerical experiments shows the compromise is promising.
VL  - 2
IS  - 3
ER  -

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