A composition projection method for feasibility problems and applications to equilibrium problems
-
1786
Downloads
-
3275
Views
Authors
Jiawei Chen
- School of Mathematics and Statistics, Southwest University, Chongqing 400715, China.
- College of Computer Science, Chongqing University, Chongqing 400044, China.
Yeong-Cheng Liou
- Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan.
Suhel Ahmad Khan
- Department of Mathematics, BITS-Pilani, Dubai Campus, Dubai-345055, UAE.
Zhongping Wan
- School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China.
Ching-Feng Wen
- Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
Abstract
In this article, we propose a composition projection algorithm for solving feasibility problem in Hilbert space.
The convergence of the proposed algorithm are established by using gap vector which does not involve the
nonempty intersection assumption. Moreover, we provide the sufficient and necessary condition for the
convergence of the proposed method. As an application, we investigate the split feasibility equilibrium
problem.
Share and Cite
ISRP Style
Jiawei Chen, Yeong-Cheng Liou, Suhel Ahmad Khan, Zhongping Wan, Ching-Feng Wen, A composition projection method for feasibility problems and applications to equilibrium problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 2, 461--470
AMA Style
Chen Jiawei, Liou Yeong-Cheng, Khan Suhel Ahmad, Wan Zhongping, Wen Ching-Feng, A composition projection method for feasibility problems and applications to equilibrium problems. J. Nonlinear Sci. Appl. (2016); 9(2):461--470
Chicago/Turabian Style
Chen, Jiawei, Liou, Yeong-Cheng, Khan, Suhel Ahmad, Wan, Zhongping, Wen, Ching-Feng. "A composition projection method for feasibility problems and applications to equilibrium problems." Journal of Nonlinear Sciences and Applications, 9, no. 2 (2016): 461--470
Keywords
- Feasibility problem
- gap vector
- projection
- split feasibility equilibrium problem.
MSC
References
-
[1]
H. H. Bauschke, J. M. Borwein , On the convergence of von Neumann's alternating projection algorithm for two sets, Set-Valued Anal., 1 (1993), 185-212.
-
[2]
H. H. Bauschke, J. Chen, X. Wang, A projection method for approximating fixed points of quasi nonexpansive mappings without the usual demiclosedness conditions, J. Nonlinear Convex Anal., 15 (2014), 129-135.
-
[3]
H. H. Bauschke, J. Chen, X.Wang, A Bregman projection method for approximating fixed points of quasi-Bregman nonexpansive mappings, Appl. Anal., 94 (2015), 75-84.
-
[4]
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2011)
-
[5]
E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-146.
-
[6]
J. Chen, Y. C. Liou, Z. Wan, J. C. Yao, A proximal point method for a class of monotone equilibrium problems with linear constraints, Operat. Res., (2015), 275-288.
-
[7]
C. S. Chuang, L. J. Lin, New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces, J. Global Optim., 57 (2013), 533-547.
-
[8]
X. P. Ding, Y. C. Liou, J. C. Yao, Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces, J. Global Optim., 53 (2012), 331-346.
-
[9]
K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, New York (1990)
-
[10]
R. Hu, Y. P. Fang, A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems, Positivity, 17 (2013), 431-441.
-
[11]
Y. C. Liou, L. J. Zhu, Y. Yao, C. C. Chyu, Algorithmic and analytical approaches to the split feasibility problems and fixed point problems, Taiwanese J. Math., 17 (2013), 1839-1853.
-
[12]
A. N. Iusem, G. Kassay, W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 259-273.
-
[13]
G. Kassay, S. Reich, S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.
-
[14]
K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
-
[15]
S. Reich, S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal., 73 (2010), 122-135.
-
[16]
R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ (1970)
-
[17]
S. Sabach, Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1289-1308.
-
[18]
X. Wang, H. H. Bauschke, Compositions and averages of two resolvents: Relative geometry of fixed points sets and a partial answer to a question by C. Byrne, Nonlinear Anal., 74 (2011), 4550-4572.