The split variational inequality problem and its algorithm iteration
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Authors
Yonghong Yao
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Xiaoxue Zheng
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Limin Leng
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Shin Min Kang
- Center for General Education, China Medical University, Taichung 40402, Taiwan.
- Department of Mathematics and the RINS, Gyeongsang National University, Jinju 52828, Korea.
Abstract
The split variational inequality problem under a nonlinear transformation has been considered. An iterative algorithm is
presented to solve this split problem. Strong convergence results are obtained.
Share and Cite
ISRP Style
Yonghong Yao, Xiaoxue Zheng, Limin Leng, Shin Min Kang, The split variational inequality problem and its algorithm iteration, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 5, 2649--2661
AMA Style
Yao Yonghong, Zheng Xiaoxue, Leng Limin, Kang Shin Min, The split variational inequality problem and its algorithm iteration. J. Nonlinear Sci. Appl. (2017); 10(5):2649--2661
Chicago/Turabian Style
Yao, Yonghong, Zheng, Xiaoxue, Leng, Limin, Kang, Shin Min. "The split variational inequality problem and its algorithm iteration." Journal of Nonlinear Sciences and Applications, 10, no. 5 (2017): 2649--2661
Keywords
- Split variational inequality
- iterative algorithm
- nonlinear transformation.
MSC
References
-
[1]
M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199–277.
-
[2]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
-
[3]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633–642.
-
[4]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116–2125.
-
[5]
L.-C. Ceng, M. Teboulle, J.-C. Yao, Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems, J. Optim. Theory Appl., 146 (2010), 19–31.
-
[6]
L.-C. Ceng, J.-C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J. Math., 10 (2006), 1293–1303.
-
[7]
Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
-
[8]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
-
[9]
Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071–2084.
-
[10]
Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301–323.
-
[11]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600.
-
[12]
F. Cianciaruso, G. Marino, L. Muglia, Y.-H. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities, J. Inequal. Appl., 2009 (2009), 13 pages.
-
[13]
R. Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York (1984)
-
[14]
Z.-H. He, W.-S. Du, On hybrid split problem and its nonlinear algorithms, Fixed Point Theory Appl., 2013 (2013), 20 pages.
-
[15]
Z.-H. He, W.-S. Du, On split common solution problems: new nonlinear feasible algorithms, strong convergence results and their applications, Fixed Point Theory Appl., 2014 (2014), 16 pages.
-
[16]
A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103–114.
-
[17]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) ´Ekonom. i Mat. Metody, 12 (1976), 747–756.
-
[18]
P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469–479.
-
[19]
X.-L. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488–502.
-
[20]
X.-L. Qin, J.-C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 9 pages.
-
[21]
R.-T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898.
-
[22]
W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417–428.
-
[23]
H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 2 (2002), 240–256.
-
[24]
H.-K. Xu, A variable Krasnoselski˘ı-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021–2034.
-
[25]
H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages.
-
[26]
Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 183 (2014), 14 pages.
-
[27]
Y.-H. Yao, R.-D. Chen, H.-K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447–3456.
-
[28]
Y.-H. Yao, W. Jigang, Y.-C. Liou, Regularized methods for the split feasibility problem, Abstr. Appl. Anal., 2012 (2012), 13 pages.
-
[29]
Y.-H. Yao, Y.-C. Liou, S. M. Kang, Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method, Comput. Math. Appl., 59 (2010), 3472–3480.
-
[30]
Y.-H. Yao, Y.-C. Liou, J.-C. Yao, Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithm construction, Fixed Point Theory Appl., 2015 (2015), 19 pages.
-
[31]
Y.-H. Yao, M. A. Noor, Y.-C. Liou, Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities, Abstr. Appl. Anal., 2012 (2012), 9 pages.
-
[32]
Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multivalued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 10 pages.
-
[33]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 12 pages.
-
[34]
H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.
-
[35]
L. J. Zhang, J. M. Chen, Z. B. Hou, Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities, (Chinese) Acta Math. Sinica (Chin. Ser.), 53 (2010), 691–698.