Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities
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Authors
Lu-Chuan Ceng
- Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China.
Yeong-Cheng Liou
- Department of Healthcare Administration and Medical Informatics, Center for Big Data Analytics \& Intelligent Healthcare, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
- Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan.
Ching-Feng Wen
- Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80708, Taiwan.
Ching-Hua Lo
- Center for Big Data Analytics \& Intelligent Healthcare, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
Abstract
The purpose of this paper is to find a solution of a general system of variational inequalities (for short,
GSVI), which is also a unique solution of a hierarchical variational inequality (for short, HVI) for an infinite family of
nonexpansive mappings in Banach spaces. We introduce general implicit and
explicit iterative algorithms, which are based on the hybrid steepest-descent method and the Mann iteration method. Under
some appropriate conditions, we prove the strong convergence of the sequences generated by the proposed iterative algorithms
to a solution of the GSVI, which is also a unique solution of the HVI.
Share and Cite
ISRP Style
Lu-Chuan Ceng, Yeong-Cheng Liou, Ching-Feng Wen, Ching-Hua Lo, Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4574--4596
AMA Style
Ceng Lu-Chuan, Liou Yeong-Cheng, Wen Ching-Feng, Lo Ching-Hua, Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities. J. Nonlinear Sci. Appl. (2017); 10(9):4574--4596
Chicago/Turabian Style
Ceng, Lu-Chuan, Liou, Yeong-Cheng, Wen, Ching-Feng, Lo, Ching-Hua. "Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4574--4596
Keywords
- System of variational inequalities
- nonexpansive mapping
- fixed point
- hybrid steepest-descent method
- global convergence.
MSC
References
-
[1]
K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006), 13 pages.
-
[2]
N. Buong, N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399–411.
-
[3]
L. C. Ceng, Q. H. Ansari, J. C. Yao , Mann-type steepest-descent and modified steepest-descent methods for variational inequalities in Banach spaces, Numer. Funct. Anal. Optim., 29 (2008), 987–1033.
-
[4]
L. C. Ceng, H. Gupta, Q. H. Ansari , Implicit and explicit algorithms for a system of nonlinear variational inequalities in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 965–984.
-
[5]
L. C. Ceng, S. M. Guu, J. C. Yao, Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems , Fixed Point Theory Appl., 2012 (2012), 19 pages.
-
[6]
L. C. Ceng, C. F. Wen, Y. Yao, Iteration approaches to hierarchical variational inequalities for infinite nonexpansive mappings and finding zero points of m-accretive operators, J. Nonlinear Var. Anal., 1 (2017), 213–235.
-
[7]
S. Y. Cho, B. A. Bin Dehaish, X. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
-
[8]
R. Glowinski, J. L. Lions, R. Tremolieres , Numerical Analysis and Variational Inequalities, North-Holland, Amsterdam-New York (1981)
-
[9]
K. Goebel, W. A. Kirk , Topics on Metric Fixed-Point Theory , Cambridge University Press , England (1990)
-
[10]
A. N. Iusem, B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309–321.
-
[11]
M. Kikkawa, W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Hilbert spaces, RIMS Kokyuroku, 1484 (2006), 105–113.
-
[12]
M. Kikkawa, W. Takahashi , Strong convergence theorems by the viscosity approximation methods for a countable family of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 583–598.
-
[13]
N. Kikuchi, J. T. Oden , Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Society for Industrial and Applied Mathematics, Philadelphia (1988)
-
[14]
D. Kinderlehrer, G. Stampacchia , An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980)
-
[15]
I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in economics and Mathematical Systems, Springer-Verlag, Berlin (2001)
-
[16]
I. V. Konnov , Equilibrium Models and Variational Inequalities, Elsevier B. V., Amsterdam (2007)
-
[17]
G. M. Korpelevich , An extragradient method for finding saddle points and for other problems, (Russian) konom. i Mat. Metody, 12 (1976), 747–756.
-
[18]
Y. Liu, A modified hybrid method for solving variational inequality problems in Banach spaces , J. Nonlinear Funct. Anal., 2017 (2017), 12 pages.
-
[19]
Z. Opial, Weak convergence of successive approximations for nonexpansive mappings , Bull. Amer. Math. Soc., 73 (1967), 591–597.
-
[20]
X. 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.
-
[21]
S. Reich , Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
-
[22]
D. R. Sahu, J. C. Yao, A generalized hybrid steepest descent method and applications, J. Nonlinear Var. Anal., 1 (2017), 111–126.
-
[23]
N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641–3645.
-
[24]
M. V. Solodov, B. F. Svaiter , A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765–776.
-
[25]
T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals , J. Math. Anal. Appl., 305 (2005), 227–239.
-
[26]
K. K. Tan, H. K. Xu , Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308.
-
[27]
H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.
-
[28]
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
-
[29]
H. K. Xu , Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
-
[30]
Y.-H. Yao, R.-D. Chen, H.-K. Xu , Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447–3456.
-
[31]
Y.-H. Yao, Y.-C. Liou, S.-M. Kang, Y. Yu, Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces , Nonlinear Anal., 74 (2011), 6024–6034.
-
[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, N. Shahzad, An algorithmic approach to the split variational inequality and fixed point problem, J. Nonlinear Convex Anal., 18 (2017), 977–991.
-
[34]
H. Zegeye, N. Shahzad and Y.-H. Yao , Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.
-
[35]
L. C. Zeng, J. C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal., 64 (2006), 2507–2515.
-
[36]
H. Y. Zhou, L. Wei, Y. J. Cho, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., 173 (2006), 196–212.