Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

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Published: Dec 22, 2015
DOI: https://doi.org/10.5556/j.tkjm.46.2015.1917
Keywords:
Hybrid extragradient method with regularization Triple hierarchical variational inequality problem General mixed equilibrium General system of variational inequalities Nonexpansive mapping Strict pseudocontraction.

Main Article Content

Lu-Chuan Ceng

Abstract

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

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How to Cite
Ceng, L.-C. (2015). Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints. Tamkang Journal of Mathematics, 46(4), 453–503. https://doi.org/10.5556/j.tkjm.46.2015.1917
Issue
Vol. 46 No. 4 (2015)
Section
Papers
Author Biography

Lu-Chuan Ceng

Department ofMathematics, Shanghai Normal University, Shanghai 200234, China.

References

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.

I. Yamada, The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, edited by D. Butnariu, Y. Censor and S. Reich, North--Holland, Amsterdam, Holland, 473--504, 2001.

H. K. Xu, Iterative methods for the split feasibility problem in infinite--dimensional Hilbert spaces, Inverse Problems, 26(2010), 105018, 17 pp.

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8(1994), 221--239.

L. C. Ceng, A. Petrusel and J. C. Yao, Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, J. Optim. Theory Appl., 143(2009), 37--58.

C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18(2002), 441--453.

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20(2004), 103--120.

L. C. Ceng, Q. H. Ansari and J. C. Yao, Iterative methods for triple hierarchical variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 151(2011), 489--512.

Y. Censor, A. Motova and A. Segal, Perturbed projections and subgradient projections for the multi-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244--1256.

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128(2006), 191--201.

L. C. Ceng and J. C. Yao, Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasibility problem constraint, Fixed Point Theory Appl., 2013, 2013:43, 50 pp.

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon., 12(1976), 747--756.

L. C. Ceng, M. M. Wong and J. C. Yao, A hybrid extragradient--like approximation method with regularization for solving split feasibility and fixed point problems, J. Nonlinear Convex Anal., 14(2013), 1--20.

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, Wiley, New York, 1980.

F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vols. I and II, Springer, Berlin, 2003.

A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal., 9(2008), 37--43.

L. C. Ceng and J. C. Yao, A relaxed extragradient--like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal., 72(2010), 1922--1937.

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.

R. T. Rockafellar, Monotone operators and the proximal point algorithms}, SIAM J. Control Optim., 14(1976), 877--898.

A. Moudafi, P.-E. Mainge, Towards viscosity approximations of hierarchical fixed point problems, Fixed Point Theory Appl., 2006 (2006), Article ID 95453, 10 pp.

A. Moudafi and P.-E. Mainge, Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim., 3(2007), 529--538.

A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241(2000), 46--55.

H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298(2004),279--291.

A. Latif, L. C. Ceng and Q. H. Ansari, Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems, Fixed Point Theory Appl., 2012, 2012:78,28 pp.

H. K. Xu and T. H. Kim, Convergence of hybrid steepest--descent methods for variational inequalities, J. Optim. Theory. Appl., 119(2003), 185--201.

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

W. Oettli, A remark on vector--valued equilibria and generalized monotonicity, Acta Math. Vietnam. 22(1997), 215--221.

Y. Yao, Y.-C. Liou and G. Marino, Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems, J. Appl. Math. Comput., 31 (2009), 433--445.

S. Atsushiba and W. Takahashi, Strong convergence theorems for a finite family of nonexpansive mappings and applications, in: B. N. Prasad Birth Centenary Commemoration Volume, Indian J. Math., 41(1999), 435--453.

Y. Censor, T. Bortfeld, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity--modulated radiation therapy, Physics in Medicine and Biology, 51(2006), 2353--2365.

L. C. Ceng, Z. R. Kong and C. F. Wen, On general systems of variational inequalities, Comput. Math. Appl., 66(2013), 1514--1532.

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20(2004), 103--120.

V. Colao, G. Marino and H. K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 344(2008), 340--352.

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118(2003), 417--428.

H. Iiduka, Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem, Nonlinear Anal., 71(2009), 1292--1297.

H. Iiduka, Iterative algorithm for solving triple-hierarchical constrained optimization problem, J. Optim. Theory Appl., 148(2011), 580--592.

G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Math. Appl., 329(2007), 336--346.

Y. Yao, Y. C. Liou and 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.

R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot--Line, 3(1999), 65--68.

F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory Appl., 2010(2010), Article ID 383740, 19 pp.

G. Marino, L. Muglia and Y. Yao, Viscosity methods for common solutions of equilibrium and variational inequality problems via multi--step iterative algorithms and common fixed points, Nonlinear Anal., 75(2012), 1787--1798.

L. C. Ceng, Q. H. Ansari, M. M. Wong and J. C. Yao, Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems, Fixed Point Theory,13(2012), 403--422.

K. Goebel and W. A. Kirk, Topics on Metric Fixed--Point Theory, Cambridge University Press, Cambridge, England, 1990.

S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal., 69(2008), 1025--1033.

L. C. Ceng, Q. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64(2012), 633--642.

L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient methods for finding minimum--norm solutions of the split feasibility problem, Nonlinear Anal., 75(2012), 2116--2125.

L. C. Ceng, Q. H. Ansari and J. C. Yao, Relaxed extragradient iterative methods for variational inequalities, Appl. Math. Comput., 218(2011), 1112--1123.

L. C. Ceng, Q. H. Ansari, N. C. Wong and J. C. Yao, An extragradient--like approximation method for variational inequalities and fixed point problems, Fixed Point Theory Appl., 2011, 2011:22, 18 pp.

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240--256.

P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization 53(2004), 475--504.