Multi-step hybrid steepest-descent methods for split feasibility problems with hierarchical variational inequality problem constraints

Volume 9, Issue 6, pp 4148--4166 http://dx.doi.org/10.22436/jnsa.009.06.58

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Authors

L. C. Ceng - Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China. Y. C. Liou - Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, Kaohsiung 80708, Taiwan. D. R. Sahu - Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India.

Abstract

In this paper, we introduce and analyze a multi-step hybrid steepest-descent algorithm by combining Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann's iteration method and gradient-projection method (GPM) with regularization in the setting of infinite-dimensional Hilbert spaces. Strong convergence was established.

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Keywords

• Hybrid steepest-descent method
• split feasibility problem
• generalized mixed equilibrium problem
• variational inclusion
• maximal monotone mapping
• nonexpansive mapping.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

References

• [1] A. E. Al-Mazrooei, B. A. Bin Dehaish, A. Latif, J. C. Yao, On general system of variational inequalities in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 639-658.


• [2] A. S. M. Alofi, A. Latif, A. E. Al-Mazrooei, J. C. Yao, Composite viscosity iterative methods for general systems of variational inequalities and fixed point problem in Hilbert spaces, J. Nonlinear Convex Anal., 17 (2016), 669-682.


• [3] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leiden (1976)


• [4] A. Bnouhachem, Q. H. Ansari, J. C. Yao , An iterative algorithm for hierarchical fixed point problems for a finite family of nonexpansive mappings, Fixed Point Theory Appl., 2015 (2015), 13 pages.


• [5] A. Bnouhachem, Q. H. Ansari, J. C. Yao, Strong convergence algorithm for hierarchical fixed point problems of a finite family of nonexpansive mappings, Fixed Point Theory, 17 (2016), 47-62.


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


• [7] L. C. Ceng, S. Al-Homidan, Algorithms of common solutions for generalized mixed equilibria, variational inclusions, and constrained convex minimization, Abstr. Appl. Anal., 2014 (2014), 25 pages.


• [8] L. C. Ceng, Q. H. Ansari, A. Petrusel, J. C. Yao, Approximation methods for triple hierarchical variational inequalities (II), Fixed Point Theory, 16 (2015), 237-260.


• [9] L. C. Ceng, Q. H. Ansari, S. Schaible, Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems, J. Global Optim., 53 (2012), 69-96.


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


• [11] 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.


• [12] 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.


• [13] 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.


• [14] L. C. Ceng, H. Y. Hu, M. M. Wong , Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings, Taiwanese J. Math., 15 (2011), 134-1367.


• [15] L. C. Ceng, A. Latif, Q. H. Ansari, J. C. Yao, Hybrid extragradient method for hierarchical variational inequalities, Fixed Point Theory Appl., 2014 (2014), 35 pages.


• [16] L. C. Ceng, A. Latif, J. C. Yao, On solutions of system of variational inequalities and fixed point problems in Banach spaces, Fixed Point Theory Appl., 2013 (2013), 34 pages.


• [17] L. C. Ceng, M. M. Wong, A. Petrusel, J. C. Yao , Relaxed implicit extragradient-like methods for finding minimum- norm solutions of the split feasibility problem , Fixed Point Theory, 14 (2013), 327-344.


• [18] L. C. Ceng, J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214 (2008), 186-201.


• [19] L. C. Ceng, 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.


• [20] L. C. Ceng, J. C. Yao, Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasibility problem constraint, Fixed Point Theory App., 2013 (2013), 50 pages.


• [21] L. C. Ceng, J. C. Yao , On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria, Tamkang J. Math., 45 (2014), 297-334.


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


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


• [24] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, England (1990)


• [25] N. J. Huang, A new completely general class of variational inclusions with noncompact valued mappings, Comput. Math. Appl., 35 (1998), 9-14.


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


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


• [28] J. W. Peng, J. C. Yao , A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese J. Math., 12 (2008), 1401-1432.


• [29] R. T. Rockafellar, Monotone operators and the proximal point algorithms, SIAM J. Control Optimization, 14 (1976), 877-898.


• [30] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001), 387-404.


• [31] W. Takahashi, Introduction to nonlinear and convex analysis, Yokohama Publishers, Yokohama (2009)


• [32] W. Takahashi, H. K. Xu, J. C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205-221.


• [33] X. M. Wang, C. Li, J. C. Yao, Subgradient projection algorithms for convex feasibility on Riemannian manifolds with lower bounded curvatures, J. Optim. Theory Appl., 164 (2015), 202-217.


• [34] H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.


• [35] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (2010), 17 pages.


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


• [37] Y. Yao, R. Chen, H. K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447-3456.


• [38] Y. Yao, W. Jigang, Y. C. Liou, Regularized methods for the split feasibility problem, Abstr. Appl. Anal., 2012 (2012), 15 pages.


• [39] Y. Yao, Y. C. Liou, S. M. Kang , Two-step projection methods for a system of variational inequality problems in Banach spaces, J. Global Optim., 55 (2013), 801-811.


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


• [41] Y. Yao, Y. C. Liou, J. C. Yao , Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory Appl., 2007 (2007), 12 pages.


• [42] Y. 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.


• [43] Y. 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.


• [44] Y. 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.


• [45] L. C. Zeng, S. M. Guu, J. C. Yao, Characterization of H-monotone operators with applications to variational inclusions, Comput. Math. Appl., 50 (2005), 329-337.