Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem

Volume 9, Issue 6, pp 4214--4225 http://dx.doi.org/10.22436/jnsa.009.06.63

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1987 Downloads
• 3701 Views

Authors

Jong Soo Jung - Department of Mathematics, Dong-A University, Busan 49315, Korea.

Abstract

In this paper, we introduce two iterative algorithms based on the hybrid steepest descent method for solving the split feasibility problem. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a solution of the split feasibility problem, which is a solution of a certain variational inequality. In particular, the minimum norm solution of the split feasibility problem is obtained.

Share and Cite

Keywords

• Split feasibility problem
• nonexpansive mapping
• variational inequality
• minimum-norm
• projection
• bounded linear operator
• \(\rho\)-Lipschitzian
• \(\eta\)-strongly monotone operator.

MSC

•  47J20
•  47J25
•  47J05
•  47H09
•  47H10
•  47H05

References

• [1] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse problems, 18 (2002), 441-453.


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


• [3] Y. Censor, T. Bortfeld, B. Martin, A. Tronfimov, A unifed approach for inversion problems in intensity modulated radiation therapy, Phy. Med. Biol., 51 (2006), 2353-2365.


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


• [5] 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.


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


• [7] Y. Dang, Y. Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (2011), 9 pages.


• [8] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990)


• [9] A. Latif, D. R. Sahu, Q. H. Ansari, Variable KM-like algorithms for fixed point problems and split feasibility problems, Fixed Point Theory Appl., 2014 (2014), 20 pages.


• [10] G. J. Minty , On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315-321.


• [11] B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655-1665.


• [12] T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one parameter nonexpansive semi- groups without Bochner integral, J. Math. Anal. Appl., 305 (2005), 227-239.


• [13] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan (2009)


• [14] F. Wang, H.-K. Xu, Approximating curve and strong convergence of the CQ algorithms for the split feasibility problem , J. Inequal. Appl., 2010 (2010), 13 pages.


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


• [16] H.-K. Xu , A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.


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


• [18] I. Yamada, The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings, D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, North-Holland, Amsterdam, Holland, (2001), 473-504.


• [19] Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261-1266.


• [20] Y. Yao, Y. J. Cho, Y.-C. Liou, Iterative algorithms for hierarchical fixed points problems and variational inequalities, Math. Comput. Modelling, 52 (2010), 1697-1705.


• [21] Y. Yao, Y. J. Cho, Y.-C. Liou, Iterative algorthims for variational inclusions, mixed equilibrium problems and fixed point problems approach optimization problems, Cent. Eur. J. Math., 9 (2011), 640-656.


• [22] Y. Yao, Y. J. Cho, Y.-C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems , European J. Oper. Res., 212 (2011), 242-250.


• [23] Y. Yao, P.-X. Yang, S. M. Kang, Composite projection algorithms for the split feasibility problem, Math. Comput. Modeling, 57 (2013), 693-700.


• [24] J. Zhao, Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791- 1799.