Abstract
In this paper, we introduce a modified relaxed projection algorithm and a modified variable-step relaxed projection algorithm for the split feasibility problem in infinite-dimensional Hilbert spaces. The weak convergence theorems under suitable conditions are proved. Finally, some numerical results are presented, which show the advantage of the proposed algorithms.
Similar content being viewed by others
References
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Byrne, C.L.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Cegielski, A.: Generalized relaxations of nonexpansive operators and convex feasibility problems. Contemp. Math. 513, 111–123 (2010)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensitymodulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Wang, F., Xu, H.K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011)
Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)
Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2011)
Yu, X., Shahzad, N., Yao, Y.: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. doi:10.1007/s11590-011-0340-0
Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)
Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)
Wang, Z., Yang, Q., Yang, Y.: The relaxed inexact projection methods for the split feasibility problem. Appl. Math. Comput. 217, 5347–5359 (2011)
Toint, PhL: Global convergence of a class of trust region methods for nonconvex minimization in Hilbert space. IMA J. Numer. Anal. 8, 231–252 (1988)
Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984)
Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)
Osilike, M.O., Aniagbosor, S.C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Math. Comput. Model. 32, 1181–1191 (2000)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
Acknowledgments
The authors express their thanks to the reviewers, whose constructive suggestions led to improvements in the presentation of the results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136) and Fundamental Research Funds for the Central Universities (No. ZXH2011C002).
Rights and permissions
About this article
Cite this article
Dong, QL., Yao, Y. & He, S. Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces. Optim Lett 8, 1031–1046 (2014). https://doi.org/10.1007/s11590-013-0619-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-013-0619-4