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Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 4123168, 7 pages
https://doi.org/10.1155/2018/4123168
Research Article

The Split Feasibility Problem and Its Solution Algorithm

Institute of Operations Research, Qufu Normal University, Rizhao, Shandong 276826, China

Correspondence should be addressed to Biao Qu; moc.361@100oaibuq

Received 20 August 2017; Revised 7 November 2017; Accepted 5 December 2017; Published 8 January 2018

Academic Editor: Shuming Wang

Copyright © 2018 Biao Qu and Binghua Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “The split feasibility model leading to a unified approch for inversion problem in intensity-modulated radiation therapy,” Tech. Rep., April Depaetment of Mathematics, University of Haifa, Haifa, 2005. View at Google Scholar
  4. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proceedings of the IEEE, vol. 71, no. 3, pp. 409–419, 1983. View at Publisher · View at Google Scholar · View at Scopus
  6. H. X. Ding, Two kinds of iterative algorithms for solving the split feasibility problems and the corresponding relaxed algorithms [Master thesis], Nankai University, 2011.
  7. H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. Dang and Y. Gao, “The strong convergence of a three-step algorithm for the split feasibility problem,” Optimization Letters, vol. 7, no. 6, pp. 1325–1339, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Y. Dang, Y. Gao, and L. Li, “Inertial projection algorithms for convex feasibility problem,” Journal of Systems Engineering and Electronics, vol. 23, no. 5, pp. 734–740, 2012. View at Publisher · View at Google Scholar
  10. G. López, V. Martín-Márquez, F. Wang, and H.-K. Xu, “Solving the split feasibility problem without prior knowledge of matrix norms,” Inverse Problems, vol. 28, no. 8, Article ID 085004, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. P.-E. Mainge, “Convergence theorems for inertial KM-type algorithms,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 223–236, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  12. B. Qu and N. Xiu, “A note on the CQ algorithm for the split feasibility problem,” Inverse Problems, vol. 21, no. 5, pp. 1655–1665, 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. B. Qu and N. Xiu, “A new halfspace-relaxation projection method for the split feasibility problem,” Linear Algebra and its Applications, vol. 428, no. 5-6, pp. 1218–1229, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. B. Qu, B. Liu, and N. Zheng, “On the computation of the step-size for the CQ-like algorithms for the split feasibility problem,” Applied Mathematics and Computation, vol. 262, pp. 218–223, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Qu, C. Wang, and N. Xiu, “Analysis on Newton projection method for the split feasibility problem,” Computational optimization and applications, vol. 67, no. 1, pp. 175–199, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. E. H. Zarantonello, “PROjections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets,” Contributions to nonlinear functional analysis (PROc. Sympos., Math. Res. Center, Univ. WISconsin, MADison, WIS., 1971), pp. 237–341, 1971. View at Google Scholar · View at MathSciNet
  18. J. Zhao, Y. Zhang, and Q. Yang, “Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1644–1653, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. C. Byrne, “Bregman-Legendre multidistance projection algorithms for convex feasibility and optimization,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), vol. 8 of Stud. Comput. Math., pp. 87–99, North-Holland, Amsterdam, Amsterdam, The Netherlands, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. Wang and N. Xiu, “Convergence of the gradient projection method for generalized convex minimization,” Computational optimization and applications, vol. 16, no. 2, pp. 111–120, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  21. R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, USA, 1970. View at MathSciNet
  22. D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Belmont, Mass, USA, 2003. View at MathSciNet
  23. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. View at MathSciNet
  24. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at Scopus