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A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem

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Abstract

In this paper, we present a simultaneous subgradient algorithm for solving the multiple-sets split feasibility problem. The algorithm employs two extrapolated factors in each iteration, which not only improves feasibility by eliminating the need to compute the Lipschitz constant, but also enhances flexibility due to applying variable step size. The convergence of the algorithm is proved under suitable conditions. Numerical results illustrate that the new algorithm has better convergence than the existing one.

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References

  1. H. H. Bauschke, J. M. Borwein: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38 (1996), 367–426.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Byrne: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18 (2002), 441–453.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Y. Censor, T. Elfving, N. Kopf, T. Bortfeld: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21 (2005), 2071–2084.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. Censor, A. Segal: Sparse string-averaging and split common fixed points. Nonlinear Analysis and Optimization I. Nonlinear Analysis. A conference in celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th birthdays, Haifa, Israel, June 18–24, 2008 (A. Leizarowitz et al., eds.). Contemporary Mathematics 513, American Mathematical Society, Providence, 2010, pp. 125–142.

    Google Scholar 

  8. Y. Censor, A. Segal: The split common fixed point problem for directed operators. J. Convex Anal. 16 (2009), 587–600.

    MATH  MathSciNet  Google Scholar 

  9. P. L. Combettes: Convex set theoretic image reconvery by extrapolated iterations of parallel subgradient projections. IEEE Transactions on Image Processing 6 (1997), 493–506.

    Article  Google Scholar 

  10. Y. Dang, Y. Gao: Non-monotonous accelerated parallel subgradient projection algorithm for convex feasibility problem. Optimization (electronic only) (2012).

  11. Y. Dang, Y. Gao: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27 (2011), Article ID 015007.

    Article  MathSciNet  Google Scholar 

  12. E. Masad, S. Reich: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8 (2007), 367–371.

    MATH  MathSciNet  Google Scholar 

  13. G. Pierra: Decomposition through formalization in a product space. Math. Program. 28 (1984), 96–115.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Pierra: Parallel constraint decomposition for minimization of a quadratic form. Optimization Techniques. Modeling and Optimization in the Service of Man Part 2. Proceedings, 7th IFIP conference, Nice, September 8–12, 1975 (J. Cea, ed.). Lecture Notes in Computer Science 41, Springer, Berlin, 1976, pp. 200–218. (In French.)

    Google Scholar 

  15. H.-K. Xu: A variable Krasnosel’skiĭ-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22 (2006), 2021–2034.

    Article  MATH  Google Scholar 

  16. Q. Yang: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20 (2004), 1261–1266.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Management, University of Shanghai for Science and Technology, 200093, Shanghai, People’s Republic of China

    Yazheng Dang & Yan Gao

Authors
  1. Yazheng Dang
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  2. Yan Gao
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Correspondence to Yazheng Dang.

Additional information

This work was supported by the National Science Foundation of China under Grant 11171221, Shanghai Leading Academic Discipline Project under Grant XTKX2012, Basic and Frontier Research Program of Science and Technology Department of Henan Province under Grant 112300410277, Innovation Program of Shanghai Municipal Education Commission under Grant 14YZ094, Doctoral Program Foundation of Institutions of Higher Education of China under Grant 20123120110004, Doctoral Starting Projection of the University of Shanghai for Science and Technology under Grant ID-10-303-002, and Young Teacher Training Projection Program of Shanghai for Science and Technology.

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Dang, Y., Gao, Y. A new simultaneous subgradient projection algorithm for solving a multiple-sets split feasibility problem. Appl Math 59, 37–51 (2014). https://doi.org/10.1007/s10492-014-0040-z

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  • DOI: https://doi.org/10.1007/s10492-014-0040-z

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