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Relaxed two points projection method for solving the multiple-sets split equality problem

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Abstract

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Censor, Y., Segal, A.: On string-averaging for sparse problems and on the split common fixed point problem. Contemp. Math. 513, 125–142 (2010)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhang, W., Han, D., Li, Z.: A self-adaptive projection method for solving the multiple-sets split feasibility problem. Inverse Prob. 25, 115001 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PEDs. J. Convex Anal. 15, 485–506 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Censor, Y., Bortfel, D., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Moudafi, A.: Alternating CQ-algorithm for convex feasibility and split fixed point problems, J. Nonlinear Convex Anal.

  11. Moudafi, A.: A relaxed alternating CQ-algorithm for convex feasibility problems. Nonlinear Anal. 79, 117–121 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Gao, Y.: Nonsmooth Optimization (In Chinese). Science Press, Beijing (2008)

    Google Scholar 

  14. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer-Verlag (2011)

  15. Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Contributions to Nonlinear Functional Analysis. In: Zararntonello, E.H. (ed.), vol. 20. Academic, New York (1971)

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Acknowledgements

This work was supported by the Natural Science Foundation of Shanghai (14ZR1429200) and Innovation Program of Shanghai Municipal Education Commission (15ZZ073, 15ZZ074).

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Authors and Affiliations

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

    Ya-zheng Dang, Jian Yao & Yan Gao

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  1. Ya-zheng Dang
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  2. Jian Yao
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  3. Yan Gao
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Correspondence to Jian Yao.

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Dang, Yz., Yao, J. & Gao, Y. Relaxed two points projection method for solving the multiple-sets split equality problem. Numer Algor 78, 263–275 (2018). https://doi.org/10.1007/s11075-017-0375-0

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  • DOI: https://doi.org/10.1007/s11075-017-0375-0

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