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Polyak’s gradient method for split feasibility problem constrained by level sets

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Abstract

In this paper, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak’s gradient method, we get a new and simple algorithm for such a problem. Under standard assumptions, we prove that the whole sequence generated by the algorithm weakly converges to a solution. We also modify the proposed algorithm and state the strong convergence without regularity conditions on the sets involved. Numerical experiments are included to illustrate its applications in signal processing.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Boikanyo, O. A.: A strongly convergent algorithm for the split common fixed point problem. Appl. Math. Comput. 265, 844–853 (2015)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensitymodulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2003)

    Article  Google Scholar 

  6. Censor, Y., Elfving, T.: A multiprojection algorithms using Bregman projection in a product space. Numer. Algorithm 8, 221–239 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheney, W., Goldstein, A. A.: Proximity maps for convex sets. Proc. Amer. Math. Soc. 10, 448–450 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  8. Combettes, P. L.: Hilbertian convex feasibility problem: Convergence of projection methods. Appl. Math. Optim. 35, 311–330 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Combettes, P. L., Wajs, V. R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Daubechies, I., Fornasier, M., Loris, I.: Accelerated projected gradient method for linear inverse problems with sparsity constraints. J. Fourier Anal. Appl. 14, 764–92 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Duchi, J. C., Shalevshwartz, S., Singer, Y., et al.: Efficient projections onto the ℓ 1 -ball for learning in high dimensions. In: International Conference on Machine Learning, pp. 272–279 (2008)

  12. Figueiredo, M. A., Nowak, R. D., Wright, S. J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process 1, 586–598 (2007)

    Article  Google Scholar 

  13. Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gubin, L. G., Polyak, B. T., Raik, E. V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7, 1–24 (1967)

    Article  Google Scholar 

  15. Halperin, I.: The product of projection operators. Acta. Sci. Math. (Szeged) 23, 96–99 (1962)

    MathSciNet  MATH  Google Scholar 

  16. López, G., Matín, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Prob. 28, 085004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kraikaew, P., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Polyak, B. T.: Minimization of unsmooth functionals. UUSSR Comput. Math. Math. Phys. 9, 14–29 (1969)

    Article  MATH  Google Scholar 

  19. Schöpfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Prob. 24, 055008 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Starck, J. L., Murtagh, F., Fadili, J. M.: Sparse image and signal processing. Wavelets, curvelets, morphological diversity. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  21. Stiles, W. J.: Closest point maps and their product II. Nieuw Arch Wisk 13, 212–225 (1965)

    MathSciNet  MATH  Google Scholar 

  22. van den Berg, E., Friedlander, M. P.: Probing the Pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31, 890–912 (2008)

  23. von Neumann, J.: On rings of operators. Reduction theory. Ann. Math. 50, 401–485 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, F.: On the convergence of CQ algorithm with variable steps for the split equality problem, 74:927–935 (2017)

  25. Xu, H. K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  27. Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Prob. 26, 105018 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xu, H. K.: Properties and iterative methods for the Lasso and its variants. Chin. Ann. Math. Ser. B 35, 501–518 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, Q.: On variable-step relaxed projection algorithm for variational inequalities. J. Math. Anal. Appl. 302, 166–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the referees for their constructive suggestions and comments, which greatly improve the manuscript. This work was supported by Program for Science and Technology Innovation Talents in the Universities of Henan Province (Grant No. 15HASTIT013) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (Grant No. CXTD20150027).

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Authors and Affiliations

  1. Department of Mathematics, Luoyang Normal University, Luoyang, 471022, People’s Republic of China

    Fenghui Wang

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  1. Fenghui Wang
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Correspondence to Fenghui Wang.

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Wang, F. Polyak’s gradient method for split feasibility problem constrained by level sets. Numer Algor 77, 925–938 (2018). https://doi.org/10.1007/s11075-017-0347-4

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  • DOI: https://doi.org/10.1007/s11075-017-0347-4

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