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“Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem

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Abstract

In this paper, first, we review the projection and contraction methods for solving the split feasibility problem (SFP), and then by using the inverse strongly monotone property of the underlying operator of the SFP, we improve the “optimal” step length to provide the modified projection and contraction methods. Also, we consider the corresponding relaxed variants for the modified projection and contraction methods, where the two closed convex sets are both level sets of convex functions. Some convergence theorems of the proposed methods are established under suitable conditions. Finally, we give some numerical examples to illustrate that the modified projection and contraction methods have an advantage over other methods, and improve greatly the projection and contraction methods.

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References

  1. Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekon. Mat. Metody 12, 1164–1173 (1976)

    Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Motonone Operator Theory in Hilbert Spaces. Springer, London (2011)

    Book  MATH  Google Scholar 

  3. 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 

  4. Bnouhachem, A., Noor, M.A., Khalfaoui, M., Zhaohan, S.: On descent-projection method for solving the split feasibility problems. J. Glob. Optim. 54, 627–639 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, X., Gu, G., He, B.: On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optim. 65, 2217–2226 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dong, Q.L., 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Gibali, A., Liu, L., Tang, Y.C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. (2017). https://doi.org/10.1007/s11590-017-1148-3

  18. He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mate. Metody 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  20. Latif, A., Qin, X.: A regularization algorithm for a splitting feasibility problem in Hilbert spaces. J. Nonlinear Sci. Appl. 10, 3856–3862 (2017)

    Article  MathSciNet  Google Scholar 

  21. Latif, A., Vahidi, J., Eslamian, M.: Strong convergence for generalized multiple-set split feasibility problem. Filomat 30(2), 459–467 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. López, G., Martín-Márquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 27, 085004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qu, B., Xiu, N.: A new halfspace-relaxation projection method for the split feasibility problem. Linear Algebra Appl. 428, 1218–1229 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655–1665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  26. Suantai, S., Cholamjiak, P., Cho, Y.J., Cholamjiak, W.: On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces. Fixed Point Theory Appl. 2016, 35 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123–140 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tang, Y., Zhu, C., Yu, H.: Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size. Fixed Point Theory Appl. 2015, 178 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B. Stat. Methodol. 58, 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  30. Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, F.: Polyak’s gradient method for split feasibility problem constrained by level sets. Numer. Algor. 77, 925–938 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Z., Yang, Q., Yang, Y.: The relaxed inexact projection methods for the split feasibility problem. Appl. Math. Comput. 217, 5347–5359 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Wen, M., Peng, J., Tang, Y.C.: A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166, 844–860 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Yao, Y., Yao, Z., Abdou, A.A.N., Cho, Y.J.: Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis. Fixed Point Theory Appl. 2015, 205 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yen, L.H., Muu, L.D., Huyen, N.T.T.: An algorithm for a class of split feasibility problems: application to a model in electricity production. Math. Methods Oper. Res. 84, 549–565 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhao, J., Yang, Q.: A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21(3), 537–546 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhao, J., Zhang, Y., Yang, Q.: Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem. Appl. Math. Comput. 219, 1644–1653 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Zhao, J., Yang, Q.: Self-adaptive projection methods for the multiple-sets split feasibility problem. Inverse Probl. 27, 035009 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We wish to thank anonymous referees and an associate editor for the thorough analysis and review. Their comments and suggestions helped tremendously in improving the quality of this paper.

The first author is supported by National Natural Science Foundation of China (No. 71602144) and Open Fund of Tianjin Key Lab for Advanced Signal Processing (No. 2016ASP-TJ01) and the second author is supported by Visiting Scholarship of Academy of Mathematics and Systems Science, Chinese Academy of Sciences (AM201622C04), the National Natural Science Foundations of China (11401293, 11661056), the Natural Science Foundations of Jiangxi Province (20151BAB211010). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).

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Authors and Affiliations

  1. Tianjin Key Laboratory for Advanced Signal Processing and College of Science, Civil Aviation University of China, Tianjin, 300300, People’s Republic of China

    Q. L. Dong

  2. Department of Mathematics, NanChang University, Nanchang, 330031, People’s Republic of China

    Y. C. Tang

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Y. J. Cho

  4. Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, 660-701, Korea

    Y. J. Cho

  5. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Th. M. Rassias

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  1. Q. L. Dong
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  2. Y. C. Tang
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  3. Y. J. Cho
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  4. Th. M. Rassias
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Correspondence to Q. L. Dong or Y. J. Cho.

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Dong, Q.L., Tang, Y.C., Cho, Y.J. et al. “Optimal” choice of the step length of the projection and contraction methods for solving the split feasibility problem. J Glob Optim 71, 341–360 (2018). https://doi.org/10.1007/s10898-018-0628-z

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  • DOI: https://doi.org/10.1007/s10898-018-0628-z

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