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Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we propose inertial forward-backward splitting algorithm to approximate the solution of common variational inclusion problems. By using the inertial technique with parallel monotone hybrid methods we prove strong convergence results under some suitable conditions in Hilbert spaces. We then give some applications and numerical experiments for supporting our main results which shows that our proposed inertial hybrid method has better convergence rate than existing algorithms. Further, we apply our result to solve a common convex minimization problem and a common split feasibility problem. Finally, we use our proposed algorithm to solve the unconstrained image restoration problems and we can show that our algorithm is flexibility and good quality to use for common types of blur effects.

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References

  1. Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert spaces. SIAM. J. Optim. 14(3), 773–782 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with dam**. Set-Valued Anal. 9, 3–11 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive map**s. J. Appl. Math. Comput. 48(1), 241–263 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44(2), 351–374 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Baillon, J.B., Haddad, G.: Quelques proprietes des operateurs angle-bornes et cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)

    MathSciNet  MATH  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics. Springer, New York (2011)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projection in a product space. Numer. Algor. 71, 915–932 (2016)

    MATH  Google Scholar 

  10. Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set Val. Var. Anal. 20, 229–247 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Cholamjiak, P.: A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numer. Algor. 8, 221–239 (1994)

    Google Scholar 

  12. Cholamjiak, W., Cholamjiak, P., Suantai, S.: An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J. Fixed Point Theory Appl. 20, 42 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Cholamjiak, P., Thong, D.V., Cho, Y.J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl. Math. (2019). https://doi.org/10.1007/s10440-019-00297-7

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Dang, Y., Sun, J., Xu, H.: Inertial accelerated algorithms for solving a split feasibility problem. J. Ind. Manag. Optim. 13(3), 1383–1394 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Dong, Q., Jiang, D., Cholamjiak, P., Shehu, Y.: A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions. J. Fixed Point Theory Appl. 19(4), 3097–3118 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, M.: Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive map**s. Optim. Lett. 12(1), 87–102 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Douglas, J., Rachford, H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Amer. Math. Soc. 82, 421–439 (1956)

    MathSciNet  MATH  Google Scholar 

  19. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory, vol. 28. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  20. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  21. Hansen, P.C.: Discrete Inverse Problems: Insight and Algorithms. SIAM, Philadelphia (2010)

    MATH  Google Scholar 

  22. Hansen, P.C.: Rank-Deficient and Discrete Ill-posed Problems. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  23. Hieu, D.V.: Parallel and cyclic hybrid subgradient extragradient methods for variational inequalities. Afr. Mat. 28(5–6), 677–692 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Hieu, D.V.: A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive map**s in Hilbert space. J. Korean Math. Soc. 52(2), 373–388 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Hieu, D.V.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive map**s. J. Appl. Math. Comput. 53, 531–554 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive map**s. Numer. Algor. 73, 197–217 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Hieu, D.V.: New subgradient extragradient methods for common solutions to equilibrium problems. Comput. Optim. Appl. 67, 571–594 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. (2019). https://doi.org/10.1007/s10589-019-00093-x

    Article  MathSciNet  MATH  Google Scholar 

  29. Kesornprom, S., Cholamjiak, P.: Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications. Optimization 68, 2365–2391 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Khan, S.A., Suantai, S., Cholamjiak, W.: Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems. Rev. Real Acad. Cienc. Exact. 113(2), 645–656 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Kim, T.H., Xu, H.K.: Strongly convergence of modified Mann iterations for with asymptotically nonexpansive map**s and semigroups. Nonlinear Anal. 64, 1140–1152 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)

    MathSciNet  MATH  Google Scholar 

  33. Lopez, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012, 25 (2012)

  34. Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51, 311–325 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)

    MathSciNet  MATH  Google Scholar 

  36. Nakajo, K., Takahashi, W.: Strongly convergence theorems for nonexpansive map**s and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    MathSciNet  MATH  Google Scholar 

  37. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)

    MathSciNet  MATH  Google Scholar 

  38. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)

    Google Scholar 

  39. Rockafellar, R.T.: On the maximality of subdifferential map**s. Pac. J. Math. 33, 209–216 (1970)

    MATH  Google Scholar 

  40. Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in Hilbert space. Math. Progr. 87, 189–202 (2000)

    MathSciNet  MATH  Google Scholar 

  41. Sto\(\breve{s}\)i\(\acute{c}\), M., Xavier, J., Dodig, M.: Projection on the intersection of convex sets. Linear Algebra Appl. 09, 191–205 (2016)

  42. Suantai, S., Peeyada, P., Yambangwai, D., Cholamjiak, W.: A parallel-viscosity-type subgradient extragradient-line method for finding the common solution of variational inequality problems applied to image restoration problems. Mathematics 8(2), 248 (2020)

    Google Scholar 

  43. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  44. Tseng, P.: A modified forward-backward splitting method for maximal monotone map**s. SIAM J. Control. Optim. 38, 431–446 (2000)

    MathSciNet  MATH  Google Scholar 

  45. Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank University of Phayao, Phayao, Thailand (Grant no. UoE62001) and Thailand Science Research and Innovation under the project IRN62W0007.

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Authors and Affiliations

  1. School of Science, University of Phayao, Phayao, 56000, Thailand

    Watcharaporn Cholamjiak & Damrongsak Yambangwai

  2. Department of Mathematics, BITS-Pilani, Dubai Campus, P.O. Box 345055, Dubai, United Arab Emirates

    Suhel Ahmad Khan

  3. Department of Mathematics, Faculty of Science and Arts–Rabigh, King Abdulaziz University, P.O. Box 344, Rabigh, 21911, Kingdom of Saudi Arabia

    Kaleem Raza Kazmi

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  1. Watcharaporn Cholamjiak
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  2. Suhel Ahmad Khan
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  3. Damrongsak Yambangwai
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  4. Kaleem Raza Kazmi
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Correspondence to Suhel Ahmad Khan.

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Cholamjiak, W., Khan, S.A., Yambangwai, D. et al. Strong convergence analysis of common variational inclusion problems involving an inertial parallel monotone hybrid method for a novel application to image restoration . RACSAM 114, 99 (2020). https://doi.org/10.1007/s13398-020-00827-1

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