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Modified general splitting method for the split feasibility problem

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Abstract

Based on the equivalent optimization problems of the splitting feasibility problem, we investigate this problem by using modified general splitting method in this paper. One is a relaxation splitting method with linearization, and the other combines the former with alternated inertial extrapolation step. The strong convergence of our algorithms is analyzed when related parameters are properly chosen. Compared with most existing results where inertial factor must be less than 1, inertial factor can be taken 1 in our alternated inertial-type algorithm. The efficiency of our methods are illustrated by some numerical examples.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

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

    Article  MathSciNet  Google Scholar 

  2. Censor, Y., Bortfeld, T., 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 

  3. 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  Google Scholar 

  4. Wang, J.H., Hu, Y.H., Li, C., Yao, J.C.: Linear convergence of CQ algorithms and applications in gene regulatory network inference. Inverse Probl. 33, 055017 (2017)

    Article  MathSciNet  Google Scholar 

  5. Yen, L.H., Huyen, N.T.T., Muu, L.D.: A subgradient algorithm for a class of nonlinear split feasibility problems: application to jointly constrained Nash equilibrium models. J. Global Optim. 73, 849–868 (2019)

    Article  MathSciNet  Google Scholar 

  6. Shehu, Y., Vuong, P.T., Cholamjiak, P.: A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems. J. Fix. Point Theory Appl. 21, 1–24 (2019)

    MathSciNet  Google Scholar 

  7. Brooke, M., Censor, Y., Gibali, A.: Dynamic string averaging CQ methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning. Int. Trans. Oper. Res. (2020)

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

    Article  MathSciNet  Google Scholar 

  9. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Gibali, A., Liu, L.W., Tang, Y.C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. 12, 817–830 (2018)

    Article  MathSciNet  Google Scholar 

  13. Dong, Q.L., Tang, Y.C., Cho, Y.J.: Rassias, ThM: ‘Optimal’ choice of the step length of the projection and contraction methods for solving the split feasibility problem. J. Global Optim. 71, 341–360 (2018)

    Article  MathSciNet  Google Scholar 

  14. Gibali, A., Ha, N.H., Thuong, N.T., Trang, T.H., Vinh, N.T.: Polyaks gradient method for solving the split convex feasibility problem and its applications. J. Appl. Numer. Optim. 1, 145–156 (2019)

    Google Scholar 

  15. López, G., Martin-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  Google Scholar 

  16. Moudafi, A., Thakur, B.S.: Solving proximal splitfeasibility problems without prior knowledge of operator norms. Optim. Lett. 8, 2099–2110 (2014)

    Article  MathSciNet  Google Scholar 

  17. Dong, Q.L., He, S., Zhao, J.: Solving the split equality problem without prior knowledge of operator norms. Optim. 64, 1887–1906 (2015)

    Article  MathSciNet  Google Scholar 

  18. Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive map**s without prior knowledge of operators norms. Optim. 64, 2619–2630 (2015)

    Article  MathSciNet  Google Scholar 

  19. 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  Google Scholar 

  20. Zhang, W., Han, D., Yuan, X.: An efficient simultaneous method for the constrained multiple-sets split feasibility problem. Comput. Optim. Appl. 52, 825–843 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MathSciNet  Google Scholar 

  23. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  Google Scholar 

  24. Gibali, A., Sabach, S., Voldman, S.: Non-convex split feasibility problems: models, algorithms and theory. Open J. Math. Optim. 1, 1–15 (2020)

    Article  MathSciNet  Google Scholar 

  25. He, H., Ling, C., Xu, H.K.: An implementable splitting algorithm for the \(\ell _1\)-norm regularized split feasibility problem. J. Sci. Comput. 67, 281–298 (2016)

    Article  MathSciNet  Google Scholar 

  26. He, H., Xu, H.K.: Splitting methods for split feasibility problems with application to Dantzig selectors. Inverse Probl. 33, 055003 (2017)

    Article  MathSciNet  Google Scholar 

  27. Dong, Q.L., He, S., Rassias, M.T.: General splitting methods with linearization for the split feasibility problem. J. Global Optim. 79, 813–836 (2021)

    Article  MathSciNet  Google Scholar 

  28. Maingé, P.E.J.: Convergence theorems for inertial KM-type algorithms. Comput. Appl. Math. 219, 223–236 (2008)

    Article  MathSciNet  Google Scholar 

  29. Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)

    MathSciNet  Google Scholar 

  30. Vong, S., Liu, D.: An inertial Mann algorithm for nonexpansive map**s. J. Fix. Point Theory Appl. 20, 1–16 (2018)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. Sahu, D.R., Cho, Y.J., Dong, Q.L., Kashyap, M.R., Li, X.H.: Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces. Numer. Algorithms 87, 1075–1095 (2021)

    Article  MathSciNet  Google Scholar 

  33. Wang, F., Yu, H.: An inertial relaxed CQ algorithm with an application to the LASSO and elastic net. Optim. 70, 1101–1119 (2021)

    Article  MathSciNet  Google Scholar 

  34. Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2010)

    Article  MathSciNet  Google Scholar 

  35. He, S., Zhao, Z., Luo, B.: A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem. Optim. 64, 1907–1918 (2015)

    Article  MathSciNet  Google Scholar 

  36. Suantai, S., Pholasa, N., Cholamjiak, P.: Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 113, 1081-1099 (2019)

  37. Iutzeler, F., Malick, J.: On the proximal gradient algorithm with alternated inertia. J. Optim. Theory Appl. 176, 688–710 (2018)

    Article  MathSciNet  Google Scholar 

  38. Iutzeler, F., Hendrickx, J.M.: A generic online acceleration scheme for optimization algorithms via relaxation and inertia. Optim. Methods Softw. 34, 383–405 (2019)

    Article  MathSciNet  Google Scholar 

  39. Shehu, Y., Gibali, A.: New inertial relaxed method for solving split feasibilities. Optim. Lett. 1–18 (2020)

  40. Shehu, Y., Dong, Q.L., Liu, L.L.: Global and linear convergence of alternated inertial methods for split feasibility problems. Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Serie A. Matematicas 115, 1–26 (2021)

    Google Scholar 

  41. Dong, Q.L., Liu, L., Qin, X.,Yao, J.C.: An alternated inertial general splitting method with linearization for the split feasibility problem. Optim. 1–23 (2022)

  42. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017)

    Book  Google Scholar 

  43. Yao, Z., Vong, S.: Two inertial-type algorithms for solving the split feasibility problem. Optim. (2022). https://doi.org/10.1080/02331934.2022.2070066

    Article  Google Scholar 

  44. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China

    Seakweng Vong & Zhongsheng Yao

  2. School of Computer Science and Engineering, Yangjiang Campus, Guangdong Ocean University, Yangjiang, China

    Zhongsheng Yao

Authors
  1. Seakweng Vong
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  2. Zhongsheng Yao
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Correspondence to Zhongsheng Yao.

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This research is funded by The Science and Technology Development Fund, Macau SAR (File No. 0151/2022/A), University of Macau (File no. MYRG2020-00035-FST, MYRG2022-00076-FST), and Guangdong Ocean University (File No. 360302012205).

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Vong, S., Yao, Z. Modified general splitting method for the split feasibility problem. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01399-9

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