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A New Inertial Self-adaptive Gradient Algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem

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Abstract

In this paper, by combining the inertial technique and the gradient descent method with Polyak’s stepsizes, we propose a novel inertial self-adaptive gradient algorithm to solve the split feasibility problem in Hilbert spaces and prove some strong and weak convergence theorems of our method under standard assumptions. We examine the performance of our method on the sparse recovery problem beside an example in an infinite dimensional Hilbert space with synthetic data and give some numerical results to show the potential applicability of the proposed method and comparisons with related methods emphasize it further.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ansari, Q. H., Rehan, A.: Split feasibility and fixed point problems. In: Ansari, Q. (eds), Nonlinear Analysis. Trends in Mathematics. Birkhäuser, New Delhi, 2014, pp. 281–322

    Chapter  Google Scholar 

  3. Aubin, J. P.: Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Ceng, L. C., Ansari, Q. H., Yao, J. C.: An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl., 64, 633–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ceng, L. C., Ansari, Q. H., Yao, J. C.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal., 75, 2116–2125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Censor, Y., Bortfeld, T., Martin, B., et al.: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol., 51, 2353–2365 (2006)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Problems, 27, Art. ID 015007, 9 pp. (2011)

  12. Dang, Y., Sun, J., Xu, H. K.: Inertial accelerated algorithms for solving a split feasibility problem. J. Indust. Manage. Optim. 13, 1383–1394 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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  MATH  Google Scholar 

  14. Gibali, A., Mai, D. T., Vinh, N. T.: A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. J. Indust. Manage. Optim., 15, 963–984 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984

    MATH  Google Scholar 

  16. Kesornprom, S., Pholasa, N., Cholamjiak, P.: On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem. Numer. Algor., 84, 997–1017 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lopez, G., Martfn-Marquez, V., Wang, F., Xu, H. K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Problems, 28, Art. ID 085004, 18 pp. (2012)

  18. Ma, X., Liu, H.: An inertial Halpern-type CQ algorithm for solving split feasibility problems in Hilbert spaces. J. Appl. Math. Comput., 68, 1699–1717 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mainge, P. E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl., 325, 469–479 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mainge, P. E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal., 16, 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc., 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. Polyak, B. T.: Introduction to Optimization, Optimization Software, New York, 1987

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Indust. Manage. Optim., 14, 1595–1615 (2018)

    Article  MathSciNet  Google Scholar 

  25. Takahashi, W.: The split feasibility problem in Banach spaces. J. Nonlinear Convex Anal., 15, 1349–1355 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Tan, K. K., Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl., 178, 301–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, F., Xu, H. K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal., 74, 4105–4111 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, H. K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 26, Art. ID 105018, 17 pp. (2010)

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

    Article  MathSciNet  MATH  Google Scholar 

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

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Acknowledgements

The authors would like to thanks the editor and the referee for valuable remarks and helpful suggestions which improved the quality of the paper. This research is funded by University of Transport and Communications (UTC) under Grant Number T2023-CB-001.

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Authors and Affiliations

  1. Department of Mathematical Analysis, University of Transport and Communications, 3 Cau Giay Street, Hanoi, Vietnam

    Nguyen The Vinh

  2. Department of Applied Mathematics, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Hanoi, Vietnam

    Pham Thi Hoai

  3. Department of Mathematics and Informatics, Hanoi University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

    Le Anh Dung

  4. Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Republic of Korea

    Yeol Je Cho

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  1. Nguyen The Vinh
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  2. Pham Thi Hoai
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  3. Le Anh Dung
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  4. Yeol Je Cho
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Correspondence to Yeol Je Cho.

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Vinh, N.T., Hoai, P.T., Dung, L.A. et al. A New Inertial Self-adaptive Gradient Algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem. Acta. Math. Sin.-English Ser. 39, 2489–2506 (2023). https://doi.org/10.1007/s10114-023-2311-7

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  • DOI: https://doi.org/10.1007/s10114-023-2311-7

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