A Bregman projection for split feasibility problem with multiple output sets: A self-adaptive inertial extragradient-type algorithm.
Nội dung chính của bài viết
Tóm tắt
The purpose of this paper is to investigate a Bregman projection algorithm for solving split feasibility problem with multiple output sets. The proposed algorithm is motivated by the ideas of the Halpern method, the CQ method and the Tseng method. Our proposed algorithm employs the inertial technique and a self-adaptive step size to guarantee a high rate of convergence. The strong convergence theorem is established without prior knowledge of the operator norm and the Lipschitz continuous assumption on the operators involved. Numerical experiments with graphical illustrations are presented to demonstrate the effectiveness and the performance of our proposed algorithm in comparison with some existing ones.
Từ khóa
Hilbert spaces, split variational inequality problem, inertial term, self-adaptive, Bregman projection
Chi tiết bài viết
Tài liệu tham khảo
[1] Bauschke, H.H., Combettes, P.L., Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Berlin: Springer (2011).
[2] Bauschke, H.H., Wang, X., Ye, J., Yuan, X., Bregman distances and Chebyshev sets, J. Approx. Theory 159: 3–25 (2009).
[3] Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl. 18: 441–453 (2002).
[4] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20: 103–120 (2004).
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[10] Chbani, Z., Riahi, H., Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities, Optimization Letters, 7: 185–206 (2013).
[11] Hayashi, M., Bregman Divergence Based Em Algorithm and its Application to Classical and Quantum Rate Distortion Theory, IEEE Transactions on Information Theory, 69(6), June (2023).
[12] Hieu, D.V., Cholamjiak, P., Modified extragradient method with Bregman distance for variational inequalities, Appl. Anal.
https://doi.org/10.1080/00036811.2020.1757078 (2020).
[13] Hieu, D. V., Reich, S., Two Bregman projection methods for solving variational inequalities, Optimization 71, 1777–1802 (2022).
[14] Landweber, L., An iterative formula for Fredholm integral equations of the first kind, Am. J. Math. 73: 615–624 (1951).
[15] Liu, L., Cho, S.Y., A Bregman Projection Algorithm with SelfAdaptive Step Sizes for Split Variational Inequality Problems Involving Non-Lipschitz Operators, J. Nonlinear Var. Anal. 8, No. 3, 396-417 (2024).
[16] Ly, L.X., Thuy, N.T.T., Anh, N.Q., Relaxed two-step inertial method for solving a class of split variational inequality problems with application to traffic analysis, Optimization. doi: 10.1080/02331934.2025.2459191 (2025).
[17] Ly, L.X., Thuy, N.T.T., Tung, T.T., Two novel two-step inertial algorithms for a class of bilevel variational inequalities, Optimization, doi: 10.1080/02331934.2024.2345763 (2024).
[18] Lohit, H., Kumar, D., Modified total Bregman divergence driven picture fuzzy clustering with local information for brain MRI image segmentation, Applied Soft Computing, 144: 110460 (2023).
[19] Mainge, P. E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued Anal. 16, 899–912 (2008).
[20] Marquez, V. M., Reich, S., Sabach, S., Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discret. Contin. Dyn. Syst. Ser. S. 6, 1043–1063 (2013).
[21] Polyak, B.T., Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4:1–17 (1964).
[22] Reich, S., Truong, M.T., Mai, T.N.H., The split feasibility problem with multiple output sets in Hilbert spaces, Optim. Lett. 14, 2335–2353 (2020).
[23] Sunthrayuth, P., Jolaoso, L. O., Cholamjiak, P., New Bregman projection methods for solving pseudo-monotone variational inequality problem, J. Appl. Math. Comput. 68, 1565–1589 (2022).
[24] Thuy, N.T.T., Nghia, N.T., A parallel algorithm for generalized multiple-set split feasibility with application to optimal control problems, Taiwan J Math.; 26(5): 1069–1092 (2022).
[25] Thuy, N.T.T., Nghia, N.T., A new iterative method for solving the multiple-set split variational inequality problem in Hilbert spaces, Optimization, 72: 1549–1575 (2023).
[26] Thuy, N.T.T., Nghia, N.T., A hybrid projection method for solving the multiple-sets split feasibility problem, Comput. Appl. Math.; 42(6): 292 (2023).
[27] Thuy, N.T.T., Nghia, N.T., An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints, Optimization, doi: 10.1080/02331934 (2023).
[28] Thuy, N.T.T., Nghia, N.T., Some novel inertial ball-relaxed CQ algorithms for solving the split feasibility problem with multiple output sets, Journal of Applied Analysis and Computation ;14(3): 1485–1507 (2023).
[29] Thuy, N.T.T., Tung, T.T., A self adaptive inertial algorithm for solving variational inequalities over the solution set of the split variational inequality problem, Optimization Letters. 18, 1619–1645 (2024).
[30] Thuy, N.T.T., Tung, T.T., Two relaxed CQ methods for the split feasibility problem with multiple output sets, Bulletin of the Malaysian Mathematical Sciences Society ; 47(68) (2024).
[31] Thuy, N.T.T., Tung, T.T., Strong convergence of one-step inertial algorithm for a class of bilevel variational inequalities, Optimization, doi: 10.1080/02331934.2024.2448737 (2025).
[32] Thuy, N.T.T., Tung, T.T., Ly, L.X., Strongly convergent two-step inertial algorithm for a class of bilevel variational inequalities, Comp. Appl. Math. 44:117 (2025).
[33] Wega, G.B., Zegeye, H., Convergence results of forward-backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comput. Appl. Math. 39(3): 223. doi: 10.1007/s40314-020-01246-z
(2020).
[2] Bauschke, H.H., Wang, X., Ye, J., Yuan, X., Bregman distances and Chebyshev sets, J. Approx. Theory 159: 3–25 (2009).
[3] Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl. 18: 441–453 (2002).
[4] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20: 103–120 (2004).
[5] Bregman, L.M., The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7: 200–217 (1967).
[6] Butnariu, D., Resmerita, E., Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006, 84919 (2006).
[7] Censor, Y., Elfving, T., A multiprojection algorithm using Bregman projections in product space, Numer. Algorithms. 8, 221–239 (1994).
[8] Censor, Y., Elfving, T., Kopf, N., Bortfeld, T., The multiple-sets split feasibility problem and its application, Inverse Probl. 21(6):2071–2084 (2005).
[9] 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).
[10] Chbani, Z., Riahi, H., Weak and strong convergence of an inertial proximal method for solving Ky Fan minimax inequalities, Optimization Letters, 7: 185–206 (2013).
[11] Hayashi, M., Bregman Divergence Based Em Algorithm and its Application to Classical and Quantum Rate Distortion Theory, IEEE Transactions on Information Theory, 69(6), June (2023).
[12] Hieu, D.V., Cholamjiak, P., Modified extragradient method with Bregman distance for variational inequalities, Appl. Anal.
https://doi.org/10.1080/00036811.2020.1757078 (2020).
[13] Hieu, D. V., Reich, S., Two Bregman projection methods for solving variational inequalities, Optimization 71, 1777–1802 (2022).
[14] Landweber, L., An iterative formula for Fredholm integral equations of the first kind, Am. J. Math. 73: 615–624 (1951).
[15] Liu, L., Cho, S.Y., A Bregman Projection Algorithm with SelfAdaptive Step Sizes for Split Variational Inequality Problems Involving Non-Lipschitz Operators, J. Nonlinear Var. Anal. 8, No. 3, 396-417 (2024).
[16] Ly, L.X., Thuy, N.T.T., Anh, N.Q., Relaxed two-step inertial method for solving a class of split variational inequality problems with application to traffic analysis, Optimization. doi: 10.1080/02331934.2025.2459191 (2025).
[17] Ly, L.X., Thuy, N.T.T., Tung, T.T., Two novel two-step inertial algorithms for a class of bilevel variational inequalities, Optimization, doi: 10.1080/02331934.2024.2345763 (2024).
[18] Lohit, H., Kumar, D., Modified total Bregman divergence driven picture fuzzy clustering with local information for brain MRI image segmentation, Applied Soft Computing, 144: 110460 (2023).
[19] Mainge, P. E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued Anal. 16, 899–912 (2008).
[20] Marquez, V. M., Reich, S., Sabach, S., Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discret. Contin. Dyn. Syst. Ser. S. 6, 1043–1063 (2013).
[21] Polyak, B.T., Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4:1–17 (1964).
[22] Reich, S., Truong, M.T., Mai, T.N.H., The split feasibility problem with multiple output sets in Hilbert spaces, Optim. Lett. 14, 2335–2353 (2020).
[23] Sunthrayuth, P., Jolaoso, L. O., Cholamjiak, P., New Bregman projection methods for solving pseudo-monotone variational inequality problem, J. Appl. Math. Comput. 68, 1565–1589 (2022).
[24] Thuy, N.T.T., Nghia, N.T., A parallel algorithm for generalized multiple-set split feasibility with application to optimal control problems, Taiwan J Math.; 26(5): 1069–1092 (2022).
[25] Thuy, N.T.T., Nghia, N.T., A new iterative method for solving the multiple-set split variational inequality problem in Hilbert spaces, Optimization, 72: 1549–1575 (2023).
[26] Thuy, N.T.T., Nghia, N.T., A hybrid projection method for solving the multiple-sets split feasibility problem, Comput. Appl. Math.; 42(6): 292 (2023).
[27] Thuy, N.T.T., Nghia, N.T., An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints, Optimization, doi: 10.1080/02331934 (2023).
[28] Thuy, N.T.T., Nghia, N.T., Some novel inertial ball-relaxed CQ algorithms for solving the split feasibility problem with multiple output sets, Journal of Applied Analysis and Computation ;14(3): 1485–1507 (2023).
[29] Thuy, N.T.T., Tung, T.T., A self adaptive inertial algorithm for solving variational inequalities over the solution set of the split variational inequality problem, Optimization Letters. 18, 1619–1645 (2024).
[30] Thuy, N.T.T., Tung, T.T., Two relaxed CQ methods for the split feasibility problem with multiple output sets, Bulletin of the Malaysian Mathematical Sciences Society ; 47(68) (2024).
[31] Thuy, N.T.T., Tung, T.T., Strong convergence of one-step inertial algorithm for a class of bilevel variational inequalities, Optimization, doi: 10.1080/02331934.2024.2448737 (2025).
[32] Thuy, N.T.T., Tung, T.T., Ly, L.X., Strongly convergent two-step inertial algorithm for a class of bilevel variational inequalities, Comp. Appl. Math. 44:117 (2025).
[33] Wega, G.B., Zegeye, H., Convergence results of forward-backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comput. Appl. Math. 39(3): 223. doi: 10.1007/s40314-020-01246-z
(2020).