VISCOSITY-BASED GRADIENT TECHNIQUES FOR SPLIT FEASIBILITY PROBLEMS
Keywords:
Fixed Point Problem, Iterative algorithm, Nonlinear Mappings, Weak and Strong Convergent
Abstract
The Split Feasibility Problem (SFP) is a key optimization model with diverse applications in fields such as inverse problems, signal processing, and medical imaging. This paper presents novel viscosity algorithms for solving the SFP in infinite-dimensional Hilbert spaces. Building on existing methods, we introduce new inertial techniques to improve convergence properties. The proposed algorithms feature adaptive step size strategies, which eliminate the need for prior knowledge of operator norms, ensuring greater computational efficiency. Strong convergence results are demonstrated under mild assumptions. These results generalize many existing findings in the literature.
References
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[9] Lpez, G., Martn-Mrquez, V., Wang, F., & Xu, H. K. Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Problems, 28(2012), 085004.
[10] Ma, X., & Liu, H. An inertial Halpern-type CQ algorithm for solving split feasibility problems in Hilbert spaces. Journal of Applied Mathematics and Computing, (2022) 1–19.
[11] Maing, P. E. Strong convergence of projected subgradient methods for nonsmooth and non- strictly convex minimization. Set-valued Analysis, 16(2008), 899–912.
[12] Vinh, N. T., Hoai, P. T., Dung, L. A., & Cho, Y. J. A New Inertial Self-adaptive Gradient Algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem. Acta Mathematica Sinica, English Series, 39(2023), 2489–2506.
[13] Nesterov, Y. (1983). A method for unconstrained convex minimization problem with the rate of convergence O (1/k2). In Dokl. Akad. Nauk. SSSR 269(1983) 543.
[14] Gubin, L. G., Polyak, B. T., & Raik, E. V. The method of projections for finding the common point of convex sets. USSR Computational Mathematics and Mathematical Physics, 7(1967), 1–24.
[15] Suantai, S., Pholasa, N., Cholamjiak, P. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 14 (2018) 1595–1615.
[16] 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. Numerical Algorithms, 87(2021) 1075–1095.
[17] Shehu, Y., & Gibali, A. New inertial relaxed method for solving split feasibilities. Optimization Letters, 15(2021), 2109–2126.
[18] Xu, H. K. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 26(2010), 105018.
[19] Xu, H. K. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66(2002), 240–256.
[20] Yao, Y., Postolache, M., & Liou, Y. C. Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory and Applications, (2013) 1–12.
[2] Aubin, J. P. Optima and equilibria: An introduction to nonlinear analysis. Springer Science & Business Media. 140 (2013)
[3] Byrne, C. Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problems, 18(2002) 441–453.
[4] Censor, Y., Elfving, T., Kopf, N., & Bortfeld, T. The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems, 21(2005), 2071–2084.
[5] Censor, Y., Motova, A., & Segal, A. Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. Journal of Mathematical Analysis and Applica- tions, 327(2007), 1244–1256.
[6] Censor, Y., Elfving, T. A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms, 8(1994) 221–239.
[7] Klman, A., & Mohammed, L. B. Iterative methods for solving split feasibility problem in Hilbert space. Malaysian Journal of Mathematical Sciences, 10(2016) 127–143.
[8] Kiliman, A., & Mohammed, L. B. A Note on the Split Common Fixed Point Problem and its Variant Forms. Mathematical Analysis and Applications: Selected Topics, (2018) 283–340.
[9] Lpez, G., Martn-Mrquez, V., Wang, F., & Xu, H. K. Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Problems, 28(2012), 085004.
[10] Ma, X., & Liu, H. An inertial Halpern-type CQ algorithm for solving split feasibility problems in Hilbert spaces. Journal of Applied Mathematics and Computing, (2022) 1–19.
[11] Maing, P. E. Strong convergence of projected subgradient methods for nonsmooth and non- strictly convex minimization. Set-valued Analysis, 16(2008), 899–912.
[12] Vinh, N. T., Hoai, P. T., Dung, L. A., & Cho, Y. J. A New Inertial Self-adaptive Gradient Algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem. Acta Mathematica Sinica, English Series, 39(2023), 2489–2506.
[13] Nesterov, Y. (1983). A method for unconstrained convex minimization problem with the rate of convergence O (1/k2). In Dokl. Akad. Nauk. SSSR 269(1983) 543.
[14] Gubin, L. G., Polyak, B. T., & Raik, E. V. The method of projections for finding the common point of convex sets. USSR Computational Mathematics and Mathematical Physics, 7(1967), 1–24.
[15] Suantai, S., Pholasa, N., Cholamjiak, P. The modified inertial relaxed CQ algorithm for solving the split feasibility problems. Journal of Industrial and Management Optimization, 14 (2018) 1595–1615.
[16] 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. Numerical Algorithms, 87(2021) 1075–1095.
[17] Shehu, Y., & Gibali, A. New inertial relaxed method for solving split feasibilities. Optimization Letters, 15(2021), 2109–2126.
[18] Xu, H. K. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 26(2010), 105018.
[19] Xu, H. K. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66(2002), 240–256.
[20] Yao, Y., Postolache, M., & Liou, Y. C. Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory and Applications, (2013) 1–12.
Published
2025-12-29
How to Cite
MOHAMMED, L. B., & KILICMAN, A. (2025). VISCOSITY-BASED GRADIENT TECHNIQUES FOR SPLIT FEASIBILITY PROBLEMS. Unilag Journal of Mathematics and Applications, 5(2), 32 - 44. Retrieved from https://lagjma.unilag.edu.ng/article/view/2826
Section
Articles
Copyright (c) 2025 LAWAN BULAMA MOHAMMED, ADEM KILICMAN
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