Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications

Abdulwahab Ahmad; Poom Kumam; Thidaporn Seangwattana

doi:10.53391/mmnsa.1595657

EN

Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications

Abstract

In this work, we construct four efficient multi-step inertial relaxed algorithms based on the monotonic step-length criterion which does not require any information about the norm of the underlying operator or the use of a line search procedure for split feasibility problems in infinite-dimensional Hilbert spaces. The first and the third are the general multi-step inertial-type methods, which unify two steps of the multi-step inertial terms with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to improve the speed of convergence of their sequences of iterates to a solution of the problem, while the second and the fourth are the three-term conjugate gradient-like and multi-step inertial-type methods, which integrate both the three-term conjugate gradient-like direction and a multi-step inertial term with the golden ratio-based and an alternating golden ratio-based extrapolation steps, respectively, to accelerate their sequences of iterates toward a solution of the problem. Under some simple and weaker assumptions, we formulate and prove some strong convergence theorems for each of these algorithms based on the convergence theorem of a golden ratio-based relaxed algorithm with perturbations and the alternating golden ratio-based relaxed algorithm with perturbations in infinite-dimensional real Hilbert spaces. Moreover, we analyze their applications in classification problems for an interesting real-world dataset based on the extreme learning machine (ELM) with the $\ell_{1}-\ell_{2}$ hybrid regularization approach and in solving constrained minimization problems in infinite-dimensional Hilbert spaces. In all the experiments, our proposed algorithms, which generalizes several algorithms in the literature, comparatively achieve better performance than some related algorithms.

Keywords

Split feasibility problem, Golden ratio algorithm, Multi-step inertial method, Three-term conjugate gradient method, Classification problem

References

[1] Censor, Y. and Elfving, T. A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms, 8, 221–239, (1994).
[2] Penfold, S., Zalas, R., Casiraghi, M., Brooke, M., Censor, Y. and Schulte, R. Sparsity constrained split feasibility for dose-volume constraints in inverse planning of intensity-modulated photon or proton therapy. Physics in Medicine & Biology, 62(9), 3599, (2017).
[3] Suantai, S., Peeyada, P., Fulga, A. and Cholamjiak, W. Heart disease detection using inertial Mann relaxed CQ algorithms for split feasibility problems. AIMS Mathematics, 8(8), 18898–18918, (2023).
[4] Dong, Q.L., Li, X.H. and Rassias, T.M. Two projection algorithms for a class of split feasibility problems with jointly constrained Nash equilibrium models. Optimization, 70(4), 871–897, (2021).
[5] Tan, B., Qin, X. and Wang, X. Alternated inertial algorithms for split feasibility problems. Numerical Algorithms, 95, 773–812, (2024).
[6] Ahmad, A., Kumam, P., Yahaya, M.M., Sitthithakerngkiet, K. and Cholamjiak, W. Two-step alternated inertial algorithm for split feasibility problems with some applications. Journal of Nonlinear and Convex Analysis, 25(12), 3165-3191, (2024).
[7] Byrne, C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems, 18(2), 441, (2002).
[8] Yang, Q. The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems, 20(4), 1261, (2004).
[9] Bauschke, H.H. and Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer Cham: New York, (2017).
[10] Qu, B. and Xiu, N. A note on the CQ algorithm for the split feasibility problem. Inverse Problems, 21(5), 1655, (2005).

[11] Dong, Q.L., Tang, Y.C., Cho, Y.J. and Rassias, Th.M. "Optimal" choice of the step length of the projection and contraction methods for solving the split feasibility problem. Journal of Global Optimization, 71, 341-360, (2018).
[12] Gibali, A., Liu, L.W. and Tang, Y.C. Note on the modified relaxation CQ algorithm for the split feasibility problem. Optimization Letters, 12, 817-830, (2018).
[13] Dong, Q.L., Liu, L. and Yao, Y. Self-adaptive projection and contraction methods with alternated inertial terms for solving the split feasibility problem. Journal of Nonlinear and Convex Analysis, 23(3), 591–605, (2022).
[14] Kratuloek, K., Kumam, P., Sriwongsa, S. and Abubarkar, J. A relaxed splitting method for solving variational inclusion and fixed point problems. Computational and Applied Mathematics, 43, 70, (2024).
[15] Sumalai, P., Abubakar, J., Kumam, P. and Salisu, S. A unified scheme for solving split inclusions with applications. Mathematical Methods in the Applied Sciences, 46(13), 14622-14639, (2023).
[16] Ma, X. and Liu, H. An inertial Halpern-type CQ algorithm for solving split feasibility problems in Hilbert spaces. Journal of Applied Mathematics and Computing, 68, 1699–1717, (2022).
[17] Shehu, Y., Dong, Q.L. and Liu, L.L. Global and linear convergence of alternated inertial methods for split feasibility problems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115, 53, (2021).
[18] Abubakar, J., Kumam, P., Taddele, G.H., Ibrahim, A.H. and Sitthithakerngkiet, K. Strong convergence of alternated inertial CQ relaxed method with application in signal recovery. Computational and Applied Mathematics, 40, 310, (2021).
[19] Suantai, S., Panyanak, B., Kesornprom, S. and Cholamjiak, P. Inertial projection and contraction methods for split feasibility problem applied to compressed sensing and image restoration. Optimization Letters, 16, 1725-1744, (2022).
[20] Dang, Y., Sun, J. and Xu, H. Inertial accelerated algorithms for solving a split feasibility problem. Journal of Industrial and Management Optimization, 13(3), 1383-1394, (2017).
[21] Sahu, D.R., Cho, Y.J., Dong, Q.L., Kashyap, M.R. and Li, X.H. Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces. Numerical Algorithms, 87, 1075-1095, (2021).
[22] Vinh, N.T., Hoai, P.T., Dung, L.A. and 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, 2489-2506, (2023).
[23] Reich, S., Tuyen, T.M. and Van Huyen, P.T. Inertial proximal point algorithms for solving a class of split feasibility problems. Journal of Optimization Theory and Applications, 200, 951–977, (2024).
[24] Chen, P., Huang, J. and Zhang, X. A primal-dual fixed point algorithm for convex separable minimization with applications to image restoration. Inverse Problems, 29(2), 025011, (2013).
[25] Iiduka, H. Fixed point optimization algorithms for distributed optimization in networked systems. SIAM Journal on Optimization, 23(1), 1–26, (2013).
[26] Polyak, B.T. Some methods of speeding up the convergence of iteration methods. USSR Computational Mathematics and Mathematical Physics, 4(5), 1–17, (1964).
[27] Ahmad, A., Kumam, P., Harbau, M.H. and Sitthithakerngkiet, K. Inertial hybrid algorithm for generalized mixed equilibrium problems, zero problems, and fixed points of some nonlinear mappings in the intermediate sense. Mathematical Methods in the Applied Sciences, 47(11), 8527–8550, (2024).
[28] Harbau, M.H., Ahmad, A., Ali, B. and Ugwunnadi, G.C. Inertial residual algorithm for fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. International Journal of Nonlinear Analysis and Applications, 13(2), 2257–2269, (2022).
[29] Alvarez, F. and Attouch, H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Analysis, 9, 3–11, (2001).
[30] Adamu, A., Kitkuan, D. and Seangwattana, T. An accelerated Halpern-type algorithm for solving variational inclusion problems with applications. Bangmod International Journal of Mathematical and Computational Science, 8, 37–55, (2022).
[31] Thammasiri, P. and Ungchittrakool, K. Accelerated hybrid Mann-type algorithm for fixed point and variational inequality problems. Nonlinear Convex Analysis and Optimization: An International Journal on Numerical, Computation and Applications, 1(1), 97–111, (2022).
[32] Beck, A. and Teboulle, M. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences, 2(1), 183–202, (2009).
[33] Lorenz, D.A. and Pock, T. An inertial forward-backward algorithm for monotone inclusions. Journal of Mathematical Imaging and Vision, 51, 311–325, (2015).
[34] Ortega, J.M. and Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics: USA, (2020).
[35] Polyak, B.T. Introduction to Optimization. Optimization Software Inc: New York, (1987).
[36] Liang, J. Convergence Rates of First-Order Operator Splitting Methods. Ph.D. Thesis, Department of Applied Mathematics, The University of Normandie, (2016). [https://hal.science/tel01388978/]
[37] Combettes, P.L. and Glaudin, L.E. Quasi-nonexpansive iterations on the affine hull of orbits: from Mann’s mean value algorithm to inertial methods. SIAM Journal on Optimization, 27(4), 2356–2380, (2017).
[38] Dong, Q.L., Cho, Y.J. and Rassias, T.M. General inertial Mann algorithms and their convergence analysis for nonexpansive mappings. In Applications of Nonlinear Analysis (pp.175-191). Springer Optimization and Its Applications, New York, USA: Springer, (2018).
[39] Dong, Q.L., Huang, J.Z., Li, X.H., Cho, Y.J. and Rassias, T.M. MiKM: multi-step inertial Krasnosel’skiˇı–Mann algorithm and its applications. Journal of Global Optimization, 73, 801–824, (2019).
[40] Malitsky, Y. Golden ratio algorithms for variational inequalities. Mathematical Programming,184, 383-410, (2020).
[41] Zhang, C. and Chu, Z. New extrapolation projection contraction algorithms based on the golden ratio for pseudo-monotone variational inequalities. AIMS Mathematics, 8(10), 23291- 23312, (2023).
[42] Iiduka, H. Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping. Mathematical Programming, 149, 131-165, (2015).
[43] Iiduka, H. Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping. Applied Mathematics and Computation, 217(13), 6315-6327, (2011).
[44] Iiduka, H. and Yamada, I. A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM Journal on Optimization, 19(4), 1881-1893, (2009).
[45] Harbau, M.H., Ugwunnadi, G.C., Jolaoso, L.O. and Abdulwahab, A. Inertial accelerated algorithm for fixed point of asymptotically nonexpansive mapping in real uniformly convex Banach spaces. Axioms, 10(3), 147, (2021).
[46] Dong, Q.L. and Yuan, H.B. Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping. Fixed Point Theory and Applications, 2015, 125, (2015).
[47] Ahmad, A., Kumam, P. and Harbau, M.H. Convergence theorems for common solutions of nonlinear problems and applications. Carpathian Journal of Mathematics, 40(2), 207-241, (2024).
[48] Kiri, A.I. and Abubakar, A.B. A family of conjugate gradient projection method for nonlinear monotone equations with applications to compressive sensing. Nonlinear Convex Analysis and Optimization: An International Journal on Numerical, Computation and Applications, 1(1), 47-65, (2022).
[49] Abubakar, J., Chaipunya, P., Kumam, P. and Salisu, S. A generalized scheme for split inclusion problem with conjugate like direction. Mathematical Methods of Operations Research, 101, 51-71, (2025).
[50] Salihu, N., Kumam, P. and Salisu, S. Two efficient nonlinear conjugate gradient methods for Riemannian manifolds. Computational and Applied Mathematics, 43, 415, (2024).
[51] Yang, Q. On variable-step relaxed projection algorithm for variational inequalities. Journal of Mathematical Analysis and Applications, 302(1), 166-179, (2005).
[52] Che, H., Zhuang, Y., Wang, Y. and Chen, H. A relaxed inertial and viscosity method for split feasibility problem and applications to image recovery. Journal of Global Optimization, 87, 619-639, (2023).
[53] Goebel, K. and Reich, S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (Vol. 83). Marcel Dekker: New York, (1984).
[54] Xu, H.K. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 26(10), 105018, (2010).
[55] Ceng, L.C., Ansari, Q.H. and Yao, J.C. An extragradient method for solving split feasibility and fixed point problems. Computers & Mathematics with Applications, 64(4), 633-642, (2012).
[56] He, S. and Yang, C. Solving the variational inequality problem defined on intersection of finite level sets. Abstract and Applied Analysis, 2013(1), 942315, (2013).
[57] Gibali, A., Thong, D.V. and Vinh, N.T. Three new iterative methods for solving inclusion problems and related problems. Computational and Applied Mathematics, 39, 187, (2020).
[58] Huang, G.B., Zhu, Q.Y. and Siew, C.K. Extreme learning machine: a new learning scheme of feedforward neural networks. In Proceedings, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541), pp. 985-990, Budapest, Hungary, (2004, July).
[59] Serre, D. Matrices, Theory and Applications (Vol. 216). Springer: New York, (2002).
[60] Ye, H., Cao, F. and Wang, D. A hybrid regularization approach for random vector functionallink networks. Expert Systems with Applications, 140, 112912, (2020).
[61] Wolberg, W., Mangasarian, O., Street, N. and Street, W. Breast Cancer Wisconsin (Diagnostic). UCI Machine Learning Repository, (1993).
[62] Demšar, J. Statistical comparisons of classifiers over multiple data sets. Journal of Machine Learning Research, 7, 1-30, (2006).
[63] Shukla, S. Game Theory for Security Investments in Cyber and Supply Chain Networks. Ph.D. Thesis, Department of Philosophy, The University of Massachusetts Amherst, (2017).
[64] Abd-Elhameed, W.M., Youssri, Y.H. and Atta, A.G. Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials. Journal of Mathematical Modeling, 12(2), 277–299, (2024).
[65] Kirk, D.E. Optimal Control Theory: An Introduction. Courier Corporation: USA, (2004).
[66] Youssri, Y.H. and Atta, A.G. Modal spectral Tchebyshev Petrov-Galerkin stratagem for the time-fractional nonlinear Burgers’ equation. Iranian Journal of Numerical Analysis and Optimization, 14(1), 172–199, (2024).

Details

Primary Language

English

Subjects

Mathematical Optimisation, Numerical and Computational Mathematics (Other)

Journal Section

Research Article

Authors

Abdulwahab Ahmad
0009-0000-0460-0204
Nigeria

Poom Kumam ^*
0000-0002-5463-4581
Thailand

Thidaporn Seangwattana
0000-0002-1240-1181
Thailand

Early Pub Date

July 15, 2025

Publication Date

June 30, 2025

Submission Date

December 3, 2024

Acceptance Date

May 12, 2025

Published in Issue

Year 2025 Volume: 5 Number: 2

DOI

https://doi.org/10.53391/mmnsa.1595657

IZ

https://izlik.org/JA52PK44FH

APA

Ahmad, A., Kumam, P., & Seangwattana, T. (2025). Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications. Mathematical Modelling and Numerical Simulation With Applications, 5(2), 307-347. https://doi.org/10.53391/mmnsa.1595657

AMA

1.Ahmad A, Kumam P, Seangwattana T. Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications. MMNSA. 2025;5(2):307-347. doi:10.53391/mmnsa.1595657

Chicago

Ahmad, Abdulwahab, Poom Kumam, and Thidaporn Seangwattana. 2025. “Strong Convergence Multi-Step Inertial Golden Ratio-Based Algorithms for Split Feasibility Problems With Applications”. Mathematical Modelling and Numerical Simulation With Applications 5 (2): 307-47. https://doi.org/10.53391/mmnsa.1595657.

EndNote

Ahmad A, Kumam P, Seangwattana T (June 1, 2025) Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications. Mathematical Modelling and Numerical Simulation with Applications 5 2 307–347.

IEEE

[1]A. Ahmad, P. Kumam, and T. Seangwattana, “Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications”, MMNSA, vol. 5, no. 2, pp. 307–347, June 2025, doi: 10.53391/mmnsa.1595657.

ISNAD

Ahmad, Abdulwahab - Kumam, Poom - Seangwattana, Thidaporn. “Strong Convergence Multi-Step Inertial Golden Ratio-Based Algorithms for Split Feasibility Problems With Applications”. Mathematical Modelling and Numerical Simulation with Applications 5/2 (June 1, 2025): 307-347. https://doi.org/10.53391/mmnsa.1595657.

JAMA

1.Ahmad A, Kumam P, Seangwattana T. Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications. MMNSA. 2025;5:307–347.

MLA

Ahmad, Abdulwahab, et al. “Strong Convergence Multi-Step Inertial Golden Ratio-Based Algorithms for Split Feasibility Problems With Applications”. Mathematical Modelling and Numerical Simulation With Applications, vol. 5, no. 2, June 2025, pp. 307-4, doi:10.53391/mmnsa.1595657.

Vancouver

1.Abdulwahab Ahmad, Poom Kumam, Thidaporn Seangwattana. Strong convergence multi-step inertial golden ratio-based algorithms for split feasibility problems with applications. MMNSA. 2025 Jun. 1;5(2):307-4. doi:10.53391/mmnsa.1595657