An accelerated conjugate method for split variational inclusion problems with applications

Yu Zhang; Xiaojun Ma; Yu Zhang; Xiaojun Ma

doi:10.3934/math.2025522

AIMS Mathematics

2025, Volume 10, Issue 5: 11465-11487. doi: 10.3934/math.2025522

Previous Article Next Article

Research article

An accelerated conjugate method for split variational inclusion problems with applications

Yu Zhang ¹,
Xiaojun Ma ^{2
,
,}

1.
School of Mathematics and Statistics, Big Data Modeling and Intelligent Computing Research Institute, Hubei University of Education, Wuhan 430205, Hubei, China
2.
School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, Shanxi, China

Received: 12 February 2025 Revised: 06 May 2025 Accepted: 09 May 2025 Published: 20 May 2025
MSC : 35A15, 47J20, 47J25, 49J40

In this work, split variational inclusion problems were investigated by combining new stepsizes and inertia with conjugate gradient methods in real Hilbert spaces, in which the inertial steps were used to speed up the convergent rate of the methods, and the new stepsizes not only avoided computing the operator norm, but also ensured that the strong convergence of the methods holds without Lipschitz continuity of the monotone operator. Also, the proximal operator was computed less than that in the original method. Further, the split feasibility and split minimization problems were considered. Finally, several examples were used for illustration and comparison.
- split variational inclusion problem,
- inertial conjugate gradient method,
- new stepsize,
- strong convergence
Citation: Yu Zhang, Xiaojun Ma. An accelerated conjugate method for split variational inclusion problems with applications[J]. AIMS Mathematics, 2025, 10(5): 11465-11487. doi: 10.3934/math.2025522

Related Papers:

Abstract

In this work, split variational inclusion problems were investigated by combining new stepsizes and inertia with conjugate gradient methods in real Hilbert spaces, in which the inertial steps were used to speed up the convergent rate of the methods, and the new stepsizes not only avoided computing the operator norm, but also ensured that the strong convergence of the methods holds without Lipschitz continuity of the monotone operator. Also, the proximal operator was computed less than that in the original method. Further, the split feasibility and split minimization problems were considered. Finally, several examples were used for illustration and comparison.

References

[1]	F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. https://doi.org/10.1023/A:1011253113155 doi: 10.1023/A:1011253113155
[2]	T. O. Alakoya, V. A. Uzor, O. T. Mewomo, A new projection and contraction method for solving split monotone variational inclusion, pseudomonotone variational inequality, and common fixed point problems, Comp. Appl. Math., 42 (2023), 3. https://doi.org/10.1007/s40314-022-02138-0 doi: 10.1007/s40314-022-02138-0
[3]	J. Abubakar, P. Chaipunya, P. Kumam, S. Salisu, A generalized scheme for split inclusion problem with conjugate like direction, Math. Meth. Oper. Res., 101 (2025), 51–71. https://doi.org/10.1007/s00186-024-00882-z doi: 10.1007/s00186-024-00882-z
[4]	T. O. Alakoya, O. T. Mewomo, Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems, Comp. Appl. Math., 41 (2022), 39. https://doi.org/10.1007/s40314-021-01749-3 doi: 10.1007/s40314-021-01749-3
[5]	F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773–782. https://doi.org/10.1137/S1052623403427859 doi: 10.1137/S1052623403427859
[6]	K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal.-Theor., 67 (2007), 2350–2360. https://doi.org/10.1016/j.na.2006.08.032 doi: 10.1016/j.na.2006.08.032
[7]	C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl., 18 (2002), 441–453. https://doi.org/10.1088/0266-5611/18/2/310 doi: 10.1088/0266-5611/18/2/310
[8]	C. Byrne, Y. Censor, A. Gibali, S. Reich, Weak and strong convergence of algorithms for the split common null point problem, arXiv: 1108.5953.
[9]	D. Butnariu, A. N. Iusem, Totally convex functions for fixed points computation and infinite dimensional optimization, Dordrecht: Springer, 2000. https://doi.org/10.1007/978-94-011-4066-9
[10]	Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071–2084. https://doi.org/10.1088/0266-5611/21/6/017 doi: 10.1088/0266-5611/21/6/017
[11]	Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301–323. https://doi.org/10.1007/s11075-011-9490-5 doi: 10.1007/s11075-011-9490-5
[12]	C.-S. Chuang, I.-J. Lin, New strong convergence theorems for split variational inclusion problems in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 176. https://doi.org/10.1186/s13660-015-0697-1 doi: 10.1186/s13660-015-0697-1
[13]	X. Chang, L. Xu, J. Cao, A splitting preconditioned primal-dual algorithm with interpolation and extrapolation for bilinear saddle point problem, Numer. Algor., in press. https://doi.org/10.1007/s11075-024-01974-x
[14]	P. Cholamjiak, S. Suantai, S. Kesornprom, Linearized splitting algorithm with inertial extrapolation for solving split variational inclusion problem and applications, Optimization, in press. https://doi.org/10.1080/02331934.2024.2444629
[15]	S.-C. Chang, J. C. Yao, M. Liu, L. C. Zhao, J. H. Zhu, Shrinking projection algorithm for solving a finite family of quasi-variational inclusion problems in Hadamard manifold, Rev. Real Acad. Cienc, Exactas Fis. Nat. Ser. A-Mat., 115 (2021), 166. https://doi.org/10.1007/s13398-021-01105-4 doi: 10.1007/s13398-021-01105-4
[16]	C.-S. Chuang, Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications, Optimization, 66 (2017), 777–792. https://doi.org/10.1080/02331934.2017.1306744 doi: 10.1080/02331934.2017.1306744
[17]	Z. Chen, H. Shao, P. Liu, G. Li, X. Rong, An efficient hybrid conjugate gradient method with an adaptive strategy and applications in image restoration problems, Appl. Numer. Math., 204 (2024), 362–379. https://doi.org/10.1016/j.apnum.2024.06.020 doi: 10.1016/j.apnum.2024.06.020
[18]	S. Dey, C. Izuchukwu, A. Taiwo, S. Reich, New iterative algorithms for solving split variational inclusions, J. Glob. Optim., 91 (2025), 587–609. https://doi.org/10.1007/s10898-024-01444-7 doi: 10.1007/s10898-024-01444-7
[19]	K. Z. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge: Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511526152
[20]	S. Husain, M. U. Khairoowala, M. Furkan, Inertial modified S-iteration method for Cayley inclusion problem and fixed point problem, J. Appl. Math. Comput., 70 (2024), 5443–5457. https://doi.org/10.1007/s12190-024-02185-2 doi: 10.1007/s12190-024-02185-2
[21]	S. Kesornprom, P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optimization, 68 (2019), 2369–2395. https://doi.org/10.1080/02331934.2019.1638389 doi: 10.1080/02331934.2019.1638389
[22]	P. Liu, Z. Yuan, Y. Zhuo, H. Shao, Two efficient spectral hybrid CG methods based on memoryless BFGS direction and Dai-Liao conjugacy condition, Optim. Method. Soft., 39 (2024), 1445–1463. https://doi.org/10.1080/10556788.2024.2364203 doi: 10.1080/10556788.2024.2364203
[23]	A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275–283. http://doi.org/10.1007/s10957-011-9814-6 doi: 10.1007/s10957-011-9814-6
[24]	X. Ma, H. Liu, X. Li, The iterative method for solving the proximal split feasibility problem with an application to LASSO problem, Comp. Appl. Math., 41 (2022), 5. https://doi.org/10.1007/s40314-021-01703-3 doi: 10.1007/s40314-021-01703-3
[25]	X. Ma, H. Liu, X. Li, Two optimization approaches for solving split variational inclusion problems with applications, J. Sci. Comput., 91 (2022), 58. https://doi.org/10.1007/s10915-022-01832-9 doi: 10.1007/s10915-022-01832-9
[26]	X. Ma, Z. Jia, Q. Li, On inertial non-lipschitz stepsize algorithms for split feasibility problems, Comp. Appl. Math., 43 (2024), 431. https://doi.org/10.1007/s40314-024-02922-0 doi: 10.1007/s40314-024-02922-0
[27]	G. Marino, H.-K. Xu, Convergence of generalized proximal point algorithm, Commun. Pure Appl. Anal., 3 (2004), 791–808. http://doi.org/10.3934/cpaa.2004.3.791 doi: 10.3934/cpaa.2004.3.791
[28]	P.-E. Maing$\acute{e}$, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912. https://doi.org/10.1007/s11228-008-0102-z doi: 10.1007/s11228-008-0102-z
[29]	Y. Nesterov, A method for solving the convex programming problem with convergence rate $O(1/k^2)$, Proceedings of the USSR Academy of Sciences, 269 (1983), 543–547.
[30]	G. N. Ogwo, C. Izuchukwu, O. T. Mewomo, Relaxed inertial methods for solving split variational inequality problems without product space formulation, Acta Math. Sci., 42 (2022), 1701–1733. https://doi.org/10.1007/s10473-022-0501-5 doi: 10.1007/s10473-022-0501-5
[31]	M. O. Osilike, S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Model., 32 (2000), 1181–1191. https://doi.org/10.1016/S0895-7177(00)00199-0 doi: 10.1016/S0895-7177(00)00199-0
[32]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Computational Mathematics and Mathematical Physics, 4 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
[33]	H. Shao, H. Guo, X. Wu, P. Liu, Two families of self-adjusting spectral hybrid DL conjugate gradient methods and applications in image denoising, Appl. Math. Model., 118 (2023), 313–411. https://doi.org/10.1016/j.apm.2023.01.018 doi: 10.1016/j.apm.2023.01.018
[34]	B. Tan, X. Qin, J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), 20. https://doi.org/10.1007/s10915-021-01428-9 doi: 10.1007/s10915-021-01428-9
[35]	T. M. Tuyen, N. S. Ha, An algorithm for approximating solutions of the split variational inclusion problem, Optimization, 74 (2025), 871–982. https://doi.org/10.1080/02331934.2023.2269981 doi: 10.1080/02331934.2023.2269981
[36]	D. V. Thong, V. T. Dung, Y. J. Cho, A new strong convergence for solving split variational inclusion problems, Numer. Algor., 86 (2021), 565–591. https://doi.org/10.1007/s11075-020-00901-0 doi: 10.1007/s11075-020-00901-0
[37]	Y. Tang, A. Gibali, New self-adaptive step size algorithms for solving split variational inclusion problems and its applications, Numer. Algor., 83 (2020), 305–331. https://doi.org/10.1007/s11075-019-00683-0 doi: 10.1007/s11075-019-00683-0
[38]	W. Takahashi, Nonlinear functional analysis: fixed point theory and its applications, Yokohama: Yokohama Publishers, 2000.
[39]	D. Van Hieu, S. Reich, P. K. Anh, N. H. Ha, A new proximal-like algorithm for solving split variational inclusion problems, Numer. Algor., 89 (2022), 811–837. https://doi.org/10.1007/s11075-021-01135-4 doi: 10.1007/s11075-021-01135-4
[40]	P. Xia, G. Cai, Q.-L. Dong, A strongly convergent viscosity-type inertial algorithm with self adaptive stepsize for solving split variational inclusion problems in Hilbert spaces, Netw. Spat. Econ., 23 (2023), 931–952. https://doi.org/10.1007/s11067-023-09600-4 doi: 10.1007/s11067-023-09600-4

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)