Abstract
This paper studies the multiple-sets split feasibility problem in Hilbert spaces. To solve this problem, we propose a new algorithm and establish a strong convergence theorem for it. Our scheme combines the hybrid projection method with the proximal point algorithm. We show that the iterative method converges strongly under weaker assumptions than the ones used recently by Yao et al. (Optimization 69(2):269–281, 2020). We also study an application to the split feasibility problem and give a strong convergence result to the minimum-norm solution to the problem. Thereafter, some numerical examples are conducted in order to illustrate the convergence analysis of the considered methods as well as compare our results to the related ones introduced by Buong (Numer Algorithms 76:783–798, 2017) and Yao et al. (2020). We end this paper by considering an application of our method to a class of optimal control problems and compare our result with the one introduced by Anh et al. (Acta Math Vietnam 42:413–429, 2017).
Similar content being viewed by others
Data availability
No data was used for the research described in the article.
References
Agarwal RP, O’Regan D, Sahu DR (2009) Fixed point theory for Lipschitzian-type map**s with applications. Springer, New York
Anh PK, Anh TV, Muu LD (2017) On bilevel split pseudo monotone variational inequality problems with applications. Acta Math Vietnam 42:413–429
Bauschke HH, Combettes PL (2011) Convex analysis and monotone operator theory in Hilbert spaces. Springer, New York
Buong N (2017) Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces. Numer Algorithms 76:783–798
Byrne C (2002) Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl 18:441–453
Censor Y, Elfving T (1994) A multi-projection algorithm using Bregman projections in a product space. Numer Algorithms 8:221–239
Censor Y, Segal A (2008) Iterative projection methods in biomedical inverse problems. Mathematical methods in biomedical imaging and intensity-modulate therapy, IMRT. Springer, Berlin
Censor Y, Elfving T, Kopf N, Bortfeld T (2005) The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl 21:2071–2084
Censor Y, Bortfeld T, Martin B, Trofimov A (2006) A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol 51:2353–2365
Censor Y, Motova A, Segal A (2007) Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J Math Anal Appl 327:1244–1256
Censor Y, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Algorithms 59:301–323
Goebel K, Kirk WA (1990) Topics in metric fixed point theory. Cambridge studies in advanced mathematics, vol 28. Cambridge University Press, Cambridge
Goebel K, Reich S (1984) Uniform convexity, hyperbolic geometry, and nonexpansive map**s. Marcel Dekker, New York
Masad E, Reich S (2007) A note on the multiple-set split convex feasibility problem in Hilbert space. J Nonlinear Convex Anal 8:367–371
Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 149:75–88
Thuy NTT (2022) A strong convergence theorem for an iterative method for solving the split variational inequalities in Hilbert spaces. Vietnam J Math 50:69–86
Thuy NTT, Nghia NT (2023) A new iterative method for solving the multiple-set split variational inequality problem in Hilbert spaces. Optimization 72(6):1549–1575
Tuyen TM, Thuy NTT, Trang NM (2019) A strong convergence theorem for a parallel iterative method for solving the split common null point problem in Hilbert spaces. J Optim Theory Appl 183:271–291
Xu HK (2006) A variable Krasnoselskii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl 22:2021–2034
Xu HK (2010) Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26(10):105018
Yao Y, Postolache M, Zhu Z (2020) Gradient methods with selection technique for the multiple-sets split feasibility problem. Optimization 69(2):269–281
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Communicated by Andreas Fischer.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Nguyen Thi Thu, T., Nguyen Trung, N. A hybrid projection method for solving the multiple-sets split feasibility problem. Comp. Appl. Math. 42, 292 (2023). https://doi.org/10.1007/s40314-023-02416-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02416-5