Abstract
Using iterative regularizations, we introduce a new proximal-like algorithm for solving split variational inclusion problems in Hilbert space and establish a strong convergence theorem for it. As applications, we also investigate the split feasibility and split optimization problems. Several numerical experiments are presented in support of our theoretical findings.
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Alber, Y., Butnariu, D., Kassay, G.: Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications. Nonconvex Optimization and Its Applications. https://doi.org/10.1007/0-387-24276-7-9, p 79. Springer, Boston (2005)
Alber, Y., Butnariu, D., Ryazantseva, I.: Regularization methods for ill-posed inclusions and variational inequalities with domain perturbations. J Nonlinear Convex Anal. 2, 53–79 (2001)
Alber, Y., Ryazantseva, I.: Nonlinear Ill-posed Problems of Monotone Type. Springer, Dordrecht (2006)
Brézis, H.: Operateurs Maximaux Monotones. North-Holland. Math. Stud. 5, 1–183 (1973)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problems. Inverse Prob. 18, 441–453 (2002)
Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J Nonlinear Convex Anal. 13, 759–775 (2012)
Censor, Y., Bortfeld, T., Martin, B., Trofmov, A.: A unifed approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Censor, Y., Elfving, T.: A multiprojection algorithms using Bragman projection in a product space. Numer Algorithm. 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithm. 59, 301–323 (2012)
Censor, Y., Gibali, A., Reich, S.: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal. 75, 4596–4603 (2012)
Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set Valued Var Anal. 20, 229–247 (2012)
Censor, Y., Segal, A.: Iterative projection methods in biomedical inverse problems. In: Censor, Y., Jiang, M., Louis, A. K. (eds.) Mathematical methods in biomedical imaging and intensity-modulated therapy, IMRT, pp 65–96. Edizioni della Norale, Pisa (2008)
Chuang, C.-S.: Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem. Optimization 65, 859–876 (2016)
Chuang, C. -S.: Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl. 2013, 350 (2013)
Combettes, P. L.: The convex feasibility problem in image recovery. Adv Imaging Electron Phys. 95, 155–270 (1996)
Cottle, R. W., Yao, J. C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)
Hieu, D. V.: Projection methods for solving split equilibrium problems. J. Ind. Manag. Optim. 16, 2331–2349 (2020)
Hieu, D. V.: Two hybrid algorithms for solving split equilibrium problems. Inter. J. Comput. Math. 95, 561–583 (2018)
Hieu, D. V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)
Hieu, D. V., Cho, Y. J., Xiao, Y -B, Kumam, P.: Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces. Vietnam J. Math. https://doi.org/10.1007/s10013-020-00447-7 (2020)
Hurt, N. E.: Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction. Kluwer Academic, Dordrecht (1989)
Long, L. V., Thong, D. V., Dung, V. T.: New algorithms for the split variational inclusion problems and application to split feasibility problems. Optimization 68, 2339–2367 (2019)
Moudafi, A.: A Regularized Hybrid Steepest Descent Method for Variational Inclusions. Numer. Funct. Anal. Optim. 33(1), 39–47 (2012)
Moudafi, A.: Split monotone variational inclusions. J Optim Theory Appl. 150, 275–283 (2011)
Stark, H.: Image Recovery: Theory and Applications. Academic Press, Orlando (1987)
Sitthithakerngkiet, K., Deepho, J., Kumam, P.: A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems. Appl. Math. Comput. 250, 986–1001 (2015)
Xu, H.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65, 109–113 (2002)
Funding
The research of the first author is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06. The second author was partially supported by the Israel Science Foundation (Grant no. 820/17), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.
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Van Hieu, D., Reich, S., Anh, P.K. et al. A new proximal-like algorithm for solving split variational inclusion problems. Numer Algor 89, 811–837 (2022). https://doi.org/10.1007/s11075-021-01135-4
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DOI: https://doi.org/10.1007/s11075-021-01135-4