Abstract
The paper concerns with a new iterative method for solving a monotone variational inclusion problem in a Hilbert space. The method is of the proximal contraction type incorporated with the regularization technique. Under the prediction stepsize conditions, we establish the strong convergence of the iterative sequences generated by the method to a particular solution of the problem satisfying a variational inequality problem. Finally, we give some numerical examples to illustrate the behavior of the new method in comparison with existing ones.
Similar content being viewed by others
References
Alber YI, Ryazantseva I (2006) Nonlinear Ill-posed Problems of Monotone Type. Springer, Dordrecht
Anh PK, Buong Ng., Hieu DV (2014) Parallel methods for regularizing systems of equations involving accretive operators. Appl Anal 93:2136-2157
Bakushinsky AB (1977) Methods for solving monotonic variational inequalities based on the principle of iterative regularization. Comput Math Math Phys 17:12-24
Barbara K, Neubauer A, Scherzer O (2008) Iterative regularization methods for nonlinear Ill-Posed problems. Walter de Gruyter, Berlin
Bauschke HH, Combettes PL (2011) Convex analysis and monotone operator theory in hilbert spaces. Springer, New York
Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2(1):183-202
Bot RI, Csetnek ER, Hendrich C (2015) Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl Math Comput 256:472-487
Brezis H, Chapitre II (1973) Operateurs maximaux monotones. North-holland Math Stud 5:19-51
Bruck RE (1974) Strong convergent iterative method for the solution 0 ?Ux for a maximal monotone operator U in Hilbert spaces. J Math Anal Appl 48:114-126
Cai X, Gu G, He B (2014) On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput Optim Appl 57:339-363
Cholamjiak P (2016) A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces. Numer Algorithm 71:915-932
Cholamjiak P, Thong DV, Cho YJ (2019) A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl Math. https://doi.org/10.1007/s10440-019-00297-7
Cottle RW, Yao JC (1992) Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl 75:281-295
Davis D, Yin WT (2017) A three-operator splitting scheme and its optimization applications. Set Valued Var Anal 25:829-858
Dempe S (2002) Foundations of bilevel programming. Kluwer Academic Publishers, Berlin
Dempe S, Kalashnikov V, Perez-Valdes GA, Kalashnykova N (2015) Bilevel programming problems. Springer, Berlin
Dong Q-L, Cho YJ, Rassias TM (2018a) The projection and contraction methods for finding common solutions to variational inequality problems. Optim Lett 12:1871-1896
Dong QL, Jiang D, Gibali A (2018b) A modified subgradient extragradient method for solving the variational inequality problem. Numer Algorithm 79:927-940
Dong QL, Cho JY, Zhong LL, Rassias M. T. h. (2018c) Inertial projection and contraction algorithms for variational inequalities. J Glob Optim 70:687-704
Dong QL, Yang J, Yuan HB (2019) The projection and contraction algorithm for solving variational inequality problems in Hilbert spaces. J Nonlinear Convex Anal 20:111-122
He BS (1997) A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim 35:69-76
He X, Huang N, Li X (2019) Modified projection methods for solving multi-valued variational inequality without monotonicity. Networks and Spatial Economics. https://doi.org/10.1007/s11067-019-09485-2
Hieu DV, Anh PK, Muu LD (2020) Modified forward - backward splitting method for variational inclusions. 4OR - Q J Oper Res. https://doi.org/10.1007/s10288-020-00440-3
Hieu DV, Anh PK, Muu LD, Strodiot JJ (2021) Iterative regularization methods with new stepsize rules for solving variational inclusions. J Appl Math Comput. https://doi.org/10.1007/s12190-021-01534-9
Hieu DV, Vy LV, Quy PK (2019) Three-operator splitting algorithm for a class of variational inclusion problems. Bull. Iranian Math Soc. https://doi.org/10.1007/s41980-019-00312-5
Hieu DV, Cho YJ, Xiao YB, Kumam P (2020) Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces. Vietnam J Math. https://doi.org/10.1007/s10013-020-00447-7
Huang NJ (1998) A new completely general class of variational inclusions with noncompact valued mappings. Comput Math Appl 35(10):9-14
Khoroshilova EV (2013) Extragradient-type method for optimal control problem with linear constraints and convex objective function. Optim. Lett. 7:1193-1214
Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekon Mat Metody 12:747-756
Lions PL, Mercier B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal 16:964-979
Lopez G, Martin-Marquez V, Wang F, Xu H (2012) Forward-backward splitting methods for accretive operators in Banach spaces. Abstr. Appl. Anal. 2012(109236)
Rockafellar RT (1976) Monotone operators and the proximal point algorithms. SIAM J Control Optim 14(5):877-898
Seydenschwanz M (2015) Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions. Comput Optim Appl 629:731-760
Solodov MV, Svaiter BF (1999) A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator. Set-Valued Anal 7(4):323-345
Shehu Y (2019) Convergence results of forward-backward algorithms for sum of monotone operators in Banach spaces. Results Math 74(138):1-24
Shehu Y, Iyiola OS (2020) Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Appl Numer Math 157:315-337
Shehu Y, Dong QL, Jiang D (2019) Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68:385-409
Sun DF (1996) A class of iterative methods for solving nonlinear projection equations. J Optim Theory Appl 91:123-140
Takahashi W (2000) Nonlinear functional Analysis-Fixed point theory and its applications. Yokohama Publishers, Yokohama
Thong DV, Cholamjiak P (2019) Strong convergence of a forward-backward splitting method with a new stepsize for solving monotone inclusions. Comput Appl Math 38(94)
Thong DV, Li X-H, Dong Q-L, Cho YJ, Rassias TM (2020) A projection and contraction method with adaptive step sizes for solving bilevel pseudo-monotone variational inequality problems. Optimization. https://doi.org/10.1080/02331934.2020.1849206
Tseng P (2000) A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim 38:431-446
van den Berg E, Friedlander MP (2007) SPGL1: a solver for large-scale sparse reconstruction. Version 1.9. Accessed 2015. http://www.cs.ubc.ca/labs/scl/spgl1
Xu H (2002) Another control condition in an iterative method for nonexpansive mappings. Bull Austral Math Soc 65:109-113
Xu H (2006) A Regularization method for the proximal point algorithm. J Glob Optim 36:115-125
Zeng LC, Guu SM, Yao JC (2005) Characterization of H-monotone operators with applications to variational inclusions. Comput Math Appl 50:329-337
Zhang C, Wang Y (2018) Proximal algorithm for solving monotone variational inclusion. Optimization 67:1197-1209
Zhan-wei LV, Cui YL, Song ZP (2005) A new class of extended variational inclusions with H-monotone operator. J Yanan Univ 24(3):9-11
Acknowledgements
The authors sincerely thank the Editor and two anonymous referees for their valuable comments and suggestions which helped us to improve the original version of this paper. The research of the first author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06.
Author information
Authors and Affiliations
Corresponding author
Additional information
Disclosure Statement
There are no conflicts of interest to this work.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Van Hieu, D., Anh, P.K. & Ha, N.H. Regularization Proximal Method for Monotone Variational Inclusions. Netw Spat Econ 21, 905–932 (2021). https://doi.org/10.1007/s11067-021-09552-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-021-09552-7