Abstract
In this paper, we introduce a new class of hybrid inertial and contraction proximal point algorithm for the variational inclusion problem of the sum of two mappings in Hilbert spaces. We prove that the proposed algorithm converges strongly to a solution of the variational inclusion problem whenever its solution set is nonempty and the single-valued mapping f is Lipschitz continuous, monotone, and the set-valued mapping \({\mathscr{A}}\) is maximal monotone in infinite-dimensional real Hilbert spaces. Our work generalize and extend some related existing results in the literature. Finally, we illustrate the numerical performance of our Algorithm 1 and we give an application to the split feasibility problem.
Similar content being viewed by others
Data availability
The author acknowledges that the data presented in this study must be deposited and made publicly available in an acceptable repository, prior to publication.
References
Zhang, C., Wang, Y.: Proximal algorithm for solving monotone variational inclusion. Optimization 67(8), 1197–1209 (2018)
Rockafellar, R.T.: Monotone operators and the proximal point algorithms. SIAM J. Control Optim. 14(5), 877–898 (1976)
Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in hilbert space. SIAM J. Optim. 14(3), 773–782 (2004)
Bruck, R.E.: Asymptotic convergence of nonlinear contraction semigroups in Hilbert space. J. Funct. Anal. 18, 15–26 (1975)
Tiel, J.V.: Convex Analysis: an Introductory Text. Wiley, New York (1984)
Dey, S., Vetrivel, V., Xu, H.K.: A neural network method for monotone variational inclusions. J. Nonlinear Convex Anal. 20(11), 2387–2395 (2019)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)
Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. CR Acad. Sci. 258, 4413–4416 (1964)
Noor, M.D.: Well-posed variational inequalities. J. Appl. Math. Comput. 11(1-2), 165–172 (2003)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Dong, Q.L., Lu, Y.Y., Yang, J., He, S.: Approximately solving multi-valued variational inequalities by using a projection and contraction algorithm. Numer. Algorithms 76(3), 799–812 (2017)
Yamada, H.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithms in feasibility and optimization and their applications 8(1), 473–504 (2001)
Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011)
Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)
He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)
Dong, Q.L., Yang, J.F., Yuan, H.B.: The projection and contraction algorithm for solving variational inequality problems in Hilbert space. J. Nonlinear Convex Anal. 20(1), 111–122 (2019)
Huang, N.J.: A new completely general class of variational inclusions with noncompact valued mappings. Comput. Math. Appl. 35(10), 9–14 (1998)
Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123–140 (1996)
Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, M.: Inertial projection contraction algorithms for variational inequalities. J. Global Optim. 70, 687–704 (2018)
Thong, D.V., Triet, N.A., Li, X.H., Dong, Q.L.: Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems. Numer. Algorithms 83, 1123–1143 (2020)
Antipin, A.S.: On a method for convex programs using a symmetrical modification of the lagrange function. Ekon Matematicheskie Metody 12, 1164–1173 (1976)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon Matematicheskie Metody 12, 747–756 (1976)
Malitsky, Y.: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)
Mainge, P.E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)
Cai, X.J., Gu, G.Y., He, B.S.: On the \(o(\frac {1}{t})\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)
Moudafi, A.: Split monotone variational inclusion. J. Optim. Theory Appl. 150, 275–283 (2011)
Zeng, L.C., Guu, S.M., Yao, J.C.: Characterization of H-monotone operators with applications to variational inclusions. Comput. Math. Appl. 50 (3-4), 329–337 (2005)
Fang, Y.P., Huang, N.J.: H-monotone operator resolvent operator technique for variational inclusion. Appl. Math Comput. 145, 795–803 (2006)
Wei, L.V.Z., Cui, Y.L., Song, Z.P.: A new class of extended variational inclusions with H-monotone operator. J. Yanan Univ. 24(3), 9–11 (2005)
Verma, R.U.: Approximation solvability of a class of A-monotone variational inclusion problems. J. Korean Soc. Ind. Appl. Math. 8(1), 55–66 (2004)
Verma, R.U.: A-monotone nonlinear relaxed co-coercive variational inclusions. Cent. Eur. J. Math. 5, 386–396 (2007)
Xiao, Y.B., Huang, N.J., Wong, M.M.: Well-posedness of hemivariational inequalities and inclusion problems. Taiwan. J. Math. 15, 1261–1276 (2011)
Chuang, C.S.: Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization 66 (5), 777–792 (2017)
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(12), 2339–2367 (2019)
Geoffroy, M.H., Alexis, C.J., Piétrus, A.: A Hummel–Seebeck type method for variational inclusions. Optimization 58(4), 389–399 (2009)
Rosasco, L., Villa, S., Vu, B.C.: A stochastic inertial forward-backward splitting algorithm for multivariate monotone inclusions. Optimization 65(6), 1293–1314 (2016)
Luc, D.T., Tan, N.X.: Existence conditions in variational inclusions with constraints. Optimization 53(5-6), 505–515 (2004)
Arias, L.B., Rivera, S.L.: A projected primal–dual method for solving constrained monotone inclusions. J. Optim. Theory Appl. 180, 907–924 (2019)
Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)
Bot, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)
Majee, P., Nahak, C.: On inertial proximal algorithm for split variational inclusion problems. Optimization 67(10), 1701–1716 (2018)
Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)
Bot, R.I., Csetnek, E.R.: A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36, 951–963 (2015)
Solodov, M.V., Svaiter, B.F.: A hybrid approximate extragradient-proximal point algorithm using the enlargementof a maximal monotone operator. Set-Valued Anal. 7(4), 323–345 (1999)
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Zirilli, F., Aluffi, F., Parisi, V.: DAFNE : a differential equations algorithm for non-linear equations. ACM Trans. Math. Software 10, 317–324 (1984)
Antipin, A.S.: Minimization of convex functions on convex sets by means of differential equations. Differential Equ. 30, 1365–1375 (1994)
Cholamjiak, W., Cholamjiak, P., Suantai, S.: An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J. Fixed Point Theory Appl., vol. 20(42) (2018)
Zhang, C., Dong, Q.L., Chen, J.: Multi-step inertial proximal contraction algorithms for monotone variational inclusion problems. Carpathian J. Math. 36(1), 159–177 (2020)
Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)
Censor, Y., Elfving, T.: A multi projection algorithm using Bregman projection in a product space. Numer. Algorithms 8, 221–239 (1994)
Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1998)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces, vol. 2057, Lecture Notes in Mathematics. Springer, Berlin (2012)
Acknowledgements
The author would like to thank the referee for his valuable suggestions to improve the earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no competing interests.
Additional information
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 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
Dey, S. A hybrid inertial and contraction proximal point algorithm for monotone variational inclusions. Numer Algor 93, 1–25 (2023). https://doi.org/10.1007/s11075-022-01400-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-022-01400-0
Keywords
- Hilbert space
- Zero point
- Set-valued mapping
- Variational inclusion
- Differential inclusion
- Monotonicity
- Maximal monotonicity
- Resolvent mapping
- Inertial algorithm
- Contraction algorithm
- Lipschitz continuity
- Normal cone
- Subdifferential
- Strong convergence
- Split feasibility problem