Convergence of inertial hybrid splitting algorithms

TitleConvergence of inertial hybrid splitting algorithms
Publication TypeJournal Article
Year of Publication2018
AuthorsSemenov, VV
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2018.12.030
Issue12
SectionInformation Science and Cybernetics
Pagination30-36
Date Published12/2018
LanguageRussian
Abstract

Two new algorithms for solving the operator inclusion problems with maximal monotone operators acting in a Hilbert space are proposed. Algorithms are based on the inertial extrapolation and three well-known methods: Tseng forward-backward splitting algorithm and two hybrid algorithms for the approximation of fixed points of nonexpansive operators. Theorems on the strong convergence of the sequences generated by the algorithms are proved.
KeywordsHilbert space, hybrid algorithm, inertial method, maximal monotone operator, operator inclusion problem, strong convergence, Tseng algorithm
References: 
  1. Bauschke, H. H. & Combettes, P. L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Berlin etc.: Springer. doi: https://doi.org/10.1007/978-1-4419-9467-7
  2. Tseng, P. (2000). A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control and Optimization, 38, pp. 431-446. doi: https://doi.org/10.1137/S0363012998338806
  3. Semenov, V. V. (2014). Hybrid Splitting Methods for the System of Operator Inclusions with Monotone Operators. Cybernetics and Systems Analysis, 50, pp. 741-749. doi: https://doi.org/10.1007/s10559-014-9664-y
  4. Semenov, V. V. (2014). A Strongly Convergent Splitting Method for Systems of Operator Inclusions with Monotone Operators. J. Automation and Information Sciences, 46, No. 5, pp. 45-56. doi: https://doi.org/10.1615/JAutomatInfScien.v46.i5.40
  5. Alvarez, F. & Attouch, H. (2001). An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal., 9, pp. 3-11. doi: https://doi.org/10.1023/A:1011253113155
  6. Attouch, H., Chbani, Z. & Riahi, H. (2018). Combining fast inertial dynamics for convex optimization with Tikhonov regularization. J. Math. Anal. Appl., 457, pp. 1065-1094. doi: https://doi.org/10.1016/j.jmaa.2016.12.017
  7. Dong, Q. L., Lu, Y. Y. & Yang, J. (2016). The extragradient algorithm with inertial effects for solving the variational inequality. Optimization, 65, pp. 2217-2226. doi: https://doi.org/10.1080/02331934.2016.1239266
  8. Dong, Q. L., Yuan, H. B., Cho, Y. J. & Rassias, Th. M. (2018). Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim. Lett., 12, No. 1, pp. 87-102. doi: https://doi.org/10.1007/s11590-016-1102-9
  9. Lorenz, D. & Pock, T. (2015). An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging and Vision, 51, pp. 311-325. doi: https://doi.org/10.1007/s10851-014-0523-2
  10. Nakajo, K. & Takahashi, W. (2003). Strong convergence theorems for nonexpansive mappings and no nexpansive semigroups. J. Math. Anal. Appl., 279, pp. 372-379. doi: https://doi.org/10.1016/S0022-247X(02)00458-4
  11. Takahashi, W., Takeuchi, Y., & Kubota, R. (2008). Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl., 341, pp. 276-286. doi: https://doi.org/10.1016/j.jmaa.2007.09.062
  12. Verlan, D. A., Semenov, V. V. & Chabak, L. M. (2015). A Strongly Convergent Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators. J. Automation and Information Sciences, 47, No. 7, pp. 31-46. doi: https://doi.org/10.1615/JAutomatInfScien.v47.i7.40
  13. Denisov, S. V., Semenov, V. V. & Chabak, L. M. (2015). Convergence of the Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators. Cybernetics and Systems Analysis, 51, pp. 757-765. doi: https://doi.org/10.1007/s10559-015-9768-z