Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space
Abstract
:1. Introduction-Preliminaries
2. Results
Algorithm 1: Composite extragradient implicit method for the GSVI (2) with VIP and CFPP constraints. |
Initial Step. Given . Let be an arbitrary initial. Iteration Steps. Compute from the current as follows: Step 1. Calculate Step 2. Calculate ; Step 3. Calculate ; Step 4. Calculate , where u is a fixed element in C, , with and . Set and go to Step 1. |
Funding
Acknowledgments
Conflicts of Interest
References
- Qin, X.; An, N.T. Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput. Optim. Appl. 2019, 74, 821–850. [Google Scholar] [CrossRef] [Green Version]
- Sahu, D.R.; Yao, J.C.; Verma, M.; Shukla, K.K. Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization 2020. [Google Scholar] [CrossRef]
- Nguyen, L.V.; Qin, X. The Minimal time function associated with a collection of sets. ESAIM Control Optim. Calc. Var. 2020. [Google Scholar] [CrossRef]
- Ansari, Q.H.; Islam, M.; Yao, J.C. Nonsmooth variational inequalities on Hadamard manifolds. Appl. Anal. 2020, 99, 340–358. [Google Scholar] [CrossRef]
- An, N.T. Solving k-center problems involving sets based on optimization techniques. J. Global Optim. 2020, 76, 189–209. [Google Scholar] [CrossRef]
- Nguyen, L.V.; Qnsari, Q.H.; Qin, X. Weak sharpness and finite convergence for solutions of nonsmooth variational inequalities in Hilbert spaces. Appl. Math. Optim. 2020. [Google Scholar] [CrossRef]
- Nguyen, L.V.; Ansair, Q.H.; Qin, X. Linear conditioning, weak sharpness and finite convergence for equilibrium problems. J. Global Optim. 2020, 77, 405–424. [Google Scholar] [CrossRef]
- Korpelevich, G.M. The extragradient method for finding saddle points and other problems. Ekon. i Mat. Metod. 1976, 12, 747–756. [Google Scholar]
- Dehaish, B.A.B. Weak and strong convergence of algorithms for the sum of two accretive operators with applications. J. Nonlinear Convex Anal. 2015, 16, 1321–1336. [Google Scholar]
- Takahahsi, W.; Yao, J.C. The split common fixed point problem for two finite families of nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2019, 20, 173–195. [Google Scholar]
- Qin, X.; Wang, L.; Yao, J.C. Inertial splitting method for maximal monotone mappings. J. Nonlinear Convex Anal. 2020, in press. [Google Scholar]
- Takahashi, W.; Wen, C.F.; Yao, J.C. The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 2018, 19, 407–419. [Google Scholar] [CrossRef]
- Qin, X.; Yao, J.C. A viscosity iterative method for a split feasibility problem. J. Nonlinear Convex Anal. 2019, 20, 1497–1506. [Google Scholar]
- Liu, L. A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings. J. Nonlinear Convex Anal. 2019, 20, 471–488. [Google Scholar]
- Liu, L.; Qin, X.; Agarwal, R.P. Iterative methods for fixed points and zero points of nonlinear mappings with applications. Optimization 2019. [Google Scholar] [CrossRef]
- Chang, S.S. Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization 2018, 67, 1183–1196. [Google Scholar] [CrossRef]
- Qin, X.; Yao, J.C. Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators. J. Inequal. Appl. 2016, 2016, 232. [Google Scholar] [CrossRef] [Green Version]
- Abbas, H.A.; Aremu, K.O.; Jolaoso, L.O.; Mewomo, O.T. An inertial forward-backward splitting method for approximating solutions of certain optimization problem. J. Nonlinear Funct. Anal. 2020, 2020, 6. [Google Scholar]
- Kimura, Y.; Nakajo, K. Strong convergence for a modified forward-backward splitting method in Banach spaces. J. Nonlinear Var. Anal. 2019, 3, 5–18. [Google Scholar]
- Ceng, L.C. Asymptotic inertial subgradient extragradient approach for pseudomonotone variational inequalities with fixed point constraints of asymptotically nonexpansive mappings. Commun. Optim. Theory 2020, 2020, 2. [Google Scholar]
- Gibali, A. Polyak’s gradient method for solving the split convex feasibility problem and its applications. J. Appl. Numer. Optim. 2019, 1, 145–156. [Google Scholar]
- Tong, M.Y.; Tian, M. Strong convergence of the Tseng extragradient method for solving variational inequalities. Appl. Set-Valued Anal. Optim. 2020, 2, 19–33. [Google Scholar]
- Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces; Noordhoff: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Ceng, L.C.; Postolache, M.; Yao, Y. Iterative algorithms for a system of variational inclusions in Banach spaces. Symmetry 2019, 11, 811. [Google Scholar] [CrossRef] [Green Version]
- Aoyama, K.; Iiduka, H.; Takahashi, W. Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006, 2006, 35390. [Google Scholar] [CrossRef] [Green Version]
- Xu, H.K. Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16, 1127–1138. [Google Scholar] [CrossRef]
- Zhang, H.; Su, Y. Convergence theorems for strict pseudocontractions in q-uniformly smooth Banach spaces. Nonlinear Anal. 2009, 71, 4572–4580. [Google Scholar] [CrossRef]
- Lopez, G.; Martin-Marquez, V.; Wang, F.; Xu, H.K. Forward-backward splitting methods for accretive operators in Banach spaces. Abst. Anal. Appl. 2012, 2012, 109236. [Google Scholar] [CrossRef] [Green Version]
- Qin, X.; Cho, S.Y.; Yao, J.C. Weak and strong convergence of splitting algorithms in Banach spaces. Optimization 2020, 69, 243–267. [Google Scholar] [CrossRef]
- Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67, 2350–2360. [Google Scholar] [CrossRef]
- Bruck, R.E. Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179, 51–262. [Google Scholar] [CrossRef]
- Reich, S. Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75, 287–292. [Google Scholar] [CrossRef] [Green Version]
- Xue, Z.; Zhou, H.; Cho, Y.J. Iterative solutions of nonlinear equations for m-accretive operators in Banach spaces. J. Nonlinear Convex Anal. 2000, 1, 313–320. [Google Scholar]
- Maingé, P.E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16, 899–912. [Google Scholar] [CrossRef]
- Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 1996, 58, 267–288. [Google Scholar] [CrossRef]
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Cho, S.Y. Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space. Symmetry 2020, 12, 998. https://doi.org/10.3390/sym12060998
Cho SY. Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space. Symmetry. 2020; 12(6):998. https://doi.org/10.3390/sym12060998
Chicago/Turabian StyleCho, Sun Young. 2020. "Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space" Symmetry 12, no. 6: 998. https://doi.org/10.3390/sym12060998