Viscosity-type extragradient algorithm for finding common solution of pseudomonotone equilibrium problem and fixed point problem in Hilbert space
Tóm tắt
The primary goal of this research is to find a common solution to the equilibrium problem for pseudomonotone bi-functions satisfying the Lipschitz-type condition as well as the fixed point problem for
$$\psi -$$
strongly quasi-nonexpansive mappings in the context of real Hilbert space by combining two different approaches. A viscosity-type extragradient algorithm is presented for solving the problems listed above. Furthermore, with a set of reasonable assumptions, a strong convergence theorem is presented. The fundamental advantage of the suggested approach is that it does not require the use of a linesearch procedure or the knowledge of Lipschitz-type constants in advance, which is a significant advantage. Moreover, we give a numerical example to support and justify our proposed algorithm. In this sense, the findings of this study generalise and extend certain previously published findings.
Tài liệu tham khảo
Bauschke, H.H., Combettes, P.L., et al. 2011. Convex analysis and monotone operator theory in Hilbert spaces vol. 408. New York: Springer.
Blum, E. 1994. From optimization and variational inequalities to equilibrium problems. The Mathematics Student 63: 123–145.
Burachik, R.S., C.Y. Kaya, and M. Mammadov. 2010. An inexact modified subgradient algorithm for nonconvex optimization. Computational Optimization and Applications 45 (1): 1–24.
Ceng, Lu-Chuan, Jen-Chih, Yao, Yekini, Shehu. 2022. On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints. Journal of Inequalities and Applications 1: 1-28.
Ceng, Lu-Chuan, Meijuan, Shang. 2021. Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings. Optimization 4: 715-740.
Ceng, Lu-Chuan, Qing, Yuan. 2019. Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems. Journal of Inequalities and Applications 1: 1–20.
Ceng, L.C., et al. 2020. A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems. Fixed Point Theory 21 (1): 93–108.
Ceng, L.C., et al. 2021. Pseudomonotone variational inequalities and fixed points. Fixed Point Theory 22 (2): 543–558.
Ceng, L.C., et al. 2021. Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization 70 (5–6): 1337–1358.
Ceng, Lu-Chuan. 2021. Modified inertial subgradient extragradient algorithms for pseudomonotone equilibrium problems with the constraint of nonexpansive mappings. Journal of Nonlinear and Variational Analysis 5: 281–297.
Ceng, L.C., and C.S. Fong. 2021. On strong convergence theorems for a viscosity-type extragradient method. Filomat 35 (3): 1033–1043.
Chen, J., Y.-C. Liou, Z. Wan, and J.-C. Yao. 2015. A proximal point method for a class of monotone equilibrium problems with linear constraints. Operational Research 15 (2): 275–288.
Cho, S.Y. 2020. A monotone Bregman projection algorithm for fixed point and equilibrium problems in a reflexive Banach space. Filomat 34 (5): 1487–1497.
Fan, J., Liu, L., Qin, X. 2019. A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities. Optimization.
Fan, K. 1972. A minimax inequality and applications. Inequalities 3: 103–113.
Giandomenico, M. 2003. On auxiliary principle for equilibrium problems. In Equilibrium Problems and Variational Models, pp. 289-298. New York: Springer.
Gibali, A., et al. 2019. Gradient projection-type algorithms for solving equilibrium problems and its applications. Computational and Applied Mathematics 38 (3): 1–18.
Goebel, K., Simeon, R. 1984. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York: Marcel Dekker.
He, Long. et al. 2021. Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule. Journal of Inequalities and Applications 1: 1-37.
Husain, S., and M. Asad. 2022. Solving equilibrium problem and fixed point problem by normal s-iteration process in Hilbert space. European Journal of Mathematical Analysis 2: 1–10.
Jadamba, B., A.A. Khan, and F. Raciti. 2014. Regularization of stochastic variational inequalities and a comparison of an Lp and a sample-path approach. Nonlinear Analysis: Theory, Methods & Applications 94: 65–83.
Kreyszig, E. 1978. Introductory functional analysis with applications. New York: John Wiley & Sons. Inc.
Mainge, P.-E. 2008. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM Journal on Control and Optimization 47 (3): 1499–1515.
Mastroeni, G. 2003. Gap functions for equilibrium problems. Journal of Global Optimization 27 (4): 411–426.
Moudafi, A. 2000. Viscosity approximation methods for fixed-points problems. Journal of mathematical analysis and applications 241 (1): 46–55.
Muu, L., and W. Oettli. 1992. Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Analysis 18 (12): 1159–1166.
Reich, S. 1979. Constructive techniques for accretive and monotone operators. In Applied Nonlinear Analysis, pp. 335-345. Amsterdam: Elsevier.
Tan, B., J. Fan, and S. Li. 2021. Self-adaptive inertial extragradient algorithms for solving variational inequality problems. Computational and Applied Mathematics 40 (1): 1–19.
Tan, B., S. Xu, and S. Li. 2020. Inertial shrinking projection algorithms for solving hierarchical variational inequality problems. Journal of Nonlinear and Convex Analysis 21: 871–884.
Van Hieu, D., P.K. Quy, and L. Van Vy. 2019. Explicit iterative algorithms for solving equilibrium problems. Calcolo 56 (2): 1–21.
Xu, H.-K. 2002. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 66 (1): 240–256.
Zhao, Tu-Yan, et al. 2020. Quasi-inertial Tseng’s extragradient algorithms for pseudomonotone variational inequalities and fixed point problems of quasi-nonexpansive operators. Numerical Functional Analysis and Optimization 1: 69-90.