A Modified Subgradient Extragradient Algorithm with Inertial Effects
In this article, we introduce an inertial modified subgradient extragradient method by combining inertial type algorithm with modified subgradient extragradient method and for solving the variational inequality (VI) in a Hilbert space \(H\). Also, we establish a weak convergence theorem for proposed algorithm. Finally, we describe the performance of our proposed algorithm with the help of numerical experiment and we show the efficiency and advantage of the inertial modified subgradient extragradient method.
Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, Journal of Optimization Theory and Applications 148(2) (2010), 318 – 335, DOI: 10.1007/s10957-010-9757-3.
H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Berlin (2011).
Y. Censor, A. Gibali and S. Reich, Strong Convergence of subgradient extragradient methods for the variational inequality problem in Hilbert spaces, Optim. Methods. Soft. 26 (2011), 827 – 845, DOI: 10.1080/10556788.2010.551536.
Q. L. Dong, D. Jiang and A. Gibali, A modified subgradient extragradient method for solving the variational inequality problem, Numerical Algorithm 79(3) (2018), DOI: 10.1007/s11075-017-0467-x.
Q. L. Dong, Y. J. Cho, L. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Global Optimization 70(3) (2017), 687 – 704, DOI: 10.1007/s10898-017-0506-0.
Q. L. Dong, Y. Y. Lu and J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality, Optimization 65 (2016), 2217 – 2226, DOI: 10.1080/02331934.2016.1239266.
C. Fang and S. Chen, A subgradient extragradient algorithm for solving multi-valued variational inequality, Appl. Math. Comput. 229 (2014), 123 – 130, DOI: 10.1016/j.amc.2013.12.039.
K. Geobel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York — Basel (1984), DOI: 10.1112/blms/17.3.293.
K. Geobel and W. A. Kirk, Topics in Metric fixed point theory, in Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, Cambridge (1991), DOI: 10.1017/CBO9780511526152.
A. A. Goldstein, Convex programming in Hilbert space, Bull. Am. Math. Soc. 70 (1964), 709 – 710, DOI: 10.1090/S0002-9904-1964-11178-2.
P. T. Harker and J. S. Pang, A damped Newton method for the linear complementarity problem, in: Computational Solution of Nonlinear System of Equations, G. Allgower and K. Georg (eds.), Lectures in Appl. Math., Vol. 26, pp. 265 – 285, AMS, Providence (1990), DOI: 10.1007/978-3-662-12629-5_52.
B. S. He, A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997), 69 – 76, DOI: 10.1007/BF02683320.
S. He, C. Yang and P. Duan, Realization of the hybrid method for Mann iterations, Appl. Math. Comput. 217 (2010), 4239 – 4247, DOI: 10.1016/j.amc.2010.10.039.
D. V. Hieu, P. K. Anh and L. D. Muu, Modified Hybrid projection methods for finding common solutions to variational inequality problem, Comput. Optim. Appl. 66 (2017), 75 – 96, DOI: 10.1007/s10589-016-9857-6.
E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comput. Math. Phys. 27 (1987), 120 – 127, DOI: 10.1016/0041-5553(87)90058-9.
G. M. Korpelevich, The extragradient method for finding saddle point and other points, Ekon. Mat. Metody 12 (1976), 747 – 756, DOI: 10.3103/S0278641910030039.
R. Kraikaew and S. Saejung, Strong convergence of the halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optimz. Theory Appl. 163 (2014), 399 – 412, DOI: 10.1007/s10957-013-0494-2.
E. S. Levitin and B. T. Polyak, Constrained minimization problems, USSR Comput. Math. Phys. 6 (1966), 1 – 50, DOI: 10.1016/0041-5553(66)90114-5.
M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004), 199 – 277, DOI: 10.1016/S0096-3003(03)00558-7.
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877 – 898, DOI: 10.1137/0314056.
M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim. 37 (1999), 765 – 776, DOI: 10.1137/S0363012997317475.
D. F. Sun, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl. 91 (1996), 123 – 140, DOI: 10.1007/BF02192286.
N. T. Vinh and P. T. Hoai, Some subgradient extradientt type algorithms for solving split feasibility and fixed point problems, Math. Method Appl. Sci. 39 (2016), 3808 – 3823, DOI: 10.1002/mma.3826.
Y. Yao, G. Marino and L. Muglia, A modified Korpelevich’s method convergent to the minimum norm solution of a variational inequality, Optimization 63 (2014), 559 – 569, DOI: 10.1080/02331934.2012.674947.
H. Zhou, Y. Zhou and G. Feng, Iterative methods for solving a class of monotone variational inequality problems with applications, J. Inequal. Appl. 2015 (2015), 68, DOI: 10.1186/s13660-015-0590-y.
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