Modified subgradient extragradient algorithms for solving monotone variational inequalities

ABSTRACT In this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a new step size, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithms.

[1]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[2]  John R. Rice,et al.  General interior Hermite collocation methods for second-order elliptic partial differential equations , 1994 .

[3]  A. Iusem,et al.  A variant of korpelevich’s method for variational inequalities with a new search strategy , 1997 .

[4]  M. Solodov,et al.  A New Projection Method for Variational Inequality Problems , 1999 .

[5]  A. Moudafi.  Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[6]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[7]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[8]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[9]  Muhammad Aslam Noor,et al.  Some developments in general variational inequalities , 2004, Appl. Math. Comput..

[10]  Paul-Emile Maingé,et al.  A Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems , 2008, SIAM J. Control. Optim..

[11]  Muhammad Aslam Noor,et al.  On an Implicit Method for Nonconvex Variational Inequalities , 2010, J. Optim. Theory Appl..

[12]  Paul-Emile Maingé,et al.  The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces , 2010, Comput. Math. Appl..

[13]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[14]  Javad Balooee.  Projection Method Approach for General Regularized Non-convex Variational Inequalities , 2013, J. Optim. Theory Appl..

[15]  Yu. V. Malitsky,et al.  An Extragradient Algorithm for Monotone Variational Inequalities , 2014 .

[16]  Satit Saejung,et al.  Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces , 2014, J. Optim. Theory Appl..

[17]  Yu. V. Malitsky,et al.  A hybrid method without extrapolation step for solving variational inequality problems , 2015, J. Glob. Optim..

[18]  Yu. V. Malitsky,et al.  Projected Reflected Gradient Methods for Monotone Variational Inequalities , 2015, SIAM J. Optim..

[19]  A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces , 2015 .

[20]  Paul-Emile Maingé,et al.  Convergence of One-Step Projected Gradient Methods for Variational Inequalities , 2016, J. Optim. Theory Appl..

[21]  Yekini Shehu,et al.  Strong convergence result for monotone variational inequalities , 2017, Numerical Algorithms.

[22]  Duong Viet Thong,et al.  Weak and strong convergence theorems for variational inequality problems , 2017, Numerical Algorithms.

[23]  Duong Viet Thong,et al.  Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems , 2018 .

[24]  Hongwei Liu,et al.  Strong convergence result for solving monotone variational inequalities in Hilbert space , 2018, Numerical Algorithms.