A new modified subgradient extragradient method for solving variational inequalities

Abstract

The goal of the note is to introduce a new modified subgradient extragradient algorithm for solving variational inequalities in Hilbert spaces. A result on the strong convergence of the algorithm is proved without the knowledge of Lipschitz constant of the operator. Several numerical experiments for the proposed algorithm are presented.

Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems

“…Similar to the results of Migorski et al [34] the proof of strong convergence of the proposed algorithm is proved without knowing the Lipschitz constant of the operator F . The proposed algorithm could be seen as a modification of the methods that are appeared in [10,12,[34][35][36]. Under mild conditions, a strong convergence theorem is proved.…”

“…where [34] proposed a viscosity-type subgradient extragradient method to solve monotone variational inequalities problems. The main contribution is the presence of a viscosity scheme in the algorithm that was used to improve the convergence rate of the iterative sequence and provide strong convergence theorem.…”

“…It is important to note that our proposed scheme is effective. In particular, by comparing the results of Migorski et al [34], our algorithm can solve pseudomonotone variational inequalities. Similar to the results of Migorski et al [34] the proof of strong convergence of the proposed algorithm is proved without knowing the Lipschitz constant of the operator F .…”

“…In particular, by comparing the results of Migorski et al [34], our algorithm can solve pseudomonotone variational inequalities. Similar to the results of Migorski et al [34] the proof of strong convergence of the proposed algorithm is proved without knowing the Lipschitz constant of the operator F . The proposed algorithm could be seen as a modification of the methods that are appeared in [10,12,[34][35][36].…”

Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

“…This notion, that mainly involves some important operators, plays a key role in applied mathematics, such as obstacle problems, optimization problems, complementarity problems as a unified framework for the study of a large number of significant real-word problems arising in physics, engineering, economics and so on. For more information, the reader can refer to [1][2][3][4][5][6][7][8][9][10][11][12]. For solving VI (1) in which the involved operator f may be monotone, several iterative algorithms have been introduced and studied, see, e.g., [13][14][15][16][17][18].…”