Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators

Strong convergence of the Tseng extragradient method for solving variational inequalities

“…where h = 1 2 τ(2η − τk 2 ), which means that I − τF : H → H is a contractive mapping with constant 1 − h. Thus, P Fix(U) V I(C,A) (I − β F) is a contraction mapping. By Banach contraction principle, there exists a unique point p ∈ H such that p = P Fix(U)…”

“…However, in some cases, the Lipschitz constant of operator A is unknown or difficult to determine. Keeping the conditions that the operator is still monotone and Lipschitz continuous, many authors proposed related algorithms, see, e.g., [1,7,8,14,19] and the references therein. In these algorithms, an adaptive method is essential.…”

A new self-adaptive iterative method for variational inclusion problems on Hadamard manifolds with applications

“…One of the most important and widely used techniques in variational inequality theory to solve the variational inequality is the projection methods which play an essential role, see [8][9][10][11][12]. In general, the projection method may fail to converge to solve variational inequality problems if the operator B is not strong monotone (see [13]).…”

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].…”