Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems

Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems

“…where x * is called a fixed point of T : H → H. The set of fixed points of T is denoted by Fix(T). Recently, many iterative methods have been proposed (see [6,[11][12][13][14][15][16][17][18][19][20][21] and the references therein) for finding a common element of Fix(T) and VI(C, A) in a real Hilbert space.…”

Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

“…Based on inertial effects, Thong and Hieu [24] proposed an inertial subgradient method, and also proved the weak convergence of their algorithms. In addition, the authors [25] investigated subgradient-based fast algorithms with inertial effects.…”

“…Inspired by the above research works in [12,[24][25][26][27], we are concerned with hybrid-adaptive step-sizes Tseng's extragradient algorithms, that are more advantageous and more subtle than the above iterative algorithms because they involve solving the VIP with Lipschitzian, pseudomonotone operators, and the common fixed-point problem of a finite family of strict pseudocontractions in Hilbert spaces. By imposing some appropriate weak assumptions on parameters, one obtains a norm solution of the problems, which solves a certain hierarchical variational inequality.…”

“…Since z n = β n f (x n ) + (1 − β n )u n with u n := y n − τ n (Ay n − Aw n ), from (25) and the boundedness of {x n }, {u n }, we get…”

“…Obviously, combining (25) and (26) guarantees that w n − z n ≤ y n − w n + z n − y n , which indicates that lim n→∞ w n − z n = 0. Let vector sequence {x n k } be a subsequence of original sequence {x n }.…”