Modified subgradient extragradient method for variational inequality problems

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

“…Under some appropriate mild conditions, they proved the weak convergence of the proposed algorithms to an element of Fix(T) ∩ VI(C, A). Inspired by the research work of [23][24][25][26][27][28], we introduce two composite inertial subgradient-extragradient algorithms with a line-search process for solving the VIP with pseudomonotone and Lipschitz continuous mapping and the CFPP of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in a real Hilbert space. The proposed algorithms are based on an inertial subgradient-extragradient method with a line-search process, hybrid steepestdescent methods, viscosity approximation methods and the Mann iteration method.…”

“…, N and T 0 = T is strictly pseudocontractive. The inertial subgradient-extragradient method with line-search (which is weakly convergent) for solving the VIP and FPP in [24] is extended to develop our composite inertial subgradient-extragradient method with linesearch process (which is strongly convergent) for solving the VIP and CFPP, which is based on an inertial subgradient-extragradient method with a line-search process, a hybrid steepest-descent method, a viscosity approximation method and a Mann iteration method. It is worth pointing out that the inertial subgradient-extragradient method with a line-search process in [23] combines the inertial subgradient-extragradient method [24] with the Mann iteration method.…”

“…In 2018, by virtue of the inertial technique, Thong and Hieu [24] first proposed the inertial subgradient-extragradient method, that is, for any initial x 0 , x 1 ∈ H, the sequence {x n } is generated by…”

Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

“…Recently, numerous methods of reduced-gradient-type are focused and extensively investigated in both infinite and infinite dimensional spaces; see, for example [14][15][16][17][18][19][20][21][22][23]. 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.…”