Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Abstract

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

Convergence analysis for variational inequalities and fixed point problems in reflexive Banach spaces

“…In particular, Tseng [64] introduced a single projection extragradient method (also called forward-backward algorithm) for the variational inequalities in real Hilbert spaces. A typical disadvantage of Tseng algorithm and many other algorithms (such as [10,11,23,24,62] and the references therein) is the assumption that the Lipschitz constant of the monotone operator is known or can be estimated. In many practical problems, the Lipschitz constant is very difficult to estimate and the cost operator might even be pseudo-monotone.…”

On a General Extragradient Implicit Method and Its Applications to Optimization

“…They proved norm convergence of {x n } to x * ∈ Ω. Recently, this problem has attracted much attention from the authors working on convex believel problems; see, e.g., [13][14][15][16][17][18][19] Meantime, in order to solve the Equation (1) with the common fixed point problem constraint of a countable family of non-expansive self-mappings {S n } ∞ n=0 on C, Song and Ceng [20] found a general iterative scheme in a Banach space with both uniformly convex and q-uniformly smooth structures (whose smoothness constant is κ q , where 1 < q ≤ 2). Let Π C , A 1 , A 2 , G be the same operators as above.…”