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

Abstract

In this paper, we introduce a new algorithm with self-adaptive method for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. The algorithm is based on the subgradient extragradient method and inertial method. At the same time, it can be considered as an improvement of the inertial extragradient method over each computational step which was previously known. The weak convergence of the algorithm is studied under standard assumptions. It is worth emphasizing that the algorithm that we propose does not require one to know the Lipschitz constant of the operator. Finally, we provide some numerical experiments to verify the effectiveness and advantage of the proposed algorithm.

Two inertial linesearch extragradient algorithms for the bilevel split pseudomonotone variational inequality with constraints

“…It is known that the class of nonexpansive mappings process the demiclosed principle. However, the class of quasinonexpansive mappings may do not process the demiclosed principle; see, e.g., [23,27].…”

Two Inertial extragradient viscosity algorithms for solving variational inequality and fixed point problems

“…We test for: λ 0 = 0.6 and λ 0 = 0.2. We also compare the computational efficiency of the proposed algorithm with Algorithm 3.1 [24] and ISA-SEGM [23]. Choose x(0) = 0, x(1) = 0, µ = 0.5 and α n = 1 n .…”

“…Choose x(0) = 0, x(1) = 0, µ = 0.5 and α n = 1 n . We choose γ n = 5 for the proposed algorithm and γ n = 0.7 for ISA-SEGM [23]. In order to terminate the algorithm, we use the condition x n+1 − x * < ε where x * is the solution of the problem.…”

“…Recently, many algorithms for finding a common solution of V IP and FPP in a real Hilbert space have been studied (see, [7,8,24,27,28] and the references therein). Tian and Tong [23] proposed an algorithm with a self-adaptive method for finding the solution of V IP involving monotone operators and the fixed point problem of a quasi-nonexpansive mapping. They studied the weak convergence of the algorithm under certain assumptions.…”

Strong convergence of the Tseng extragradient method for solving variational inequalities

“…If V I(C, A) = / 0 and λ ∈ (0, 1/L), the sequence {x n } generated by (1.3) converges weakly to an element of V I(C, A). Recently, this method has attracted attentions by many authors, see, e.g., [2,12,15,16,18] and the references therein. The second method for solving the variational inequality problems, the second projection onto the feasible set C replaced by a projection onto a special constructible half-space, is the so-called subgradient extragradient method by Censor, Gibali and Reich [3].…”