Modified Extragradient Method for Pseudomonotone Variational Inequalities in Infinite Dimensional Hilbert Spaces
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Modified Extragradient Method for Pseudomonotone Variational Inequalities in Infinite Dimensional Hilbert Spaces Dang Van Hieu1
· Yeol Je Cho2,3 · Yi-Bin Xiao2 · Poom Kumam4
Received: 3 February 2020 / Accepted: 6 July 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we prove the weak convergence of a modified extragradient algorithm for solving a variational inequality problem involving a pseudomonotone operator in an infinite dimensional Hilbert space. Moreover, we establish further the R-linear rate of the convergence of the proposed algorithm with the assumption of error bound. Several numerical experiments are performed to illustrate the convergence behaviour of the new algorithm in comparisons with others. The results obtained in the paper have extended some recent results in the literature. Keywords Variational inequality · Pseudomonotone operator · Projection method · Lipschitz condition Mathematics Subject Classification (2010) 65K15 · 47H05 · 47J20 · 49J40 · 49M30
Dang Van Hieu
[email protected] Yeol Je Cho [email protected] Yi-Bin Xiao [email protected] Poom Kumam [email protected] 1
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, People’s Republic of China
3
Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea
4
Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thrung Khru, Bangkok 10140, Thailand
D.V. Hieu et al.
1 Introduction Let H be a real Hilbert space with the inner product ·, · and the induced norm · and C be a nonempty closed convex subset of H. We say that an operator A : H → H is Lipschitz continuous on C if there exists a number L > 0 such that A(x) − A(y) ≤ Lx − y ∀x, y ∈ C.
(1)
If inequality (1) holds with L = 1, then the operator A is called nonexpansive. This paper concerns an algorithm for approximating a solution of a variational inequality problem (shortly, (VIP)) [10, 20, 23, 26] in an infinite dimensional Hilbert space. Recall that problem (VIP) for the operator A on C is stated as follows: Find p ∈ C such that A(p), x − p ≥ 0
for all x ∈ C.
(VIP)
The solution set of problem (VIP) is denoted by V I (A, C). Throughout this paper, we assume that the set V I (A, C) is nonempty. As usual, an operator A is called monotone on C if A(x) − A(y), x − y ≥ 0 ∀x, y ∈ C, and pseudomonotone on C if A(x), y − x ≥ 0
=⇒
A(y), y − x ≥ 0
∀x, y ∈ C.
From the definitions mentioned above, it is seen that, if A is monotone, then A is pseudomonotone. The converse case in general is not true. Problem (VIP) is an important problem in optimization theory and it unifies many models in applied mathematics as necessary optimality conditions, complementarity problems, network equilibrium problems and
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