A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems
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A Novel Inertial Projection and Contraction Method for Solving Pseudomonotone Variational Inequality Problems Prasit Cholamjiak1 · Duong Viet Thong2 · Yeol Je Cho3,4
Received: 29 March 2019 / Accepted: 25 October 2019 © Springer Nature B.V. 2019
Abstract In this paper, we introduce a new algorithm which combines the inertial contraction projection method and the Mann-type method (Mann in Proc. Am. Math. Soc. 4:506– 510, 1953) for solving monotone variational inequality problems in real Hilbert spaces. The strong convergence of our proposed algorithm is proved under some standard assumptions imposed on cost operators. Finally, we give some numerical experiments to illustrate the proposed algorithm and the main result. Keywords Inertial contraction projection method · Mann-type method · Pseudomonotone mapping · Pseudomonotone variational inequality problem Mathematics Subject Classification (2010) 65Y05 · 65K15 · 68W10 · 47H05 · 47H10
1 Introduction Let H be a real Hilbert space with the inner product ·, · and the induced norm · , C be a nonempty closed convex subset of H and A : H → H be an operator. Dedicated to Professor Pham Ky Anh on the occasion of his 70th birthday
B D.V. Thong
[email protected] P. Cholamjiak [email protected] Y.J. Cho [email protected]
1
School of Science, University of Phayao, Phayao 56000, Thailand
2
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
4
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China
P. Cholamjiak et al.
In this paper, we consider the classical variational inequality (VI) of Fichera [26, 27]. The variational inequality problem for F on C is as follows: Find a point x ∗ ∈ C such that F x ∗ , x − x ∗ ≥ 0
for all x ∈ C.
(1)
Let us denote VI(C, F) by the solution set of the problem (VI) (1). The problem of finding solutions of the problem (VI) (1) is a fundamental problem in optimization theory. Due to this, the problem (VI) (1) has received a lot of attention by many authors. In fact, there are two general approaches to study variational inequality problems, which are the regularized method and the projection method. Based on these directions, many algorithms have been considered and proposed for solving the problem (VI) (1) (see, for example, [16–19, 28, 32, 35, 39, 40, 42, 43, 50, 58–60]). The basic idea consists of extending the projected gradient method for solving the problem of minimizing f (x) subject to x ∈ C given by xn+1 = PC xn − αn f (xn ) ,
∀n ≥ 0,
(2)
where {αn } is a positive real sequence satisfying certain conditions and PC is the metric projection onto C. For some convergence properties of this method for the case in which f : H → R is convex and differentiable function, one may see [1]. An immediate extension of the method (2) to (VI) (1) is the projected gradient method fo
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