An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem C
- PDF / 509,904 Bytes
- 16 Pages / 439.642 x 666.49 pts Page_size
- 10 Downloads / 204 Views
An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem Constraints Nguyen Minh Hai1 · Le Huynh My Van2,3 · Tran Viet Anh4 Received: 9 August 2019 / Revised: 3 May 2020 / Accepted: 5 May 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of a split variational inequality and fixed point problem. Strong convergence of the iterative process is proved. In particular, the problem of finding a common solution to a variational inequality with pseudomonotone mapping and a fixed point problem involving demicontractive mapping is also studied. Besides, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each step. A simple numerical example is given to illustrate the proposed algorithm. Keywords Split variational inequality and fixed point problem · Pseudomonotone mapping · Demicontractive mapping · Subgradient extragradient method · Strong convergence · Minimum-norm solution · Split feasibility problem Mathematics Subject Classification (2010) 49M37 · 90C26 · 65K15
Tran Viet Anh
[email protected]; [email protected] Nguyen Minh Hai [email protected] Le Huynh My Van [email protected] 1
Department of Mathematical Economics, Banking University Ho Chi Minh, Ho Chi Minh City, Vietnam
2
Department of Mathematics, University of Information Technology, Ho Chi Minh City, Vietnam
3
Vietnam National University, Ho Chi Minh City, Vietnam
4
Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam
N.M. Hai et al.
1 Introduction Let H1 and H2 be two real Hilbert spaces and let A : H1 −→ H2 be a bounded linear operator. Let C be a nonempty closed convex subset of H1 . Given mappings G : H1 −→ H1 and S : H2 −→ H2 , the split variational inequality and fixed point problem (in short, SVIFPP) is to find a solution x ∗ of the variational inequality problem in the space H1 so that the image Ax ∗ , under a given bounded linear operator A, is a fixed point of another mapping in the space H2 . More specifically, the SVIFPP can be formulated as Find x ∗ ∈ C : G(x ∗ ), x − x ∗ ≥ 0, ∀x ∈ C
(1)
such that
S(Ax ∗ ) = Ax ∗ . When G = 0 and S = PQ , the SVIFPP reduces to the split feasibility problem, shortly SFP, Find x ∗ ∈ C such that Ax ∗ ∈ Q. The SFP was first introduced by Censor and Elfving [4] in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [1]. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy [3, 5, 6], and many other practical problems. If we consider only the problem (1) then (1) is a classical variational inequality problem. If H1 = H2 and A is the identity ma
Data Loading...
