A new low-cost double projection method for solving variational inequalities
- PDF / 2,615,778 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 72 Downloads / 224 Views
A new low‑cost double projection method for solving variational inequalities Aviv Gibali1,2 · Duong Viet Thong3 Received: 24 August 2019 / Revised: 3 February 2020 / Accepted: 3 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work we are concerned with variational inequalities in real Hilbert spaces and introduce a new double projection method for solving it. The algorithm is motivated by the Korpelevich extragradient method, the subgradient extragradient method of Gibali et al. and Popov’s method. The proposed scheme combines some of the advantages of the methods mentioned above, first it requires only one orthogonal projection onto the feasible set of the problem while the next computation has a closed formula. Second, only one mapping evaluation is required per each iteration and there is also a usage of an adaptive step size rule that avoids the need to know the Lipschitz constant of the associated mapping. We present two convergence theorems of the proposed method, weak convergence result which requires pseudomonotonicity, Lipschitz and sequentially weakly continuity of the associated mapping and strong convergence theorem with rate of convergence which requires Lipschitz continuity and strongly pseudomonotone only. Primary numerical experiments and comparisons demonstrate the advantages and potential applicability of the new scheme. Keywords Popov’s method · Variational inequality problem · Pseudo-monotone mapping · Strongly pseudo-monotone mapping Mathematics Subject Classification 65Y05 · 65K15 · 68W10 · 47H06 · 47H09 · 47H10
* Duong Viet Thong [email protected] Aviv Gibali [email protected] 1
Department of Mathematics, ORT Braude College, 2161002 Karmiel, Israel
2
The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, 3498838 Haifa, Israel
3
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
13
Vol.:(0123456789)
A. Gibali, D. V. Thong
1 Introduction In this paper, we study the classical Variational Inequality (VI) of Fichera (1963, 1964) of finding a point x∗ ∈ C such that
⟨Ax∗ , x − x∗ ⟩ ≥ 0 ∀x ∈ C,
(1)
where C is a nonempty, closed and convex subset of a real Hilbert space H and A ∶ H → H is some single-valued mapping. We denote by VI(C, A) the solution set of (1). Variational inequalities are fundamental in a broad range of mathematical and applied sciences; and hence are still studied intensively with respect to the development of new theory, see e.g., Facchinei and Pang (2003), Konnov (2001) as well as applications, see in particular the special issue of Optimization and Engineering: Applications of Variational Inequality Problems (Tawhid 2012). One of the earliest and simplest iterative method for solving VIs is the following projection method, which can be seen as an extension of the projected gradient method for solving convex optimization problems:
xn+1 = PC (xn − 𝜆Axn ),
(2)
for each n ≥ 1 , where PC denot
Data Loading...
