Weak convergence of iterative methods for solving quasimonotone variational inequalities
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Weak convergence of iterative methods for solving quasimonotone variational inequalities Hongwei Liu1 · Jun Yang1,2 Received: 17 June 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, we introduce self-adaptive methods for solving variational inequalities with Lipschitz continuous and quasimonotone mapping(or Lipschitz continuous mapping without monotonicity) in real Hilbert space. Under suitable assumptions, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. The results obtained in this paper extend some recent results in the literature. Some preliminary numerical experiments and comparisons are reported. Keywords Variational inequalities · Projection · Gradient method · Quasimonotone mapping · Convex set Mathematics Subject Classification 47J20 · 90C25 · 90C30 · 90C52
1 Introduction Let C be a nonempty closed and convex set in a real Hilbert space H and F ∶ H ⟶ H is a continuous mapping, ⟨⋅, ⋅⟩ and ∥ ⋅ ∥ denote the inner product and the induced norm in H, respectively. The variational inequality (VI(C, F)) is
find
x∗ ∈ C
such that ⟨F(x∗ ), y − x∗ ⟩ ≥ 0,
∀y ∈ C.
(1)
* Jun Yang [email protected] Hongwei Liu [email protected] 1
School of Mathematics and Statistics, Xidian University, Xi’an 710126, Shaanxi, China
2
School of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000, Shaanxi, China
13
Vol.:(0123456789)
H. Liu, J. Yang
Weak converge of the sequence {xn } to a point x is denoted by xn ⇀ x while xn → x to denote the sequence {xn } converges strongly to x. Let S be the solution set of (1) and SD be the solution set of the following problem,
find
x∗ ∈ C
such that ⟨F(y), y − x∗ ⟩ ⩾ 0,
∀y ∈ C.
(2)
It is obvious that SD is a closed convex set (possibly empty). As F is continuous and C is convex, we get (3)
SD ⊆ S.
If F is a pseudomonotone and continuous mapping, then S = SD(see Lemma 2.1 in [1]). The inclusion S ⊂ SD is false, if F is a quasimonotone and continuous mapping [2]. For solving quasimonotone variational inequalities, the convergence of interior proximal algorithm [3, 4] was obtained under more assumptions than SD ≠ ∅ . Under the assumption of SD ≠ ∅ , Ye and He [2] proposed a double projection algorithm for solving quasimonotone (or without monotonicity) variational inequalities in Rn . For a nonempty closed and convex set C ⊆ H , PC is called the projection from H onto C, that is, for every element x ∈ H such that ‖x − PC (x)‖ = min{∥ y − x ∥∣ y ∈ C} . It is easy to check that problem (1) is equivalent to the following fixed point problem:
find
x∗ ∈ C
such that
x∗ = PC (x∗ − 𝜆F(x∗ ))
(4)
for any 𝜆 > 0 . Various projection algorithms have been proposed and analyzed for solving variational inequalities [2, 5–27]. Among them, the extragradientmethod was proposed by Korpelevich [5] and Antipin [18], that is
yn = PC (xn − 𝜆F(xn )),
xn+1 = PC (xn − 𝜆F(yn )),
(5)
where 𝜆 ∈ (0, and L is the Lipschitz constant of F. Tseng [8] m
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