Global Bounds for Cocoercive Variational Inequalities

By using the strong monotonicity of the perturbed ﬁxed-point map and the normal map associated with cocoercive variational inequalities, we establish two new global bounds measuring the distance between any point and the solution set for cocoercive variational inequalities. the Commons Attribution License, unrestricted dis-tribution,


Introduction
Throughout this paper, let R n be a Euclidean space, whose inner product and norm are denoted by •, • and • , respectively.Let K be a nonempty closed convex set in R n , let f : R n →R n be a continuous function.We consider the variational inequality problem associated with K and f , denoted by VIP(K, f ), which is to find a vector x * ∈ K such that f (x * ), y − x * ≥ 0, ∀y ∈ K. (1.1) Variational inequalities have many applications in different fields such as mathematical programming, game theory, economics, and engineering, see [1][2][3] and the references mentioned there.Error bounds have played an important role not only in theoretical analysis but also in convergence analysis of iterative algorithms for solving variational inequalities, see [4] for an excellent survey of the theory and application.A few error bounds have been presented for variational inequality, which mainly use the following assumptions on the map f : (i) strong monotonicity + Lipschitz continuous [5][6][7]; (ii) strong monotonicity [8,9].
When the map f is cocoercive, by using the perturbed fixed point and normal maps, and by utilizing Williamson geometric estimation of fixed points of contractive maps, Zhao and Hu [7] established global bounds for VIP(K, f ).
In this paper, by using the strong monotonicity of the perturbed fixed-point and normal maps, we establish two new global bounds measuring the distance between any point and the solution set for cocoercive variational inequalities.We need weaker restriction on the constants involved in the (perturbed) fixed point and normal maps.

Preliminaries and notations
The fixed-point and the normal equations for VIP(K, f ) are defined by respectively, where α is an arbitrary positive scalar and Π K (•) is the projection operator on the convex set K, that is, Π K (x) = min z∈K z − x .The projection operator is nonexpansive, that is, for any x,x ∈ R n , it holds that In fact, the perturbed fixed-point and normal maps also have been extensively studied, which are defined by respectively.
For the map f , we require the following concepts.
Definition 2.1.The map f : R n →R n is said to be (i) monotone if (ii) strongly monotone with modulus c if there is a scalar c > 0 such that (2.7) If L < 1, f is said to be contractive.
F. Jianghua and W. Xiaoguo 3 Definition 2.2.The map f : R n →R n is said to be cocoercive with modulus β if there exists a constant β > 0 such that Remark 2.3.Our analysis in the rest of the paper is based upon the cocoercive condition.Gabay [10] implicitly introduced the strong-f -monotonicity and Tseng [11], using the name cocoercivity, explicitly stated this condition.The cocoercive condition plays an important role in the convergence analysis of algorithms; for more details, see [12,7,[13][14][15].Notice that any cocoercive map with modulus β is monotone and Lipschitz continuous (with modulus L = 1/β), but it is not necessary to be strongly monotone (e.g., the constant map).
In some cases, the modulus β of the cocoercive map can be determined explicitly, for example, see [14,15].
Let us introduce more required notations.Let B denote the open unit ball in R n and SOL(K, f ) denotes the solution set of VIP(K, f ).Denote dist(x,D) for the distance from the vector x to any set D ⊆ R n , and denote π −1 α (0) for the zeros of π α (x).We state some lemmas, which are crucial in the proof of our main theorems.The first shows us the monotonicity of the (perturbed) fixed-point and normal maps associated with VIP(K, f ) under certain conditions.Lemma 2.4 (Zhao and Li [13]).Let K be an arbitrary closed convex set in R n and K ⊆ S ⊆ R n .Let f : R n →R n be a function.
(iii) If f is strongly monotone with modulus c > 0 and f is Lipschitz continuous with constant L > 0 on the set S, then for any fixed scalar α satisfying 0 < α < 4c/L 2 , the fixed point map π α (x) is strongly monotone with modulus α(c − αL 2 /4) on the set S.
(iv) If f is strongly monotone with modulus c > 0 and f is Lipschitz continuous with constant L > 0 on the set S, then for any α satisfying α > L 2 /(4c), the normal map Φ α (x) is strongly monotone with modulus r on the set S, where 0 < r < α/2 and 2r + L 2 /4(α − 2r) < c.
The upper-semicontinuity theorem concerning weakly univalent maps established by Ravindran and Gowda [16] is as follows.
Lemma 2.5 (Ravindran and Gowda [16]).Let g : R n →R n be weakly univalent, that is, g is continuous and there exists one-to-one continuous function g k : R n →R n such that g k →g uniformly on every bounded subset of R n .Suppose that g −1 (0) = {x ∈ R n : g(x) = 0} is nonempty and compact.Then for any given γ > 0, there exists a scalar δ > 0 such that for any weakly univalent function h : R n →R n with sup Ω h(x) − g(x) < δ, one has ∅ =h −1 (0) ⊆ g −1 (0) + γB, where Ω denotes the closure of Ω = g −1 (0) + γB.
The following lemma shows us an important property of strongly monotone maps.
Lemma 2.6.Let f : R n →R n be strongly monotone with modulus c > 0, then the following inequality holds: (2.9) Proof.Since f is strongly monotone with modulus c > 0, it holds that (2.10) On the other hand, from the Cauchy-Schwarz inequality, we have Combining (2.10) and (2.11), we obtain (2.12)

Main results
In this section, we first establish two global bounds measuring the distance between any point and the solution set for cocoercive VIP(K, f ) by using the strong monotonicity of the perturbed fixed-point and normal maps.