Gap Functions and Error Bounds for Variational–Hemivariational Inequalities

In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results.


Introduction
In the study of various complementarity and equilibrium problems occurring in operation research, economics, mechanics, mathematical programming, etc., we often naturally meet the variational inequality problem of the form: find u * ∈ K such that Au * , v − u * X ≥ 0 for all v ∈ K. (1.1) Here K is a nonempty closed convex subset of a normed space X representing constraints, A : X → X * is a given operator, and ·, · X denotes the duality pairing between X and its dual X * . Among several approaches available in the literature, it is well known that the variational inequality (1.1) can be solved by transforming it into an equivalent optimization problem for the so-called merit function π(·; α) : X → R ∪ {+∞} defined by where α is a nonnegative parameter. If X is finite dimensional, this function was first introduced by Auslender in [4] for α = 0, and by Fukushima in [12] for α > 0. The function π(·; 0) is usually called the gap function, and the function π(·; α) for α > 0 is called the regularized gap function. It is known, see [4,19], that for all α > 0, the function π(·; α) is nonnegative on K, and π(u * ; α) = 0 whenever u * satisfies the variational inequality (1.1). An advantage of this approach is that the resulting optimization problem can be solved by descent algorithms which enjoys a global convergence property. However, it turns out that even for finite dimensional space X, the gap function fails to be differentiable in general, and it may not be finite valued. In contrast, the regularized gap function for α > 0 is nicer since it is finite valued and is differentiable whenever A is differentiable, see [12] for details.
The gap functions are today very useful to investigate existence conditions, solution methods and stability conditions for optimization-related problems in order to simplify the computational aspects. Based on the idea of Fukushima [12] the regularized function of the Moreau-Yosida type has been developed by Yamashita and Fukushima in [49]. They also proposed the so-called error bounds for variational inequalities via the regularized gap functions. The notion of error bounds is known as an upper estimate of the distance between an arbitrary feasible point and the solution set of a certain problem. Such error estimates have played a vital role in convergence analysis of iterative algorithms for solving variational inequalities. In recent years, there have been many studies on gap functions for different models on different topics such as iterative algorithms [23], the Painlevé-Kuratowski convergence [2], stability of solutions [3,[20][21][22] and error bounds [6,13,[24][25][26]. We also refer the reader to [1,5,7,11,14,[27][28][29] and the references therein for a more detailed discussion of interesting topic.
On the other hand, the theory of variational-hemivariational inequalities is known as a generalization of variational inequalities and hemivariational inequalities to the case involving both the convex and the nonconvex potentials, and based on the notion of the Clarke generalized gradient for locally Lipschitz functions. Interest in the study of variationalhemivariational inequalities was originally motivated by various problems in mechanics, see e.g., [45,46]. The theory of variational-hemivariational inequalities has been extensively studied by many authors in different directions, and it has found various applications in mechanics, engineering, especially in optimization and nonsmooth analysis. Recent existence results for variational-hemivariational inequalities can be found, in e.g., [16, 31, 33-41, 43, 47, 48], the stability in the sense of convergence and the well-posedness, in e.g., [18,30,32,[50][51][52][53], and the computational issues have been addressed in, e.g., [15,17].
To the best of our knowledge, up to now, there has not been any study on the gap functions and global error bounds for the variational-hemivariational inequalities. Our goal is to fill in this gap and provide new results in this area. The novelties of the paper are as follows. First, we introduce the gap functions and regularized gap functions for a class of variational-hemivariational inequalities. Also, we treat the gap functions for the Minty version of these inequalities. Next, we study the Moreau-Yosida regularized gap functions, introduced by Yamashita and Fukushima in [49], and provide two new global error bounds for variational-hemivariational inequalities via the regularized and the Moreau-Yosida regularized gap functions. Finally, we illustrate the abstract results by an application to a nonsmooth semipermeability obstacle problem described by an elliptic variational-hemivariational inequality for which we deliver global error bounds.
The article is arranged as follows. In Sect. 2, we recall basic definitions and results which are needed in the sequel and revisit the constrained variational-hemivariational inequality of elliptic type. In Sect. 3, we study some regularized gap functions of Yamashita-Fukushima type, and establish global error bounds for the variational-hemivariational inequalities. An application to a semipermeability problem for stationary heat problem is given in Sect. 4 to illustrate our main theoretical findings.

Preliminaries
In this section we recall the notation and some preliminary material which will be needed in the sequel. For more details, we refer to [8][9][10]42].
Let (X, · X ) be a real Banach space with the dual X * , and we denote by ·, · X the duality pairing between X * and X. We begin with the following definitions.
(c) lower semicontinuous (l.s.c.) at u ∈ X, if for any sequence {u n } ⊂ X such that u n → u, it holds h(u) ≤ lim inf h(u n ). (d) upper semicontinuous (u.s.c.) at u ∈ X, if for any sequence {u n } ⊂ X such that u n → u, it holds lim sup h(u n ) ≤ h(u). (e) l.s.c (resp. u.s.c.) on X, if h is l.s.c (resp. u.s.c.) at every u ∈ X. Definition 2.2 Let f : X → R be a proper, convex and l.s.c. function. The convex subdifferential ∂ c f : X ⇒ X * of f is defined by An element u * ∈ ∂ c f (u) is called a subgradient of f at u ∈ X.

Definition 2.3
A function h : X → R is said to be locally Lipschitz, if for every u ∈ X, there exist a neighbourhood U of u and a constant L u > 0 such that Given a locally Lipschitz function h : X → R, we denote by h 0 (u; v) the Clarke generalized directional derivative of h at the point u ∈ X in the direction v ∈ X defined by The generalized gradient of h at u ∈ X, denoted by ∂h(u), is a subset of X * given by The generalized directional derivative and generalized gradient of a locally Lipschitz function enjoy many nice properties and rich calculus. Below we collect some basic and useful results, see, e.g., [42,Proposition 3.23].

Lemma 2.1
Let X be a real Banach space and h : X → R be a locally Lipschitz function. Then the following assertions hold.
(a) For each u ∈ X, the function X v → h 0 (u; v) ∈ R is finite, positively homogeneous and subadditive, and Recall that a single-valued operator A : X → X * is said to be pseudomonotone, if A is a bounded operator and for every sequence {x n } ⊆ X converging weakly to x ∈ X such that lim sup Ax n , x n − x ≤ 0, we have Ax, x − y ≤ lim inf Ax n , x n − y for all y ∈ X.
Let X be a reflexive Banach space and K be a nonempty subset of X. Given an operator A : K → X * , functions ϕ : K × K → R and J : X → R, and f ∈ X * , we are concerned with the study of the following constrained variational-hemivariational inequality.
We now impose the following hypotheses on the data of Problem 2.1.
(b) A is strongly monotone, i.e., there exists m A > 0 such that K is nonempty, closed and convex subset of X, and f ∈ X * . (2.4) Remark 2.1 Note that in some recent works, such as [43,47,50], the authors have supposed that A : X → X * and J : X → R enjoy hypotheses (2.1) and (2.3). They also required the following additional condition where α A > m J . However, this assumptions is redundant. Indeed, given u 0 ∈ K, from hypotheses (2.1), we obtain It is obvious that the above estimate guarantees the condition (2.5).
We have the following existence and uniqueness result for Problem 2.1.
If, in addition, the following smallness condition is satisfied Proof The existence and uniqueness of solution to Problem 2.1 is a direct consequence of [47, Theorem 1].
Let u ∈ K be the unique solution to Problem 2.1. First, we note that the hypothesis (2.3)(b) is equivalent to the following relaxed monotonicity condition of the generalized gradient for all v, u ∈ X. Next, the smallness condition (2.7) together with (2.9) and the strong mono- Let v ∈ K be arbitrary. Combining the above inequality, Lemma 2.1(c) and the definition of generalized gradient entails Since v ∈ K is arbitrary, therefore, u ∈ K solves the problem (2.8) too.
Conversely, let u ∈ K be a solution to the problem (2.8). For any v ∈ K and t ∈ (0, 1), where we have used the convexity of v → ϕ(u, v) and the positive homogeneity of v → J 0 (u; v). Hence, Note that A is pseudomonotone, so, it is demicontinuous, see e.g. [42,Theorem 3.69]. Passing to the upper limit as t → 0 + in (2.10), it gives where we have applied Lemma 2.1(b). Recall that v ∈ K is arbitrary, so, we conclude that u ∈ K is a solution to Problem 2.1 as well. This completes the proof.

Main Results
In this section, we are devoted to explore some global error estimates for variationalhemivariational inequality in Problem 2.1, by introducing concepts of a gap function, a regularized gap function, and the Moreau-Yosida regularized gap function associated to Problem 2.1.
Invoking the idea of Yamashita-Fukushima in [49], we now introduce the definitions of a gap function and a regularized gap function for Problem 2.1. Definition 3.1 A real-valued function π : K → R is said to be a gap function for Problem 2.1, if it satisfies the following properties: (a) π(u) ≥ 0 for all u ∈ K. (b) u * ∈ K is such that π(u * ) = 0 if and only if u * is a solution to Problem 2.1.

Consider the functions
for all u ∈ K, respectively.
The following proposition shows that functions Θ f and Θ f * are gap functions for Problem 2.1. Proof In what follows, we prove that Θ f is a gap function for Problem 2.1. In an analogous way, it is not difficult to show that the function Θ f * is also a gap function for Problem 2.1. We will verify two conditions of Definition 3.1.
(a) In fact, it is obvious that Θ f (u) ≥ 0 for all u ∈ K. This property holds since for all u ∈ K, we have This together with the fact for all v ∈ K. Therefore, we conclude that u * is a solution to Problem 2.1 if and only if Θ f (u * ) = 0.
Let γ > 0 be a fixed parameter. We consider the following functions Θ f,γ , Θ f,γ * : K → R defined by is also a gap function for Problem 2.1. We will check two conditions of Definition 3.1.
(a) For each γ > 0 fixed, it is trivial that for each u ∈ K it holds Θ f,γ (u) ≥ 0. This is due to u ∈ K and This means For any w ∈ K and t ∈ (0, 1), we insert v = v t := (1 − t)u * + tw ∈ K into the above inequality to obtain where we have used the convexity of v → ϕ(u, v) and positive homogeneity of v → J 0 (u; v). Hence, we have for all w ∈ K. Letting t → 0 + for the above inequality, it gives Au * − f, w − u * X + ϕ u * , w − ϕ u * , u * + J 0 u * ; w − u * ≥ 0 for all w ∈ K. Hence, u * is also a solution to Problem 2.1. Conversely, suppose that u * ∈ K is a solution of Problem 2.1, that is, The latter combined with the fact Θ f,γ (u) ≥ 0 for all u ∈ K reveals that Θ f,γ (u * ) = 0. This completes the proof.
Further, we will show that the regularized gap functions Θ f,γ and Θ f,γ * are both lower semicontinuous. Proof We will prove that Θ f,γ is lower semicontinuous for each γ > 0. It is not difficult to apply a similar argument to verify that Θ f,γ * has the same property. Consider the function Θ f,γ : K × K → R defined by Recall that the operator A : X → X * is demicontinuous being pseudomonotone. This means that the function u → Au, u X is continuous. The latter together with the lower semicontinuity of (u, v) → −J 0 (u; v), and the continuity of (u, v) → ϕ(u, v) and u → u X guarantees that u → Θ f,γ (u, v) is lower semicontinuous for all v ∈ K. Next, we observe that Let {u n } ⊂ K be such that u n → u as n → ∞. Then, we have for all w ∈ K. Passing to supremum with w ∈ K for the above inequality, it gives so, the function Θ f,γ is lower semicontinuous. This completes the proof.
Let γ , ζ > 0 be two parameters. Moreover, let us consider the following functions for all u ∈ K, respectively. In the sequel, we call the functions Π Θ f,γ ,ζ and Π Θ f,γ ,ζ * to be the Moreau-Yosida regularized gap functions for Problem 2.1. Subsequently, we will verify that these functions are two gap functions for Problem 2.1. Proof We will show that Π Θ f,γ ,ζ is a gap function for Problem 2.1. In an analogous way, it is possible to demonstrate that Π Θ f,γ ,ζ * is also a gap function for Problem 2.1. (a) For any γ , ζ > 0 fixed, recall that Θ f,γ is a gap function for Problem 2.1, hence Θ f,γ (u) ≥ 0 for all u ∈ K. In consequence, Π Θ f,γ ,ζ (u) ≥ 0 for all u ∈ K.
(b) Suppose that u * ∈ K is a solution to Problem 2.1. Theorem 3.1 indicates that Θ f,γ (u * ) = 0. Moreover, the inequality Therefore, there exists a minimizing sequence {w n } in K such that It is obvious that Θ f,γ (w n ) → 0 and u * − w n X → 0, as n → ∞. This implies w n → u * , as n → +∞. Invoking Lemma 3.1 and nonnegativity of Θ f,γ results in the inequality thus is, Θ f,γ (u * ) = 0. Because Θ f,γ is a gap function, therefore, u * is a solution to Problem 2.1. The proof is complete.
We now provide an example to illustrate the results of Theorems 3.1 and 3.2. 3 2 ], f = 1 and A : R → R, ϕ : K × K → R and J : R → R be the functions defined by It is obvious that J is a locally Lipschitz function and Besides, it is not difficult to verify that all assumptions of Theorem 2.1 are valid with m A = 5, m J = 2 and α ϕ = 3 2 . Using Theorem 2.1, we deduce that the following inequality has a unique solution u = 1 2 : find u ∈ K such that for all v ∈ K. Next, let γ = 1. For the problem (3.9), we consider the regularized function Θ f,γ defined in (3.4). A simple calculation gives Observe that Θ f,γ (u) ≥ 0 for all u ∈ K, and Θ f,γ (u) = 0 if and only if u = 1 2 . This means that Θ f,γ is a gap function for the problem (3.9).
Analogously, it also can proved that the regularized gap function Θ f,γ * and the Moreau-Yosida regularized gap function Π Θ f,γ ,ζ * are two gap functions for the problem (3.9).
We conclude this section with two global error bounds for Problem 2.1 associated with the regularized gap function Θ f,γ and the Moreau-Yosida regularized gap function Π Θ f,γ ,ζ , respectively. These global error estimates measure the distance between any admissible point and the unique solution to Problem 2.1.

Theorem 3.3
Let u * ∈ K be the unique solution to Problem 2.1 and γ > 0 be such that m A − α ϕ − m J > 1 2γ . Assume that the hypotheses of Theorem 2.1 hold. Then, for each u ∈ K, we have (3.10) Proof Let u * ∈ K be the unique solution to Problem 2.1, i.e., For any u ∈ K fixed, we insert v = u into the above inequality to obtain By virtue of the definition of Θ f,γ , one has It follows from the monotonicity of A, hypotheses (2.2)(b) and (2. where the last inequality is obtained by using (3.11). Combining the last two inequality, we have Hence, the desired inequality (3.10) is valid.
Proof Let u * ∈ K be the unique solution of Problem 2.1. By the definition of the function Π Θ f,γ ,ζ , it follows for all u ∈ K, which completes the proof of the theorem.
Finally, we will illustrate the results of Theorems 3.3 and 3.4 by the following example. and

Application to an Elliptic Boundary Value Problem
In the section we shall investigate a boundary value problem with the generalized gradient and an obstacle effect which illustrates the applicability of the abstract results.
Let Ω be a bounded domain in R d (d = 2, 3) with Lipschitz continuous boundary Γ . The boundary is divided into two mutually disjoint measurable parts Γ 1 and Γ 2 such that meas(Γ 1 ) > 0. Consider the following elliptic mixed boundary value problem with constraints. Here ∂g and ∂ c h denote the generalized gradient and the convex subdifferential of the functions g : Ω × R → R and h : Γ 2 × R → R respectively with respect to their second variables, while the conormal derivative ∂u ∂νa = (a(x, ∇u), ν) R d represents the heat flux through the part Γ 2 , where ν stands for the outward unit normal on Γ .

Problem 4.1 Find a function
The mathematical model (4.1)-(4.4) is motivated by the study of semipermeability phenomena which may appear in the interior and on the boundary of the body Ω, and are met, for instance, in electrostatics, magnetostatics or stationary heat transfer (the behavior of natural and artificial semipermeable membranes of finite thickness, temperature control problems, etc.), see [44][45][46]50] and the references therein. The function u represents the electric potential, magnetic potential or temperature, respectively, the function a = a(x, ∇u) is the dielectric coefficient, magnetic permeability or thermal conductivity, and f = f (x) is a given source term. The material which occupies Ω is non-isotropic and heterogeneous, and thus a effectively depends on x. Condition (4.2) represents an additional unilateral constraint for the solution. Since the function g(x, ·) is supposed to be locally Lipschitz for a.e. x ∈ Ω, but not necessary convex, the multivalued relation (4.1) is nonmonotone in general. Combining it with (4.2)-(4.4) leads to a variational formulation which is a constrained variational-hemivariational inequality. Note that in general there is no function h such that ∂ h = k ∂ c h. This means that if g ≡ 0, then the weak form of Problem 4.1, stated in Problem 4.2 below, reduces to quasi-variational inequality.
We need the following standard functional space. Let X be defined by Since meas(Γ 1 ) > 0, the space X is endowed with the inner product and corresponding norm given by 1 2 for all u, v ∈ X. Also, we denote by γ 0 : X → L 2 (Γ ) the trace operator. On the other hand, we consider the admissible set K defined by In order to provide the result on the unique solvability of Problem 4.1, we need the following hypotheses on the data. (·, z) is measurable on Ω for all z ∈ R d with a(x, 0) = 0 for a.e. x ∈ Ω.
(b) a(x, ·) is continuous on R d for a.e. x ∈ Ω.
(c) a(x, z) R d ≤ m a (1 + z R d ) for all z ∈ R d , a.e. x ∈ Ω with m a > 0.
(ii) for any ζ > 0, if γ > 0 is such that then, for each u ∈ K the following bounds holds