Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

The aim of this paper is to establish new results on the error bounds for a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix. We first propose some regularized gap functions of this problem using the concept of GA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{A}$$\end{document}-convexity of a vector-valued function. Then, we derive error bounds for vector equilibrium problems with partial order given by a polyhedral cone in terms of regularized gap functions under some suitable conditions. Finally, a real-world application to a vector network equilibrium problem is given to illustrate the derived theoretical results.


Introduction
It is well known that the theory of equilibrium problems is a generalization of various problems related to optimization such as variational inequalities, complementarity problems and optimization problems, etc. Equilibrium problems have remarkable applications in the fields of mechanics, network analysis, transportation, finance, economics, operations research and optimization. For details, we refer the readers to works [9,13,18,20,25,26] and the references therein.
A scalar equilibrium problem is defined as follows: (EP) : find x * ∈ K such that F(x * , y) ≥ 0, for all y ∈ K , where K is a given set, F : K × K → R is a bifunction satisfying F(x, x) = 0 for all x ∈ K . The problem (EP) is also known under the name of equilibrium problem in the works of Blum and Oettli [4], Muu and Oettli [30], or Ky Fan equilibrium problem in [11]. The error bound of a certain problem is known as an upper estimate of the distance between an arbitrary feasible point and the solution set. It has played a crucial role from the point of view of theoretical analysis as well as to study the convergence of iterative algorithms for solving optimization problems, complementarity problems, variational inequalities and equilibrium problems. In 2003, based on the study of gap functions of (EP), Mastroeni [29] established global error bounds for (EP) under the assumption of strong monotonicity of F. The theory of gap functions in [29] is a generalization of the gap functions for variational inequalities considered in the literature. The notion of gap functions was first introduced by Auslender (1976) [2] to reformulate the variational inequality into an equivalent optimization problem: where h : R n → R n , and ·, · is the scalar product in R n . However, in general, the Auslender gap functions are not differentiable. To overcome this non-differentiability, Fukushima [12], Yamashita and Fukushima [39] proposed the notion of regularized gap functions for variational inequalities: where α is a nonnegative parameter. Moreover, Yamashita and Fukushima [39] also established the regularized function of Moreau-Yosida type as follows: where λ is a positive parameter. Using gap functions in forms of the Fukushima regularization and the Moreau-Yosida regularization, Yamashita and Fukushima [39] established global error bounds for general variational inequalities. Thereafter, many regularized gap functions and error bounds have been studied for various kinds of equilibrium problems and variational inequalities, see e.g., [1,3,19,[21][22][23][24]27,35] and the references therein for a more detailed discussion of interesting topics. In particular, Khan and Chen [27] established regularized gap functions and error bounds for generalized mixed vector equilibrium problems under the partial order introduced by the usual positive cone in finite dimensional spaces. Anh et al. [1] and Hung et al. [23] developed the results in [27] for generalized mixed vector equilibrium problems of strong types and mixed vector variational inequalities, respectively, whose final space is partially ordered by a infinite dimensional cone.
On the other hand, the study of linear inequalities has been applied to mathematical programming and econometrics (see Stoer and Witzgall [34]). This study has led to investigate a special class of cones, the polyhedral cones (cf. Definition 2.1). The theory of polyhedral cones is considered extensively. For more details, we refer the interested readers to [5,14,34]. Using properties of polyhedral convex cones associated to the matrices, Chegancas and Burgat [6] established sufficient conditions for asymptotic stability of a linear discrete-time-varing system. A class of linear complementarity problems over a polyhedral cone was considered by Zhang et al. [40]. They also designed a Newton-type algorithm for solving it. Very recently, Gutiérrez et al. [15,16] characterized the several kind of exact and approximate efficient solutions of a class of multiobjective optimization problems with partial order provided by a polyhedral cone. From a computational point of view, they showed that the results in [15,16] based on the ordering cone generated by some matrix are attractive. Using the partial order considered in [16], Hai et al. [17] continued to investigating vector equilibrium problems with a polyhedral ordering cone. In [17], variants of the Ekeland variational principle for a type of approximate proper solutions of those vector equilibrium problems were also provided. However, to the best of our knowledge, up to now, there is no paper devoted to error bounds for vector equilibrium problems and vector network equilibrium problems, whose final space is finite dimensional partially ordered by a polyhedral cone generated by some matrix.
Inspired by the works above, in this paper, we establish some new results on the error bounds for a class of vector equilibrium problems with a partial order defined by a polyhedral cone generated by some matrix based on regularized gap functions in forms of the Fukushima regularization and the Moreau-Yosida regularization introduced in [39] by using the concept of G A -convexity. Finally, we illustrate the main results by an application to a vector network equilibrium problem for which we establish analysing error bounds.
The paper is structured as follows. In Sect. 2 we introduce the framework, notations and definitions which are needed along the paper. Based on a partial order provided by a polyhedral cone, in Sect. 3 we propose some regularized gap functions for a class of vector equilibrium problems under some suitable conditions. Then, the error bound analysis for these problems in terms of regularized gap functions is studied. An application to a vector network equilibrium problem is given to illustrate our main theoretical results in Sect. 4. Finally, some conclusions are stated in Sect. 5.

Preliminaries and notations
We first introduce a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix A as follows: for all y ∈ C, where C is a nonempty closed and convex subset of a real normed space X , A = (a i j ) is a real matrix with p rows and m columns with the positive integers p, m such that p ≥ m and rank(A) = m, G A is a polyhedral cone generated by A (cf. Definition 2.1) such that G A has non-empty interior, F : In case of C = X being X is a nontrivial complete metric space, the problem VEP(C, F, G A ) was mentioned in [17]. Next, we recall some basic concepts and their properties. Throughout the paper, let R p be the p-dimensional Euclidean space and For any two vectors a = (a 1 , . . . , a p ) and b = (b 1 , . . . , b p ) , a, b ∈ R p , we define the relationships between vectors as follows: a < b if and only if a i < b i for all i ∈ {1, . . . , p}.
As usual, a hyperplane in R p is a set associated with some (a, b) ∈ R p × R, a = 0, and defined as {x ∈ R p : a, x = b}. The closed half-space of R p is a set associated with some (a, b) ∈ R p × R, a = 0, and defined as {x ∈ R p : a, x ≤ b}. A set P ⊂ R p is said to be a polyhedral set if it can be expressed as the intersection of a finite family of closed half-spaces or hyperplanes. Proposition 2.1 (see [32]) The following statements are equivalent for a set G ⊂ R m : (i) G is a polyhedral cone; (ii) G has a representation of the form for some positive integer p and some a i ∈ R m , i = 1, . . . , p.
Denote the set of all real matrices with p rows and m columns by R p×m . Definition 2.1 (see [10]) Let A ∈ R p×m . Then which is called a cone generated by A.
The cone G A is polyhedral, and so it is also convex and closed.
Proposition 2.2 (see [31], Proposition 4 and Proposition 5) Let A ∈ R p×m . Then (ii) If the matrix A has no zero rows, then Let A ∈ R p×m be a given matrix. The mapping defined by the matrix A is also denoted by A, where A : R m → R p defined by x → Ax (or A(x)) is a bounded linear mapping. Proposition 2.3 (see [33], Proposition 4.1) Let A be a mapping defined by a matrix A ∈ R p×m . Assume that the set {x ∈ R m : Ax ≥ 0} is a pointed cone, or, equivalently, that rank(A) = m and p ≥ m. Then, the following statements hold: To end this section, we derive a useful remark to convert the problem VEP(C, F, G A ) into a usual vector equilibrium problem, which is easier to handle from a computational point of view.

Remark 2.1 Suppose that the function
for all x, y ∈ C. Let A • F denote the composition of the mapping defined by the matrix A with the function F. Hence, Then, thanks to Proposition 2.2(ii), Proposition 2.3(ii) and the linearity of mapping A, we can show that (1) is equivalent to

Main results
In this section, we shall introduce the notion of the G A -convexity of a vector-valued function based on the partial order provided by a polyhedral cone. Then regularized gap functions and error bounds for the problem VEP(C, F, G A ) will be investigated by using the property of the G A -convexity and some suitable assumptions.
Throughout the paper, unless other specified, let X be a real normed space with norm · and C be a nonempty closed and convex subset of X . For a fixed subset D ⊂ X and a ∈ X , the distance between the point a and the set Definition 3.1 A real function ρ : C → R is said to be a gap function for the problem VEP(C, F, G A ) if the following properties hold: For each fixed constant γ > 0, we now consider the following function γ : C → R defined by where : C ×C → R + is a continuously differentiable function, which satisfies the following property with the associated constants β ≥ 2α > 0.
Remark 3.1 (i) The condition (7) is equivalent to (ii) Moreover, if a ik ≥ 0 and ϕ k is a convex function for all i ∈ {1, . . . , p} and k ∈ {1, . . . , m}, then it is easy to see that the above inequality holds and so ϕ is G A -convex. However, the reverse implication does not hold, that is, assume that ϕ is G A -convex. Then it is not necessary that the function ϕ k is convex for all k ∈ {1, . . . , m}. The following example shows the converse is not true.
Then, rank(A) = 2 and we have For each i ∈ {1, 2}, it is easy to check that the inequality (8) holds for all x, y ∈ C and λ ∈ [0, 1]. Thus, it follows from Remark 3.1(i) that ϕ is G A -convex on C. However, the function ϕ 2 defined by ϕ 2 (x) = −x 2 is not convex on C.

Proposition 3.1 Let F be G A -convex in the second component and satisfy the condition (H ).
Then for any γ > 0, the function γ defined by (6) is finite-valued on C.
This implies that for all i ∈ {1, . . . , p}, the function y → m k=1 a ik F k (x, y) is convex and so the function is strongly convex. Since C is a closed and convex set, the function (9) has a unique minimizer over C. Thus γ is finite-valued on C.

Theorem 3.1 Suppose that F is G
Equivalently, Then there exits for all y ∈ C. For any x ∈ C and λ ∈ (0, 1), we set y λ := λx * + (1 − λ)x ∈ C, and so It follows from the condition (H ) that By the G A -convexity of y → F(x * , y) and F k (x * , x * ) = 0 for all k ∈ {1, . . . , m}, we have m k=1 The above inequality becomes Having in mind relations (10)- (12), it gives for all x ∈ C. In (13), Conversely, suppose that x * is a solution of VEP(C, F, G A ), that is, x * ∈ C and Hence, for every y ∈ C, there exists i 0 (y) ∈ {1, . . . , p} such that It follows from the above inequality that γ (x * ) ≤ 0. Since γ (x * ) ≥ 0, one has γ (x * ) = 0. The proof is completed.

Remark 3.2
Recently, several regularized gap functions for various kinds of vector equilibrium problems with partial order provided by the cone R m + have been studied. For instance, we reconsider a special case of the vector equilibrium problem in Khan and Chen [27], which consists in finding x * ∈ C such that F(x * , y) = (F 1 (x  *  , y), . . . , F m (x * , y)) / ∈ −int(R m + ), ∀y ∈ C.
Then, to establish the regularized gap function for problem (14), all component functions y → F 1 (x, y), y → F 2 (x, y), . . . , y → F m (x, y) are imposed to be convex on C (see [27,Theorem 3.2]). Meanwhile, our Theorem 3.1 used the characteristic of the G A -convexity in terms of a perturbation of the matrix A, that is, the convexity assumption of all component functions F 1 , F 2 , . . . , F m in the gap function is not required.

Lemma 3.1
Assume that all assumptions of Theorem 3.1 hold and, further, that C is a compact set, F k is continuous for all k ∈ {1, . . . , m} and satisfies the condition (H ). Then, for each γ > 0, the gap function γ is continuous on C.
Proof Since F k is continuous for each k ∈ {1, . . . , m}, and is continuous, we have that the function h : is continuous in C × C. Moreover, C is a compact set, so the function γ defined by is continuous on C. This completes the proof. Now, we propose a gap function based on the Moreau-Yosida regularization of γ as follows: where ψ : C×C → R + is a continuously differentiable function, which satisfies the following property with the associated constants θ ≥ 2η > 0.

Theorem 3.2 Suppose that all conditions of Lemma 3.1 hold, and assume further that ψ satisfies the condition (H ψ ).
Then the function Δ ψ γ ,δ defined by (15) for any γ, δ > 0 is a gap function for VEP(C, F, G A ).
Proof (a) For any θ, τ > 0, it is easy to show that Δ ψ γ ,δ (x) ≥ 0 for all x ∈ C. (b) Assume that x * ∈ S(C, F, G A ). It follows from Theorem 3.1 that γ (x * ) = 0 and so Then, for each n, there is a u n ∈ C such that Since ψ satisfies the condition (H ψ ), it follows from (17) that This implies that γ (u n ) → 0 and x * − u n → 0, as n → +∞, i.e., γ (u n ) → 0 and u n → x * , as n → +∞. Thanks to Lemma 3.1, the continuity of γ holds, and so Therefore, we get x * ∈ S(C, F, G A ). This completes the proof. Now, we give the following example to illustrate all the assumptions of Theorems 3.1 and 3.2 are satisfied. Then, rank(A) = 2 and we have Let , ψ : C × C → R + and F = (F 1 , F 2 ) : C × C → R 2 be defined as follows: This implies that By Remark 2.1, the problem VEP(C, F, G A ) is equivalent to finding x ∈ C such that that is, finding x ∈ [0, e 2 ] such that Hence, it is easy to see that x = 0 is the solution of the problem VEP(C, F, G A ). Thus, we have S(C, F, By the convexity of R y → F 1 (x, y) and R y → F 2 (x, y) for all x ∈ R, we can show that R y → F(x, y) is G A -convex [by Remark 3.1(ii)]. Moreover, is a continuously differentiable function and satisfies condition (H ) with β = 1, α = 1 2 . Therefore, all the assumptions of Theorem 3.1 hold. Then for any γ > 0, the function γ defined by (6) is a gap function for VEP(C, F, G A ). For example, we take γ = 1, and we obtain Hence, γ is a gap function for VEP(C, F, G A ). By a similar argument, all the assumptions of Theorem 3.2 are satisfied with θ = 1 and η = 1 2 . Then for any γ, δ > 0, the function Δ ψ γ ,δ defined by (15) is a gap function for VEP(C, F, G A ). For example, we take γ = θ = 1, and we have

Remark 3.3
As mentioned before there is no result concerning regularized gap functions for the problem VEP(C, F, G A ). Thus, our results Theorems 3.1 and 3.2 are new in establishing regularized gap functions for vector equilibrium problems, whose final space is finite dimensional and partially ordered by a polyhedral cone generated by some matrix.
for all x, y ∈ C, H : C → R n , then the functions γ and Δ ψ γ ,δ reduce to gap functions in forms of the Fukushima regularization and the Moreau-Yosida regularization introduced in [39], respectively for variational inequalities. Hence, in this sense, Theorems 3.1 and 3.2 extend the corresponding known results in [39].
To establish error bounds for the problem VEP(C, F, G A ), we now impose the following hypotheses: Next, we establish error bounds for vector equilibrium problems with partial order provided by a polyhedral cone based on the regularized gap functions studied above.
Proof Let x * ∈ S(C, F, G A ). For each x ∈ C, we have Without loss of generality, we assume that there exists i * ∈ {1, . . . , p} such that and so, (19) gives that It follows from the condition (H F ) 2 that Since the condition (H F ) 1 holds, without loss of generality, we assume that Moreover, it follows from the property (H ) that Employing (20)- (23), we obtain This implies that and the proof is completed.
Proof Let x * ∈ S(C, F, G A ). According to Theorem 3.3 and thanks to the condition (H ψ ), for each x ∈ C, we obtain Hence, Therefore, the proof is completed.
Finally, we give an illustrative example where it is shown that all assumptions of Theorem 3.3 and Theorem 3.4 hold. X , C, p, m, A, F, , ψ, α, β, γ, δ,

Example 3.3 Let
This implies that 0 Furthermore, it follows from Example 3.2 that all assumptions of Theorems 3.1 and 3.2 hold. Thus, we conclude that all the assumptions of Theorems 3.3 and 3.4 are satisfied and so, (18) and (24) hold. Indeed, for example, for all x ∈ [0, e 2 ] and for x * = 0 ∈ S(C, F, G A ), we have Hence, inequalities (18) and (24)

Application to a vector network equilibrium problem
The network equilibrium model was introduced by Wardrop [36] for a transportation network. This model has played a vital role in the traffic network planning and to optimize the traffic control. Based on vector-valued cost functions or multicriteria consideration, many network equilibrium models have been investigated, see e.g., [7,8,28,37] and the references therein. In this section, we consider a formulation of vector network equilibrium problem with partial order provided by a polyhedral cone generated by some matrix, which illustrates the applicability of the abstract results. Let v = (v 1 , v 2 , . . . , v ν ) ∈ R ν be the vector of arc flow. We say that a path flow f satisfies demands if k∈P w f k =d w for all w ∈ W. A flow f ≥ 0 satisfying the demand is called a feasible flow. Let Assume that H = ∅. It is easy to check that the set H is compact and convex. Let c e : R ν → R m be a vector-valued cost function for arc e which is in general a function of all the arc flows.
We assume that c e (v) = (c 1 e (v), c 2 e (v), . . . , c m e (v)) ∈ R m . Let T k : R N → R m be a vectorvalued cost function along the path k. For each w ∈ W and k ∈ P w , the vector cost T k is assumed to be the sum of all the arc cost of the flow f k through arcs, which belong to the path k, i.e., For each w ∈ W, k ∈ P w , j ∈ {1, 2, . . . , m}, v ∈ R ν and f ∈ H, let Then, for each f ∈ H, let Let G A be the polyhedral cone considered in Sect. 2. We now introduce the notion of G A -equilibria of a flow f ∈ H.

Definition 4.1 A flow f ∈ H is said to be in G
Remark 4.1 Let p = m. If A is the identity matrix of size m, then G A = {x ∈ R m : Ax ≥ 0} = R m + . We get that (25) becomes Then the flow f is in weak vector equilibrium, see [7, Definition 3.2].

Proposition 4.1
The path flow f * ∈ H is in G A -equilibrium if f * solves the vector variational inequality (for short, VVI(H, T , G A )) : Proof Let f * ∈ H be the solution of problem VVI(H, T , G A ). Consider a path flow vector h to be such that Hence, ∀w ∈ W, q∈P w h q = q∈P w f * q =d w , and so h ∈ H. Then, we have

Remark 4.3 (i)
To the best of our knowledge, up to now, there is no paper concerning regularized gap functions and error bounds for vector network equilibrium problem with partial order provided by a polyhedral cone generated by some matrix. Thus, Theorems 4.1 and 4.2 are new. (ii) In the case of Remark 4.1, G A = {x ∈ R m : Ax ≥ 0} = R m + . Then, our formulated vector network equilibrium problem reduces to the vector network equilibrium problem with the ordering cone R m + considered in [7,8]. In this special case, our main results in this section are still new.

Conclusions
In this work, we have studied a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix A. Using the concept of G A -convexity of a vector-valued function, we have proposed some gap functions in forms of the Fukushima regularization and the Moreau-Yosida regularization to the problem VEP(C, F, G A ) (Theorems 3.1 and 3.2). We have also provided some error bounds for the problem VEP(C, F, G A ) by virtue of these regularized gap functions (Theorems 3.3 and 3.4). Especially, to illustrate our main theoretical findings in a real-world application, we have derived a vector network equilibrium problem in Sect. 4. Our results in this section are new even in the case where the ordering cone in R m + .