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An interior approximal method for solving pseudomonotone equilibrium problems
Journal of Inequalities and Applications volume 2013, Article number: 156 (2013)
Abstract
In this paper, we present an interior approximal method for solving equilibrium problems for pseudomonotone bifunctions without Lipschitztype continuity on polyhedra. The method can be viewed as combining a special interior proximal function, which replaces the usual quadratic function, Armijotype linesearch techniques and the cutting hyperplane methods. Convergence properties of the method are established, among them the global convergences are proved under few assumptions. Finally, we present some preliminary computational results to CournotNash oligopolistic market equilibrium models.
MSC:65K10, 90C25.
1 Introduction
Let C be a nonempty closed convex subset of {\mathcal{R}}^{n} and a bifunction f:C\times C\to \mathcal{R} satisfying f(x,x)=0 for all x\in C. We consider equilibrium problems in the sense of Blum and Oettli [1] (shortly EP(f,C)), which are to find {x}^{\ast}\in C such that
Let Sol(f,C) denote the set of solutions of Problem EP(f,C). When
where F:C\to {\mathcal{R}}^{n}, Problem EP(f,C) is reduced to the variational inequalities:
In this article, for solving Problem EP(f,C), we assume that the bifunction f and C satisfy the following conditions:
A_{1}. C=\{x\in {R}^{n}:Ax\le b\}, where A is a p\times n maximal matrix (rankA=n), b\in {\mathcal{R}}^{p}, and intC=\{x:Ax<b\} is nonempty.
A_{2}. For each x\in C, the function f(x,\cdot ) is convex and subdifferentiable on C.
A_{3}. f is pseudomonotone on C\times C, i.e., for each x,y\in C, it holds
A_{4}. f is continuous on C\times C.
A_{5}. Sol(f,C)\ne \mathrm{\varnothing}.
Equilibrium problems appear in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics, and network (see [2–5]). In recent years, both theory and applications became attractive for many researchers (see [1, 6–14]).
Most of the methods for solving equilibrium problems are derived from fixed point formulations of Problem EP(f,C): A point {x}^{\ast}\in C is a solution of the problem if and only if {x}^{\ast}\in C is a solution of the following problem:
Namely, the sequence \{{x}^{k}\} is generated by {x}^{0}\in C and
To conveniently compute the point {x}^{k+1}, Mastroeni in [15] proposed the auxiliary problem principle for solving Problem EP(f,C). This principle is based on the following fixedpoint property: {x}^{\ast}\in C is a solution of Problem EP(f,C) if and only if {x}^{\ast}\in C is a solution of the problem:
where \lambda >0 and g(\cdot ) is a strongly convex differentiable function on C. Under the assumptions that f is strongly monotone with constant \beta >0 on C\times C, i.e.,
and f is Lipschitztype continuous with constants {c}_{1}>0, {c}_{2}>0, i.e.,
the author showed that the sequence \{{x}^{k}\} globally converges to a solution of Problem EP(f,C). However, the convergence depends on three positive parameters {c}_{1}, {c}_{2}, and β and in some cases, they are unknown or difficult to approximate.
Many algorithms for solving the optimization problems and variational inequalities are projection algorithms that employ projections onto the feasible set C, or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich [16] proposed an algorithm for solving the variational inequalities. In each iteration of the algorithm, in order to get the next iterate {x}^{k+1}, two orthogonal projections onto C are calculated, according to the following iterative step. Given the current iterate {x}^{k}, calculate
and then
where \lambda >0 is some positive number. Recently, Tran et al. [17] extended these projection techniques to Problem EP(f,C) involving monotone equilibrium bifunctions but it must satisfy a certain Lipschitztype continuous condition. To avoid this requirement, they proposed linesearch procedures commonly used in variational inequalities to obtain projectiontype algorithms for solving equilibrium problems.
It is well known that the interior approximal technique is a powerful tool for analyzing and solving optimization problems. This technique has been used extensively by many authors for solving variational inequalities and equilibrium problems on a polyhedron convex set (see [18–21]), where Bregmantype interior approximal function d replaces the function g in (1.1):
with \mu \in (0,1) and y\in {\mathcal{R}}_{+}^{n}=\{{({x}_{1},\dots ,{x}_{n})}^{T}\in {\mathcal{R}}^{n}:{x}_{i}>0\phantom{\rule{0.25em}{0ex}}\mathrm{\forall}i=1,\dots ,n\}. Then the interior proximal linesearch extragradient methods can be viewed as combining the function d and Armijotype linesearch techniques. Convergence of the iterative sequence is established under the weaker assumptions that f is pseudomonotone on C\times C. However, at each iteration k in the Armijotype linesearch progress of the algorithm requires the computation of a subgradient of the bifunction \partial f({x}^{k},\cdot )({y}^{k}), which is not easy in some cases. Moreover, most of current algorithms for solving Problem EP(f,C) are based on Lipschitztype continuous assumptions or the computation of subgradients of the bifunction f (see [21–25]).
Our main purpose of this paper is to give an iterative algorithm for solving a pseudomonotone equilibrium problem without Lipschitztype continuity of the bifunction and the computation of subgradients. To summarize our approach, first we use an interior proximal function d as in [22], which replaces the usual quadratic function in auxiliary problems. Next, we construct an appropriate hyperplane and a convex set, which separate the current iterative point from the solution set and we also combine this technique with the Armijotype linesearch technique. Then the next iteration is obtained as the projection of the current iterate onto the intersection of the feasible set with the convex set and the halfspace containing the solution set.
The paper is organized as follows. In Section 2, we recall the auxiliary problem principle of Problem EP(f,C) and propose a new iterative algorithm. Section 3 is devoted to the proof of its global convergence and also show the relation between the solution set of EP(f,C) and the cluster point of the iterative sequences in the algorithm. In Section 4, we apply our algorithm for solving generalized variational inequalities. Applications to the NashCournot oligopolistic market equilibrium model and the numerical results are reported in the last section.
2 Proposed algorithm
Let {a}_{i} denote the rows of the matrix A, and
where the function d is defined by (1.2). Then the gradient {\mathrm{\nabla}}_{1}D(x,y) of D(\cdot ,y) at x for every y\in C is defined by
where {X}_{y}=diag({l}_{1}(y),\dots ,{l}_{p}(y)) and log\frac{l(x)}{l(y)}=(log\frac{{l}_{1}(x)}{{l}_{1}(y)},\dots ,log\frac{{l}_{p}(x)}{{l}_{p}(y)}).
It is well known that {x}^{\ast} is a solution of the regularized auxiliary problem:
where c>0 is a regularization parameter, if and only if {x}^{\ast} is a solution of Problem EP(f,C) (see [3]). Motivated by this, first we solve the following strongly convex problem with the interior proximal function D:
for some positive constants β. It is easy to see that with f(x,y)=\u3008F(x),yx\u3009, where F:C\to {\mathcal{R}}^{n}, and D(x,y)=\frac{1}{2}{\parallel xy\parallel}^{2}, computing {y}^{k} becomes Step 1 of the extragradient method proposed in [16]. In Lemma 3.2(i), we will show that if \parallel {y}^{k}{x}^{k}\parallel =0 then {x}^{k} is a solution to Problem EP(f,C). Otherwise, a computationally inexpensive Armijotype procedure is used to find a point {z}^{k} such that the convex set {C}_{k}:=\{x\in {\mathcal{R}}^{n}:f({z}^{k},x)\le 0\} and the hyperplane {H}_{k}:=\{x\in {\mathcal{R}}^{n}:\u3008x{x}^{k},{x}^{0}{x}^{k}\u3009\le 0\} contain the solution set Sol(f,C) and strictly separates {x}^{k} from the solution. Then we compute the next iterate {x}^{k+1} by projecting {x}^{0} onto the intersection of the feasible set C with {C}_{k} and the halfspace {H}_{k}. The algorithm is described in more detail as follows.
Algorithm 2.1 Choose {x}^{0}\in C, 0<\sigma <\frac{\beta}{2{\parallel {\overline{A}}^{1}\parallel}^{2}} and \gamma \in (0,1).
Step 1.
Evaluate
If r({x}^{k})=0 then Stop. Otherwise, set {z}^{k}={x}^{k}{\gamma}^{{m}_{k}}r({x}^{k}), where {m}_{k} is the smallest nonnegative number such that
Step 2. Evaluate {x}^{k+1}={Pr}_{C\cap {C}_{k}\cap {H}_{k}}({x}^{0}), where
Increase k by 1, and return to Step 1.
3 Convergence results
In the next lemma, we show the existence of the nonnegative integer {m}_{k} in Algorithm 2.1.
Lemma 3.1 For \gamma \in (0,1), 0<\sigma <\frac{\beta}{2{\parallel {\overline{A}}^{1}\parallel}^{2}}, if \parallel r({x}^{k})\parallel >0 then there exists the smallest nonnegative integer {m}_{k} which satisfies (2.3).
Proof Assume on the contrary, (2.3) is not satisfied for any nonnegative integer i, i.e.,
Letting i\to \mathrm{\infty}, from the continuity of f, we have
Otherwise, for each t>0, we have 1\frac{1}{t}\le logt. We obtain after multiplication by \frac{{l}_{i}({y}^{k})}{{l}_{i}({x}^{k})}>0 for each i=1,\dots ,p,
Then it follows from rank\overline{A}=n that
and
Since {y}^{k} is the solution to the strongly convex program (2.2), we have
Substituting y={x}^{k}\in C and using assumptions f({x}^{k},{x}^{k})=0, D({x}^{k},{x}^{k})=0, we get
Combining (3.2) with (3.3), we obtain
Then inequalities (3.1) and (3.4) imply that
Hence, it must be either r({x}^{k})=0 or \sigma \ge \frac{\beta}{2{\parallel {\overline{A}}^{1}\parallel}^{2}}. The first case contradicts to r({x}^{k})\ne 0, while the second one contradicts to the fact \sigma <\frac{\beta}{2{\parallel {\overline{A}}^{1}\parallel}^{2}}. The proof is completed. □
Let us discuss the global convergence of Algorithm 2.1.
Lemma 3.2 Let \{{x}^{k}\} be the sequence generated by Algorithm 2.1 and Sol(f,C)\ne \mathrm{\varnothing}. Then the following hold.

(i)
If r({x}^{k})=0, then {x}^{k}\in Sol(f,C).

(ii)
{x}^{k}\notin {C}_{k}
.

(iii)
Sol(f,C)\subseteq C\cap {C}_{k}\cap {H}_{k}
.

(iv)
{lim}_{k\to \mathrm{\infty}}\parallel {x}^{k+1}{x}^{k}\parallel =0
.
Proof (i) Since {y}^{k} is the solution to problem (2.2) and an optimization result in convex programming, we have
where {N}_{C} denotes the normal cone. From {y}^{k}\in intC, it follows that {N}_{C}({y}^{k})=\{0\}. Hence,
where {\xi}^{k}\in \partial f({x}^{k},\cdot )({y}^{k}). Replacing {y}^{k}={x}^{k} in this equality, we get
Since
we have
Thus, {\xi}^{k}=0. Combining this with f({x}^{k},{x}^{k})=0, we obtain
which means that {x}^{k}\in Sol(f,C).

(ii)
Since {z}^{k}={x}^{k}{\gamma}^{{m}_{k}}r({x}^{k}), r({x}^{k})>0, f(x,x)=0 for every x\in C and f(x,\cdot ) is convex on C, we have
\begin{array}{rcl}0& =& f({z}^{k},{z}^{k})\\ =& f({z}^{k},(1{\gamma}^{{m}_{k}}){x}^{k}+{\gamma}^{{m}_{k}}{y}^{k})\\ \le & (1{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k})+{\gamma}^{{m}_{k}}f({z}^{k},{y}^{k})\\ \le & (1{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k}){\gamma}^{{m}_{k}}\sigma {\parallel r\left({x}^{k}\right)\parallel}^{2}\\ <& (1{\gamma}^{{m}_{k}})f({z}^{k},{x}^{k}).\end{array}
Hence, we have f({z}^{k},{x}^{k})>0. This means that {x}^{k}\notin {C}_{k}.

(iii)
For {x}^{\ast}\in Sol(f,C). Then since f is pseudomonotone on C and f({x}^{\ast},{z}^{k})\ge 0, we have f({z}^{k},{x}^{\ast})\le 0. So {x}^{\ast}\in {C}_{k}. To prove Sol(f,C)\subseteq {H}_{k}, we will use mathematical induction. Indeed, for k=0 we have {H}_{0}={\mathcal{R}}^{n}. This holds. Suppose that
Sol(f,C)\subseteq {H}_{m}\phantom{\rule{1em}{0ex}}\text{for}k=m\ge 0.
Then, from {x}^{\ast}\in Sol(f,C) and {x}^{m+1}={Pr}_{C\cap {C}_{m}\cap {H}_{m}}({x}^{0}), it follows that
and hence {x}^{\ast}\in {H}_{m+1}. It implies that Sol(f,C)\subseteq {H}_{m+1}. Therefore, (iii) is proved.

(iv)
Since {x}^{k} is the projection of {x}^{0} onto C\cap {C}_{k1}\cap {H}_{k1} and (iii), by the definition of projection, we have
\parallel {x}^{k}{x}^{0}\parallel \le \parallel {x}^{\ast}{x}^{0}\parallel \phantom{\rule{1em}{0ex}}\mathrm{\forall}{x}^{\ast}\in Sol(f,C).
So, \{{x}^{k}\} is bounded. Otherwise, using the definition of {H}_{k}, we have
and hence
From {x}^{k+1}\in {H}_{k}, it holds {x}^{k+1}={Pr}_{{H}_{k}}({x}^{k+1}). Combining this and (3.5), we obtain
This implies that
Thus, the sequence \{\parallel {x}^{k}{x}^{0}\parallel \} is bounded and nondecreasing, and hence there exists {lim}_{k\to \mathrm{\infty}}\parallel {x}^{k}{x}^{0}\parallel. Consequently,
□
Theorem 3.3 Suppose that assumptions A_{1} to A_{5} hold, \partial f(x,\cdot )(x) is upper semicontinuous on C, and the sequence \{{x}^{k}\} is generated by Algorithm 2.1. Then \{{x}^{k}\} globally converges to a solution {x}^{\ast} of Problem EP(f,C), where
Proof For each {\overline{w}}^{k}\in \partial f({z}^{k},\cdot )({z}^{k}), set
From {\overline{w}}^{k}\in \partial f({z}^{k},\cdot )({z}^{k}) and f(x,x)=0 for every x\in C, it follows that
Then we have
and hence
On the other hand, it follows from
that
Substituting y={y}^{k} into (3.7), we have
Combining this and (2.3), we have
From {z}^{k}=(1{\gamma}^{{m}_{k}}){x}^{k}+{\gamma}^{{m}_{k}}{y}^{k}, it implies that
Using this and (3.10), we have
From (3.9) and (3.11), it follows that
Then, since \partial f(x,\cdot )(x) is upper semicontinuous on C and \{{x}^{k}\} is bounded, there exists M>0 such that
Combining this, {x}^{k+1}\in {C}_{k} and (3.8), we have
Then, it follows from (iv) of Lemma 3.2 that
The cases remaining to consider are the following.
Case 1. {lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}{\gamma}^{{m}_{k}}>0.
This case must follow that {lim\hspace{0.17em}inf}_{k\to \mathrm{\infty}}\parallel r({x}^{k})\parallel =0. Since \{{x}^{k}\} is bounded, there exists an accumulation point \overline{x} of \{{x}^{k}\}. In other words, a subsequence \{{x}^{{k}_{i}}\} converges to some \overline{x} such that r(\overline{x})=0, as i\to \mathrm{\infty}. Then we see from Lemma 3.2(i) that \overline{x}\in Sol(f,C).
Case 2. {lim}_{k\to \mathrm{\infty}}{\gamma}^{{m}_{k}}=0.
Since \{\parallel {x}^{k}{x}^{0}\parallel \} is convergent, there is the subsequence \{{x}^{{k}_{j}}\} of \{{x}^{k}\} which converges to \overline{x} as j\to \mathrm{\infty}. Then, from the continuity of f and
there exists \overline{y} such that the sequence \{{y}^{{k}_{j}}\} converges \overline{y} as j\to \mathrm{\infty}, where
Since {m}_{k} is the smallest nonnegative integer, {m}_{k}1 does not satisfy (2.3). Hence, we have
and besides
Passing onto the limit in (3.12), as j\to \mathrm{\infty} and using the continuity of f, we have
where r(\overline{x})=\overline{x}\overline{y}. From Algorithm 2.1, we have
Since f is continuous, passing onto the limit, as j\to \mathrm{\infty}, we obtain
Using this and (3.13), we have
which implies r(\overline{x})=0, and hence \overline{x}\in Sol(f,C). So, all cluster points of \{{x}^{k}\} belong to the solution set Sol(f,C).
Set \stackrel{\u02c6}{x}={Pr}_{Sol(f,C)}({x}^{0}) and suppose that the subsequence \{{x}^{{k}_{j}}\} converges to {x}^{\ast}\in Sol(f,C) as j\to \mathrm{\infty}. By (iii) of Lemma 3.2, we have
So,
Thus,
As j\to \mathrm{\infty}, we get {x}^{{k}_{j}}\to {x}^{\ast} and
The last inequality is due to \stackrel{\u02c6}{x}={Pr}_{Sol(f,C)}({x}^{0}). So, {x}^{\ast}=\stackrel{\u02c6}{x} and the sequence \{{x}^{k}\} has an unique cluster point {Pr}_{Sol(f,C)}({x}^{0}). □
Now we consider the relation between the solution existence of Problem EP(f,C) and the convergence of \{{x}^{k}\} generated by Algorithm 2.1.
Lemma 3.4 (see [4])
Suppose that C is a compact convex subset of {\mathcal{R}}^{n} and f is continuous on C. Then the solution set of Problem EP(f,C) is nonempty.
Theorem 3.5 Suppose that assumptions A_{1} to A_{4} hold, f is continuous, \partial f(x,\cdot )(x) is upper semicontinuous on C, the sequence \{{x}^{k}\} is generated by Algorithm 2.1, and Sol(f,C)=\mathrm{\varnothing}. Then we have
Consequently, the solution set of Problem EP(f,C) is empty if and only if the sequence \{{x}^{k}\} diverges to infinity.
Proof The first, we show that C\cap {C}_{k}\cap {H}_{k}\ne \mathrm{\varnothing} for every k\ge 0. On the contrary, suppose that there exists {k}_{0}\ge 1 such that
Then there exists a positive number M such that
where B({x}^{0},M)=\{x\in {\mathcal{R}}^{n}:\parallel x{x}^{0}\parallel \le M\}. From Lemma 3.4, it implies that the solution set of Problem EP(f,\overline{C}) is nonempty, where \overline{C}=C\cap B({x}^{0},2M). Applying Algorithm 2.1 to Problem EP(f,\overline{C}). In order to avoid confusion with the sequences \{{x}^{k}\}, \{{C}_{k}\} and \{{H}_{k}\}, we denote the three corresponding sequences by \{{\overline{x}}^{k}\}, \{{\overline{C}}_{k}\} and \{{\overline{H}}_{k}\}. With {\overline{x}}^{0}={x}^{0}, the following claims hold:

(a)
The set \{{\overline{x}}^{k}\} has at least {k}_{0}+1 elements.

(b)
{x}^{k}={\overline{x}}^{k}
, {C}_{k}={\overline{C}}_{k} and {H}_{k}={\overline{H}}_{k} for every k=0,1,\dots ,{k}_{0}.

(c)
{x}^{{k}_{0}}
is not a solution to Problem EP(f,\overline{C}).
Using Sol(f,\overline{C})\ne \mathrm{\varnothing} and (iii) of Lemma 3.2, we have \overline{C}\cap {\overline{C}}_{k}\cap {\overline{H}}_{k}\ne \mathrm{\varnothing}. Then we also have C\cap {C}_{k}\cap {H}_{k}\ne \mathrm{\varnothing}, which contradicts the supposition that C\cap {C}_{k}\cap {H}_{k}=\mathrm{\varnothing}. So,
This implies that the inequality (3.6) also holds in this case, the sequence \{\parallel {x}^{k}{x}^{0}\parallel \} is still nondecreasing. We claim that
Suppose for contraction that the exists {lim}_{k\to \mathrm{\infty}}\parallel {x}^{k}{x}^{0}\parallel \in [0,+\mathrm{\infty}). Then \{{x}^{k}\} is bounded and it follows from (3.6) that
A similar discussion as above leads to the conclusion that the sequence \{{x}^{k}\} converges to {Pr}_{Sol(f,C)}({x}^{0}), which contradicts the emptiness of the solution set Sol(f,C). The theorem is proved. □
4 Applications to CournotNash equilibrium model
Now we consider the following CournotNash oligopolistic market equilibrium model (see [25–28]): There are nfirms producing a common homogenous commodity and that the price {p}_{i} of firm i depends on the total quantity {\sigma}_{x}={\sum}_{i=1}^{n}{x}_{i} of the commodity. Let {h}_{i}({x}_{i}) denote the cost of the firm i when its production level is {x}_{i}. Suppose that the profit of firm i is given by
where {h}_{i} is the cost function of firm i that is assumed to be dependent only on its production level. There is a common strategy space C\subseteq {\mathcal{R}}^{n} for all firms. Each firm seeks to maximize its own profit by choosing the corresponding production level under the presumption that the production of the other firms are parametric input. In this context, a Nash equilibrium is a production pattern in which in which no firm can increase its profit by changing its controlled variables. Thus, under this equilibrium concept, each firm determines its best response given other firms’ actions. Mathematically, a point {x}^{\ast}={({x}_{1}^{\ast},\dots ,{x}_{n}^{\ast})}^{T}\in C is said to be a Nash equilibrium point if {x}^{\ast} is a solution of the problem:
Set
and
Then the problem of finding an equilibrium point of this model can be formulated as Problem EP(f,C). It follows from Lemma 3.2 (i) that {x}^{k} is a solution of Problem EP(f,C) if and only if r({x}^{k})=0. Thus, {x}^{k} is an ϵsolution to Problem EP(f,C), if \parallel r({x}^{k})\parallel \le \u03f5. To illustrate our algorithm, we consider two academic numerical tests of the bifunction f in {\mathcal{R}}^{5}.
Example 4.1 We consider an application of CournotNash oligopolistic market equilibrium model taken from [17]. The equilibrium bifunction is defined by
where
and
In this case, the bifunction f is pseudomonotone on C and the interior approximal function (2.1) is defined through
It is easy to see that rankA=5. Take \parallel {\overline{A}}^{1}\parallel =1, \beta =4, \sigma =1.5, \gamma =0.7, \mu =0.55, we get iterates in Table 1. The approximate solution obtained after 361 iterations is
Example 4.2
The same as Example 4.1, we only change the bifunction which has the form
where arctan(xy)={(arctan({x}_{1}{y}_{1}),\dots ,arctan({x}_{5}{y}_{5}))}^{T}, the components of d are chosen randomly in (0,10).
Then the bifunction f satisfies convergent assumptions of Theorem 3.3 in this paper and Theorem 3.1 in [21]. We choose the parameters in Algorithm 2.1: \parallel {\overline{A}}^{1}\parallel =1, \beta =5, \sigma =1.2, \gamma =0.5, \mu =0.2. In the algorithm (shortly (IPLE)) proposed by Nguyen et al. [21], the parameters are chosen as follows: \theta =0.5, \tau =0.7, \alpha =0.4, \mu =2, {c}_{k}=0.5+\frac{1}{k+1} for all k\ge 1. We compare Algorithm 2.1 with (IPLE). The iteration numbers and the computational time for 5 problems are given in Table 2.
The computations are performed by Matlab R2008a running on a PC Desktop Intel(R) Core(TM)i5 650@3.2 GHz 3.33 GHz 4 Gb RAM.
5 Conclusion
This paper presented an iterative algorithm for solving pseudomonotone equilibrium problems without Lipschitztype continuity of the bifunctions. Combining the interior proximal extragradient method in [22], the Armijotype linesearch and cutting hyperplane techniques, the global convergence properties of the algorithm are established under few assumptions. Compared with the current methods such as the interior proximal extragradient method, the dual extragradient algorithm in [14], the auxiliary principle in [15], the inexact subgradient method in [29], and other methods in [4], the fundamental difference here is that our algorithm does not require the computation of subgradient of a convex function. We show that the cluster point of the sequence in our algorithm is the projection of the starting point onto the solution set of the equilibrium problems. Moreover, we also give the relation between the existence of solutions of equilibrium problems and the convergence of the iteration sequence.
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Acknowledgements
We are very grateful to the anonymous referees for their really helpful and constructive comments in improving the paper. The work was supported by National Foundation for Science and Technology Development of Vietnam (NAFOSTED), code 101.022011.07.
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The main idea of this paper is proposed by PNA. PNA and PMT prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Anh, P.N., Tuan, P.M. & Long, L.B. An interior approximal method for solving pseudomonotone equilibrium problems. J Inequal Appl 2013, 156 (2013). https://doi.org/10.1186/1029242X2013156
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DOI: https://doi.org/10.1186/1029242X2013156
Keywords
 equilibrium problem
 pseudomonotone
 interior approximal function
 cutting hyperplane method