Open Access

An interior approximal method for solving pseudomonotone equilibrium problems

Journal of Inequalities and Applications20132013:156

https://doi.org/10.1186/1029-242X-2013-156

Received: 10 July 2012

Accepted: 10 March 2013

Published: 5 April 2013

Abstract

In this paper, we present an interior approximal method for solving equilibrium problems for pseudomonotone bifunctions without Lipschitz-type continuity on polyhedra. The method can be viewed as combining a special interior proximal function, which replaces the usual quadratic function, Armijo-type 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 Cournot-Nash oligopolistic market equilibrium models.

MSC:65K10, 90C25.

Keywords

equilibrium problem pseudomonotone interior approximal function cutting hyperplane method

1 Introduction

Let C be a nonempty closed convex subset of R n and a bifunction f : C × C R satisfying f ( x , x ) = 0 for all x C . We consider equilibrium problems in the sense of Blum and Oettli [1] (shortly EP ( f , C ) ), which are to find x C such that
f ( x , y ) 0 y C .
Let Sol ( f , C ) denote the set of solutions of Problem EP ( f , C ) . When
f ( x , y ) = F ( x ) , y x x , y C ,
where F : C R n , Problem EP ( f , C ) is reduced to the variational inequalities:
Find  x C  such that, for all  y C , F ( x ) , y x 0 .

In this article, for solving Problem EP ( f , C ) , we assume that the bifunction f and C satisfy the following conditions:

A1. C = { x R n : A x b } , where A is a p × n maximal matrix ( rank A = n ), b R p , and int C = { x : A x < b } is nonempty.

A2. For each x C , the function f ( x , ) is convex and subdifferentiable on C.

A3. f is pseudomonotone on C × C , i.e., for each x , y C , it holds
f ( x , y ) 0 implies f ( y , x ) 0 .

A4. f is continuous on C × C .

A5. Sol ( f , C ) .

Equilibrium problems appear in many practical problems arising, for instance, physics, engineering, game theory, transportation, economics, and network (see [25]). In recent years, both theory and applications became attractive for many researchers (see [1, 614]).

Most of the methods for solving equilibrium problems are derived from fixed point formulations of Problem EP ( f , C ) : A point x C is a solution of the problem if and only if x C is a solution of the following problem:
min { f ( x , y ) : y C } .
Namely, the sequence { x k } is generated by x 0 C and
x k + 1 arg min { f ( x k , y ) : y C } .
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 fixed-point property: x C is a solution of Problem EP ( f , C ) if and only if x C is a solution of the problem:
min { λ f ( x , y ) + g ( y ) g ( x ) , y : y C } ,
(1.1)
where λ > 0 and g ( ) is a strongly convex differentiable function on C. Under the assumptions that f is strongly monotone with constant β > 0 on C × C , i.e.,
f ( x , y ) + f ( y , x ) β x y 2 x , y C ,
and f is Lipschitz-type continuous with constants c 1 > 0 , c 2 > 0 , i.e.,
f ( x , y ) + f ( y , z ) f ( x , z ) c 1 x y 2 c 2 y z 2 x , y , z C ,

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
y k : = Pr C ( x k λ F ( x k ) )
and then
x k + 1 : = Pr C ( x k λ F ( y k ) ) ,

where λ > 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 Lipschitz-type continuous condition. To avoid this requirement, they proposed linesearch procedures commonly used in variational inequalities to obtain projection-type 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 [1821]), where Bregman-type interior approximal function d replaces the function g in (1.1):
d ( x , y ) = { 1 2 x y 2 + μ i = 1 n y i 2 ( x i y i log x i y i x i y i + 1 ) if  x i > 0 i = 1 , , n , + otherwise,
(1.2)

with μ ( 0 , 1 ) and y R + n = { ( x 1 , , x n ) T R n : x i > 0 i = 1 , , n } . Then the interior proximal linesearch extragradient methods can be viewed as combining the function d and Armijo-type linesearch techniques. Convergence of the iterative sequence is established under the weaker assumptions that f is pseudomonotone on C × C . However, at each iteration k in the Armijo-type linesearch progress of the algorithm requires the computation of a subgradient of the bifunction f ( x k , ) ( y k ) , which is not easy in some cases. Moreover, most of current algorithms for solving Problem EP ( f , C ) are based on Lipschitz-type continuous assumptions or the computation of subgradients of the bifunction f (see [2125]).

Our main purpose of this paper is to give an iterative algorithm for solving a pseudomonotone equilibrium problem without Lipschitz-type 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 Armijo-type 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 half-space 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 Nash-Cournot 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
(2.1)
where the function d is defined by (1.2). Then the gradient 1 D ( x , y ) of D ( , y ) at x for every y C is defined by
1 D ( x , y ) = A T ( l ( x ) l ( y ) + μ X y log l ( x ) l ( y ) ) ,

where X y = diag ( l 1 ( y ) , , l p ( y ) ) and log l ( x ) l ( y ) = ( log l 1 ( x ) l 1 ( y ) , , log l p ( x ) l p ( y ) ) .

It is well known that x is a solution of the regularized auxiliary problem:
Find  x C  such that  f ( x , y ) + 1 c D ( y , x ) 0  for all  y C ,
where c > 0 is a regularization parameter, if and only if x 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:
y k = arg min { f ( x k , y ) + β D ( y , x k ) : y C } ,

for some positive constants β. It is easy to see that with f ( x , y ) = F ( x ) , y x , where F : C R n , and D ( x , y ) = 1 2 x y 2 , computing y k becomes Step 1 of the extragradient method proposed in [16]. In Lemma 3.2(i), we will show that if y k x k = 0 then x k is a solution to Problem EP ( f , C ) . Otherwise, a computationally inexpensive Armijo-type procedure is used to find a point z k such that the convex set C k : = { x R n : f ( z k , x ) 0 } and the hyperplane H k : = { x R n : x x k , x 0 x k 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 half-space H k . The algorithm is described in more detail as follows.

Algorithm 2.1 Choose x 0 C , 0 < σ < β 2 A ¯ 1 2 and γ ( 0 , 1 ) .

Step 1.

Evaluate
y k = arg min { f ( x k , y ) + β 2 D ( y , x k ) : y C } , r ( x k ) = x k y k .
(2.2)
If r ( x k ) = 0 then Stop. Otherwise, set z k = x k γ m k r ( x k ) , where m k is the smallest nonnegative number such that
f ( x k γ m k r ( x k ) , y k ) σ r ( x k ) 2 .
(2.3)
Step 2. Evaluate x k + 1 = Pr C C k H k ( x 0 ) , where
{ C k = { x R n : f ( z k , x ) 0 } , H k = { x R n : x x k , x 0 x k 0 } .

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 γ ( 0 , 1 ) , 0 < σ < β 2 A ¯ 1 2 , if r ( x k ) > 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.,
f ( x k γ i r ( x k ) , y k ) + σ r ( x k ) 2 > 0 .
Letting i , from the continuity of f, we have
f ( x k , y k ) + σ r ( x k ) 2 0 .
(3.1)
Otherwise, for each t > 0 , we have 1 1 t log t . We obtain after multiplication by l i ( y k ) l i ( x k ) > 0 for each i = 1 , , p ,
l i ( y k ) l i ( x k ) 1 l i ( y k ) l i ( x k ) log l i ( y k ) l i ( x k ) .
Then it follows from rank A ¯ = n that
x y = A ¯ 1 A ¯ ( x y ) A ¯ 1 A ¯ ( x y )
and
D ( y k , x k ) = 1 2 l ( x k ) l ( y k ) 2 + μ i = 1 n l i 2 ( x k ) ( l i ( y k ) l i ( x k ) log l i ( y k ) l i ( x k ) l i ( y k ) l i ( x k ) + 1 ) 1 2 l ( x k ) l ( y k ) 2 = 1 2 A ( x k y k ) 2 1 2 A ¯ ( x k y k ) 2 1 2 A ¯ 1 2 x k y k 2 = 1 2 A ¯ 1 2 r ( x k ) 2 .
(3.2)
Since y k is the solution to the strongly convex program (2.2), we have
f ( x k , y ) + β D ( y , x k ) f ( x k , y k ) + β D ( y k , x k ) y C .
Substituting y = x k C and using assumptions f ( x k , x k ) = 0 , D ( x k , x k ) = 0 , we get
f ( x k , y k ) + β D ( y k , x k ) 0 .
(3.3)
Combining (3.2) with (3.3), we obtain
f ( x k , y k ) + β 2 A ¯ 1 2 r ( x k ) 2 0 .
(3.4)
Then inequalities (3.1) and (3.4) imply that
σ r ( x k ) 2 f ( x k , y k ) β 2 A ¯ 1 2 r ( x k ) 2 .

Hence, it must be either r ( x k ) = 0 or σ β 2 A ¯ 1 2 . The first case contradicts to r ( x k ) 0 , while the second one contradicts to the fact σ < β 2 A ¯ 1 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 ) . Then the following hold.
  1. (i)

    If r ( x k ) = 0 , then x k Sol ( f , C ) .

     
  2. (ii)
    x k C k
    .
     
  3. (iii)
    Sol ( f , C ) C C k H k
    .
     
  4. (iv)
    lim k x k + 1 x k = 0
    .
     
Proof (i) Since y k is the solution to problem (2.2) and an optimization result in convex programming, we have
0 f ( x k , ) ( y k ) + β 1 D ( y k , x k ) + N C ( y k ) ,
where N C denotes the normal cone. From y k int C , it follows that N C ( y k ) = { 0 } . Hence,
ξ k + β 1 D ( y k , x k ) = 0 ,
where ξ k f ( x k , ) ( y k ) . Replacing y k = x k in this equality, we get
ξ k + β 1 D ( x k , x k ) = 0 .
Since
1 D ( x , y ) = A T ( l ( x ) l ( y ) + μ X y log l ( x ) l ( y ) ) x , y int C ,
we have
1 D ( x k , x k ) = 0 .
Thus, ξ k = 0 . Combining this with f ( x k , x k ) = 0 , we obtain
f ( x k , y ) ξ k , y ξ k = 0 y C ,
which means that x k Sol ( f , C ) .
  1. (ii)
    Since z k = x k γ m k r ( x k ) , r ( x k ) > 0 , f ( x , x ) = 0 for every x C and f ( x , ) is convex on C, we have
    0 = f ( z k , z k ) = f ( z k , ( 1 γ m k ) x k + γ m k y k ) ( 1 γ m k ) f ( z k , x k ) + γ m k f ( z k , y k ) ( 1 γ m k ) f ( z k , x k ) γ m k σ r ( x k ) 2 < ( 1 γ m k ) f ( z k , x k ) .
     
Hence, we have f ( z k , x k ) > 0 . This means that x k C k .
  1. (iii)
    For x Sol ( f , C ) . Then since f is pseudomonotone on C and f ( x , z k ) 0 , we have f ( z k , x ) 0 . So x C k . To prove Sol ( f , C ) H k , we will use mathematical induction. Indeed, for k = 0 we have H 0 = R n . This holds. Suppose that
    Sol ( f , C ) H m for  k = m 0 .
     
Then, from x Sol ( f , C ) and x m + 1 = Pr C C m H m ( x 0 ) , it follows that
x x m + 1 , x 0 x m + 1 0 ,
and hence x H m + 1 . It implies that Sol ( f , C ) H m + 1 . Therefore, (iii) is proved.
  1. (iv)
    Since x k is the projection of x 0 onto C C k 1 H k 1 and (iii), by the definition of projection, we have
    x k x 0 x x 0 x Sol ( f , C ) .
     
So, { x k } is bounded. Otherwise, using the definition of H k , we have
x x k , x 0 x k 0 x H k ,
and hence
x k = Pr H k ( x 0 ) .
(3.5)
From x k + 1 H k , it holds x k + 1 = Pr H k ( x k + 1 ) . Combining this and (3.5), we obtain
x k + 1 x k 2 = Pr H k ( x k + 1 ) Pr H k ( x 0 ) 2 x k + 1 x 0 2 Pr H k ( x k + 1 ) x k + 1 + x 0 Pr H k ( x 0 ) 2 = x k + 1 x 0 2 x k x 0 2 .
This implies that
x k + 1 x 0 2 x k x 0 2 + x k + 1 x k 2 .
(3.6)
Thus, the sequence { x k x 0 } is bounded and nondecreasing, and hence there exists lim k x k x 0 . Consequently,
lim k x k + 1 x k = 0 .

 □

Theorem 3.3 Suppose that assumptions A1 to A5 hold, f ( x , ) ( 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 of Problem EP ( f , C ) , where
x = Pr Sol ( f , C ) ( x 0 ) .
Proof For each w ¯ k f ( z k , ) ( z k ) , set
H k = { y R n : w ¯ k , y z k 0 } .
From w ¯ k f ( z k , ) ( z k ) and f ( x , x ) = 0 for every x C , it follows that
w ¯ k , y z k f ( z k , y ) f ( z k , z k ) = f ( z k , y ) .
(3.7)
Then we have
C k H k k 0 ,
and hence
x k Pr H k ( x k ) x k Pr C k ( x k ) .
(3.8)
On the other hand, it follows from
Pr H k ( x k ) = x k w ¯ k , y z k w ¯ k 2 w ¯ k
that
x k Pr H k ( x k ) = | w ¯ k , y z k | w ¯ k .
(3.9)
Substituting y = y k into (3.7), we have
f ( z k , y k ) w ¯ k , y k z k .
Combining this and (2.3), we have
w ¯ k , z k y k σ r ( x k ) 2 .
(3.10)
From z k = ( 1 γ m k ) x k + γ m k y k , it implies that
x k z k = γ m k 1 γ m k ( z k y k ) .
Using this and (3.10), we have
w ¯ k , x k z k = γ m k 1 γ m k w ¯ k , z k y k γ m k σ r ( x k ) 2 1 γ m k .
(3.11)
From (3.9) and (3.11), it follows that
x k Pr H k ( x k ) γ m k σ r ( x k ) 2 w ¯ k .
Then, since f ( x , ) ( x ) is upper semicontinuous on C and { x k } is bounded, there exists M > 0 such that
x k Pr H k ( x k ) γ m k σ r ( x k ) 2 M .
Combining this, x k + 1 C k and (3.8), we have
x k + 1 x k x k Pr C k ( x k ) γ m k σ r ( x k ) 2 M .
Then, it follows from (iv) of Lemma 3.2 that
lim k γ m k r ( x k ) 2 = 0 .

The cases remaining to consider are the following.

Case 1. lim sup k γ m k > 0 .

This case must follow that lim inf k r ( x k ) = 0 . Since { x k } is bounded, there exists an accumulation point x ¯ of { x k } . In other words, a subsequence { x k i } converges to some x ¯ such that r ( x ¯ ) = 0 , as i . Then we see from Lemma 3.2(i) that x ¯ Sol ( f , C ) .

Case 2. lim k γ m k = 0 .

Since { x k x 0 } is convergent, there is the subsequence { x k j } of { x k } which converges to x ¯ as j . Then, from the continuity of f and
y k j = arg min { f ( x k j , y ) + β 2 D ( y , x k j ) : y C } ,
there exists y ¯ such that the sequence { y k j } converges y ¯ as j , where
y ¯ = arg min { f ( x ¯ , y ) + β 2 D ( y , x ¯ ) : y C } .
Since m k is the smallest nonnegative integer, m k 1 does not satisfy (2.3). Hence, we have
f ( x k γ m k 1 r ( x k ) , y k ) > σ r ( x k ) 2 ,
and besides
f ( x k j γ m k j 1 r ( x k j ) , y k j ) > σ r ( x k j ) 2 .
(3.12)
Passing onto the limit in (3.12), as j and using the continuity of f, we have
f ( x ¯ , y ¯ ) σ r ( x ¯ ) 2 ,
(3.13)
where r ( x ¯ ) = x ¯ y ¯ . From Algorithm 2.1, we have
f ( x k j γ m k j r ( x k j ) , y k j ) σ r ( x k j ) 2 .
Since f is continuous, passing onto the limit, as j , we obtain
f ( x ¯ , y ¯ ) σ r ( x ¯ ) 2 .
Using this and (3.13), we have
σ r ( x ¯ ) 2 f ( x ¯ , y ¯ ) σ r ( x ¯ ) 2 ,

which implies r ( x ¯ ) = 0 , and hence x ¯ Sol ( f , C ) . So, all cluster points of { x k } belong to the solution set Sol ( f , C ) .

Set x ˆ = Pr Sol ( f , C ) ( x 0 ) and suppose that the subsequence { x k j } converges to x Sol ( f , C ) as j . By (iii) of Lemma 3.2, we have
x ˆ C C k j 1 H k j 1 .
So,
x k j x 0 x ˆ x 0 .
Thus,
x k j x ˆ 2 = x k j x 0 + x 0 x ˆ 2 = x k j x 0 2 + x 0 x ˆ 2 + 2 x k j x 0 , x 0 x ˆ x ˆ x 0 2 + x 0 x ˆ 2 + 2 x k j x 0 , x 0 x ˆ .
As j , we get x k j x and
x x ˆ 2 2 x ˆ x 0 2 + 2 x x 0 , x 0 x ˆ = x x ¯ , x 0 x ¯ 0 .

The last inequality is due to x ˆ = Pr Sol ( f , C ) ( x 0 ) . So, x = x ˆ 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 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 A1 to A4 hold, f is continuous, f ( x , ) ( x ) is upper semicontinuous on C, the sequence { x k } is generated by Algorithm 2.1, and Sol ( f , C ) = . Then we have
lim k x k x 0 = + .

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 C k H k for every k 0 . On the contrary, suppose that there exists k 0 1 such that
C C k H k = .
Then there exists a positive number M such that
{ x k : 0 k k 0 } B ( x 0 , M ) ,
where B ( x 0 , M ) = { x R n : x x 0 M } . From Lemma 3.4, it implies that the solution set of Problem EP ( f , C ¯ ) is nonempty, where C ¯ = C B ( x 0 , 2 M ) . Applying Algorithm 2.1 to Problem EP ( f , C ¯ ) . In order to avoid confusion with the sequences { x k } , { C k } and { H k } , we denote the three corresponding sequences by { x ¯ k } , { C ¯ k } and { H ¯ k } . With x ¯ 0 = x 0 , the following claims hold:
  1. (a)

    The set { x ¯ k } has at least k 0 + 1 elements.

     
  2. (b)
    x k = x ¯ k
    , C k = C ¯ k and H k = H ¯ k for every k = 0 , 1 , , k 0 .
     
  3. (c)
    x k 0
    is not a solution to Problem EP ( f , C ¯ ) .
     
Using Sol ( f , C ¯ ) and (iii) of Lemma 3.2, we have C ¯ C ¯ k H ¯ k . Then we also have C C k H k , which contradicts the supposition that C C k H k = . So,
C C k H k k 0 .
This implies that the inequality (3.6) also holds in this case, the sequence { x k x 0 } is still nondecreasing. We claim that
lim k x k x 0 = .
Suppose for contraction that the exists lim k x k x 0 [ 0 , + ) . Then { x k } is bounded and it follows from (3.6) that
lim k x k + 1 x k = 0 .

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 Cournot-Nash equilibrium model

Now we consider the following Cournot-Nash oligopolistic market equilibrium model (see [2528]): There are n-firms producing a common homogenous commodity and that the price p i of firm i depends on the total quantity σ x = 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
f i ( x 1 , , x n ) = x i p i ( σ x ) h i ( x i ) i = 1 , , n ,
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 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 = ( x 1 , , x n ) T C is said to be a Nash equilibrium point if x is a solution of the problem:
max { f i ( y , i ) : y , i = ( x 1 , , x i 1 , y i , x i + 1 , , x n ) T C } i = 1 , , n .
Set
ϕ ( x , y ) = i = 1 n f i ( x 1 , , x i 1 , y i , x i + 1 , , x n )
(4.1)
and
f ( x , y ) = ϕ ( x , y ) ϕ ( x , x ) .
(4.2)

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 ϵ-solution to Problem EP ( f , C ) , if r ( x k ) ϵ . To illustrate our algorithm, we consider two academic numerical tests of the bifunction f in  R 5 .

Example 4.1 We consider an application of Cournot-Nash oligopolistic market equilibrium model taken from [17]. The equilibrium bifunction is defined by
f ( x , y ) = M ( x + y ) + x , d B ( x + y ) + q , y x ,
(4.3)
where
and
C = { x R + 5 , 4 x 1 + 2 x 2 + x 3 + 3 x 5 12 , 7 i = 1 5 x i 15 , 6 x 2 + x 3 + 2 x 4 13 , 3 x 2 + x 3 5 .
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 rank A = 5 . Take A ¯ 1 = 1 , β = 4 , σ = 1.5 , γ = 0.7 , μ = 0.55 , we get iterates in Table 1. The approximate solution obtained after 361 iterations is
x 361 = ( 4.5397 , 0.0529 , 2.8942 , 2.0265 , 1.4868 ) T .
Table 1

Example 4.1: Iterations of Algorithm 2.1 with r ( x k ) 0.0001

Iteration (k)

x 1 k

x 2 k

x 3 k

x 4 k

x 5 k

0

1

3

1

1

1

10

4.3842

0.0001

3.4633

1.7683

1.3842

20

4.4657

0.0013

3.1371

1.9315

1.4657

50

4.4901

0.0017

3.0404

1.9796

1.4898

100

4.4976

0.0082

3.0117

1.9938

1.4969

150

4.5001

0.0099

3.0031

1.9979

1.4990

180

4.5012

0.0107

3.0005

1.9989

1.4995

200

4.5106

0.0142

2.9716

2.0071

1.4965

250

4.5309

0.0412

2.9176

2.0206

1.4897

300

4.5370

0.0494

2.9013

2.0247

1.4877

350

4.5389

0.0519

2.8963

2.0259

1.4870

355

4.5395

0.0526

2.8948

2.0263

1.4868

358

4.5396

0.0528

2.8943

2.0264

1.4868

360

4.5397

0.0529

2.8942

2.0264

1.4868

361

4.5397

0.0529

2.8942

2.0265

1.4868

Example 4.2

The same as Example 4.1, we only change the bifunction which has the form
f ( x , y ) = P x + Q y + q , y x + d , arctan ( x y ) ,

where arctan ( x y ) = ( arctan ( x 1 y 1 ) , , 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: A ¯ 1 = 1 , β = 5 , σ = 1.2 , γ = 0.5 , μ = 0.2 . In the algorithm (shortly (IPLE)) proposed by Nguyen et al. [21], the parameters are chosen as follows: θ = 0.5 , τ = 0.7 , α = 0.4 , μ = 2 , c k = 0.5 + 1 k + 1 for all k 1 . We compare Algorithm 2.1 with (IPLE). The iteration numbers and the computational time for 5 problems are given in Table 2.
Table 2

Example 4.2: The tolerance r ( x k ) 0.19

Algorithm 2.1

Algorithm (IPLE)

Problem

No. iterations

CPU times (seconds)

Problem

No. iterations

CPU time (seconds)

No. 1

547

16.5140

No. 1

421

15.1388

No. 2

379

8.0249

No. 2

372

14.4729

No. 3

1,045

21.2740

No. 3

1,026

23.9581

No. 4

781

15.7751

No. 4

1,216

32.3385

No. 5

625

14.1925

No. 5

297

9.4139

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 Lipschitz-type continuity of the bifunctions. Combining the interior proximal extragradient method in [22], the Armijo-type 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.

Declarations

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 101.02-2011.07.

Authors’ Affiliations

(1)
Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology
(2)
Academy of Military Science and Technology

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