#### Abstract

We consider bilevel pseudomonotone equilibrium problems. We use a penalty function to convert a bilevel problem into one-level ones. We generalize a pseudo--monotonicity concept from -monotonicity and prove that under pseudo--monotonicity property any stationary point of a regularized gap function is a solution of the penalized equilibrium problem. As an application, we discuss a special case that arises from the Tikhonov regularization method for pseudomonotone equilibrium problems.

#### 1. Introduction

Let be a nonempty closed-convex subset in , and let be two bifunctions satisfying for every . Such a bifunction is called an equilibrium bifunction. We consider the following bilevel equilibrium problem (BEP for short): where , that is, is the solution set of the equilibrium problems As usual, we call problem (1.1) the upper problem and (1.2) the lower one. BEPs are special cases of mathematical programs with equilibrium constraints. Sources for such problems can be found in [1–3]. Bilevel monotone variational inequality, which is a special case of problem (1.1), was considered in [4, 5]. Moudafi in [6] suggested the use of the proximal point method for monotone BEPs. Recently, Ding in [7] used the auxiliary problem principle to BEPs. In both papers, the bifunctions and are required to be monotone on . It should be noticed that under the pseudomonotonicity assumption on the solution-set of the lower problem (1.2) is a closed-convex set whenever is lower semicontinuous and convex on for each . However, the main difficulty is that, even the constrained set is convex, it is not given explicitly as in a standard mathematical programming problem, and therefore the available methods (see, e.g., [8–14] and the references therein) cannot be applied directly.

In this paper, first, we propose a penalty function method for problem (1.1). Next, we use a regularized gap function for solving the penalized problems. Under certain pseudo--monotonicity properties of the regularized bifunction, we show that any stationary point of the gap function on the convex set is a solution to the penalized subproblem. Finally, we apply the proposed method to the Tikhonov regularization method for pseudomonotone equilibrium problems.

#### 2. A Penalty Function Method

Penalty function method is a fundamental tool widely used in optimization to convert a constrained problem into unconstrained (or easier constrained) ones. This method was used to monotone variational inequalities in [5] and equilibrium problems in [15]. In this section, we use the penalty function method in the bilevel problem (1.1). First, let us recall some well-known concepts on monotonicity and continuity (see, e.g., [16]) that will be used in the sequel.

*Definition 2.1. *The bifunction is said to be as follows:(a)strongly monotone on with modulus if
(b)monotone on if
(c)pseudomonotone on if
(d)upper semicontinuous at with respect to the first argument on if
(e)lower semicontinuous at with respect to the second argument on if

Clearly, (a)*⇒*(b)*⇒*(c).

*Definition 2.2 (see [17]). * The bifunction is said to be *coercive* on if there exists a compact subset and a vector such that

Theorem 2.3 (see [18, Proposition 2.1.14]). * Let be a equilibrium bifunction such that is upper semicontinuous on for each and is lower semicontnous, convex on for each . Suppose that is compact or is coercive on , then there exists at least one such that for every .*

The following proposition tells us about a relationship between the coercivity and the strong monotonicity.

Proposition 2.4. *Suppose that the equilibrium bifunction is strongly monotone on , and is convex, lower semicontinuous with respect to the second argument for all , then for each , there exists a compact set such that and .*

*Proof. *Suppose by contradiction that the conclusion does not hold, then there exists an element such that for every compact set there is an element such that . Take as the closed ball centered at with radius . Then there exists such that . Let be the intersection of the line segment with the unit sphere centered at and radius 1. Hence, , where . By the strong monotonicity of , we have
Since is convex on , it follows that
which implies that . Thus,
However, since is lower semicontinuous on , by the well-known Weierstrass Theorem, attains its minimum on the compact set . This fact contradicts (2.9).

From this proposition, we can derive the following corollaries.

Corollary 2.5 (see [18]). * If the bifunction is strongly monotone on , and is convex, lower semicontinuous with respect to the second argument for all , then is coercive on C.*

Corollary 2.6. *Suppose that the bifunction is strongly monotone on , and is convex, lower semicontinuous with respect to the second argument for all . If the bifunction is coercive on then, for every , the bifunction is uniformly coercive on , for example, there exists a point and a compact set both independent of such that
*

*Proof. *From the coercivity of , we conclude that there exists a compact and such that . Since is strongly monotone, convex, lower semicontinuous on , by choosing , from Proposition 2.4, there exists a compact such that . Set , then is compact and .

*Remark 2.7. *It is worth to note that if both , are coercive and pseudomonotone on , then the function is not necessary coercive or pseudomonotone on .

To see this, let us consider the following bifunctions.

*Example 2.8. *Let *, *, and then we have(i) are pseudomonotone and coercive on ,(ii)for all the bifunctions are neither pseudomonotone nor coercive on *. *

Indeed, (i)if , then , thus is pseudomonotone on . By choosing and , we have , which means that is coercive on . Similarly, we can see that is coercive on ,(ii)by definition of , we have thatTake , then , whereas for is sufficiently large. So is not pseudomonotone on .

Now, we show that the bifunction is not coercive on . Suppose, by contradiction, that there exist a compact set and such that , then, by coercivity of , it follows, and . With , we have . However(i)if , then from it follows that and for is sufficiently large, which contradicts with coercivity,(ii)if , then, by choosing , we obtain and for is large enough. But this cannot happen because of the coercivity of .

Now, for each fixed , we consider the penalized equilibrium problem defined as
By , we denote the solution set of .

Theorem 2.9. *Suppose that the equilibrium bifunctions are pseudomonotone, upper semicontinuous with respect to the first argument and lower semicontinuous, convex with respect to the second argument on , then any cluster point of the sequence with is a solution to the original bilevel problem (1.1). In addition, if is strongly monotone and is coercive on , then for each the penalized problem is solvable, and any sequence with converges to the unique solution of the bilevel problem (1.1) as . *

*Proof. *Let be any sequence with , and let be any of its cluster points. Without lost of generality, we may assume that . Since , one has
For any , we have and in particular, . Then, by the pseudomonotonicity of , we have . Replacing by in (2.13), we obtain
which implies that
Let , by upper semicontinuity of , we have .

To complete the proof, we need only to show that . Indeed, for any , we have
Again, by upper semicontinuity of and , we obtain in the limit, as , that . Hence, .

Now suppose, in addition, that is strongly monotone on . By Corollary 2.6, is uniformly coercive on . Thus, problem is solvable and, for all , the solution sets of these problems are contained in a compact set . So any infinite sequence of the solutions has a cluster point, say, . By the first part, is a solution of (1.1). Note that, from the assumption on , the solution set of the lower equilibrium () is a closed, convex, compact set. Since is lower semicontinuous and convex with respect to the second argument and is strongly monotone on , the upper equilibrium problem has a unique solution. Using again the first part of the theorem, we can see that

*Remark 2.10. *In a special case considered in [6], where both and are monotone, the penalized problem (PEP) is monotone too. In this case, (PEP) can be solved by some existing methods (see, e.g., [6, 11–14, 19]) and the references therein. However, when one of these two bifunctions is pseudomonotone, the penalized problem (PEP), in general, does not inherit any monotonicity property from and . In this case, problem (PEP) cannot be solved by the above-mentioned existing methods.

#### 3. Gap Function and Descent Direction

A well-known tool for solving equilibrium problem is the gap function. The regularized gap function has been introduced by Taji and Fukushima in [20] for variational inequalities, and extended by Mastroeni in [11] to equilibrium problems. In this section, we use the regularized gap function for the penalized equilibrium problem (PEP). As we have mentioned above, this problem, even when is pseudomonotone and is strongly monotone, is still difficult to solve.

Throughout this section, we suppose that both and are lower semicontinuous, convex on with respect to the second argument. First, we recall (see, e.g., [11]) the definition of a gap function for the equilibrium problem.

*Definition 3.1. *A function is said to be a gap function for (PEP) if(i),
(ii) if and only if is a solution for (PEP).

A gap function for (PEP) is . This gap function may not be finite and, in general, is not differentiable. To obtain a finite, differentiable gap function, we use the regularized gap function introduced in [20] and recently used by Mastroeni in [11] to equilibrium problems. From Proposition 2.2 and Theorem 2.1 in [11], the following proposition is immediate.

Proposition 3.2. *Suppose that is a nonnegative differentiable, strongly convex bifunction on with respect to the second argument and satisfies*(a)*,
*(b)*.**Then the function
**
is a finite gap function for (PEP). In addition, if and are differentiable with respect to the first argument and are continuous on , then is continuously differentiable on and
**
where
*

Note that the function , where is a symmetric positive definite matrix of order that satisfies the assumptions on .

We need some definitions on -monotonicity.

*Definition 3.3. *A differentiable bifunction is called as follows:(a)strongly -monotone on if there exists a constant such that,
(b)strictly -monotone on if
(c)-monotone on if
(d)strictly pseudo--monotone on if
(e)pseudo--monotone on if

*Remark 3.4. *The definitions (a), (b), and (c) can be found, for example, in [8, 11]. The definitions (d) and (e), to our best knowledge, are not used before. From the definitions, we have
However, (c) may not imply (d) and vice versa as shown by the following simple examples.

*Example 3.5. *Consider the bifunction defined on * with *. This bifunction is not -monotone on , because
is negative for . However, is strictly pseudo*- *-monotone. Indeed, we have
It is not difficult to verify that

Hence this function is strictly pseudo*- *-monotone but is not -monotone.

Vice versa, considering the bifunction defined on , where is a matrix of order , we have the following:(i) is -monotone, because
Clearly, is not strictly--monotone,(ii) is strictly pseudo -monotone if and only if
implies
The latter inequality equivalent to is a positive definite matrix of order .

*Remark 3.6. *As shown in [8] when with a differentiable monotone operator on , is monotone on if and only if is monotone on , and in this case, monotonicity of on coincides with -monotonicity of on .

The following example shows that pseudomonotonicity may not imply pseudo--monotonicity.

*Example 3.7. *Let , defined on , . It is easy to see that
Thus, is pseudomonotone on .

We have

But

So is not pseudo--monotone on .

From the definition of the gap function , a global minimal point of this function over is a solution to problem (PEP). Since is not convex, its global minimum is extremely difficult to compute. In [8], the authors have shown that under the strict -monotonicity a stationary point is also a global minimum of gap function. By a counterexample, the authors in [8] also pointed out that the strict -monotonicity assumption cannot be relaxed to -monotonicity. The following theorem shows that the stationary property is still guaranteed under the strict pseudo--monotonicity.

Theorem 3.8. *Suppose that is strictly pseudo- -monotone on . If is a stationary point of over , that is,
**
then solves (PEP).*

*Proof. *Suppose that does not solve (PEP), then .

Since is a stationary point of on , from the definition of , we have
By strict pseudo--monotonicity of , it follows that
On the other hand, since minimizes over , we have
which is in contradiction with (3.21).

To compute a stationary point of a differentiable function over a closed-convex set, we can use the existing descent direction algorithms in mathematical programming (see, e.g., [8, 21]). The next proposition shows that if is a solution of the problem , then is a descent direction on of at . Namely, we have the following proposition.

Proposition 3.9. *Suppose that is strictly pseudo--monotone on and is not a solution to Problem (PEP), then
*

*Proof. *Let . Since is not a solution to (PEP), then . Suppose that, by contradiction, is not a descent direction on of at , then
which, by strict pseudo--monotonicity of , implies
On the other hand, since minimizes over , by the well-known optimality condition, we have
which contradicts (3.25).

Proposition 3.10. *Suppose that is strictly convex on for every and is strictly pseudo--monotone on . If is not a solution of (PEP), then there exists such that is a descent direction of on at for all .*

*Proof. *By contradiction, suppose that the statement of the proposition does not hold, then there exist and such that
Since is strictly convex differentiable on , by Theorem 2.1 in [9], the function is continuous with respect to , thus tends to as , where . Since is continuously differentiable, letting in (3.27), we obtain
By strict pseudo--monotonicity of , it follows that
On the other hand, since minimizes over , we have
Taking the limit, we obtain
which contradicts (3.29).

To illustrate Theorem 3.8, let us consider the following examples.

*Example 3.11. *Consider the bifunctions and defined on . It is not hard to verify that,(i) are monotone, strictly pseudo--monotone on ,(ii)for all the bifunction is monotone and strictly pseudo--monotone on and satisfying all of the assumptions of Theorem 3.8.

*Example 3.12. *Let and defined on it is easy to see that,(i) are pseudomonotone, strictly -monotone on ,(ii)for all the bifunction is pseudomonotone and strictly -monotosne on and satisfying all of the assumptions of Theorem 3.8.

#### 4. Application to the Tikhonov Regularization Method

The Tikhonov method [22] is commonly used for handling ill-posed problems. Recently, in [23] the Tikhonov method has been extended to the pseudomonotone equilibrium problem where, as before, is a closed-convex set in and is a pseudomonotone bifunction satisfying for every .

In the Tikhonov regularization method considered in [23], problem () is regularized by the problems where is an equilibrium bifunction on and and play the role of the regularization bifunction and regularization parameter, respectively.

In [23], the following theorem has been proved.

Theorem 4.1. *Suppose that are upper semicontinuous and are lower semicontinuous convex on for each and that is pseudomonotone on . Suppose further that is strongly monotone on satisfying the condition
**
where (plays the role of a guess solution) is given.**Then the following three statements are equivalent:*(a)*the solution set of () is nonempty for each and exists, where is arbitrarily chosen in the solution set of (),*(b)*the solution set of () is nonempty for each and , where is arbitrarily chosen in the solution set of (),*(c)*the solution set of () is nonempty.**Moreover, if any one of these statements holds, then is equal to the unique solution of the strongly monotone equilibrium problem , where denotes the solution set of the original problem ().*

Note that, when is monotone on , the regularized subproblems are strongly monotone and therefore, they can be solved by some existing methods. When is pseudomonotone, the subproblems, in general, are no longer strongly monotone, even not pseudomonotone. So solving them becomes a difficult task. However, the problem of finding the limit point of the sequences of iterates leads to the unique solution of problem .

In order to apply the penalty and gap function methods described in the preceding sections, let us take, for instant, Clearly, is both strongly monotone and strongly -monotone with the same modulus 1. Moreover, satisfies the condition (4.1). Therefore, the problem of finding the limit point in the above Tikhonov regularization method can be formulated as the bilevel equilibrium problem which is of the form (1.1). Now, for each fixed , we consider the penalized equilibrium problem defined as As before, by , we denote the solution set of .

Applying Theorems 2.9 and 3.8, we obtain the following result.

Theorem 4.2. *Suppose that the bifunction satisfies the following conditions:*(i)* is convex, lower semicontinuous for all ,*(ii)* is pseudomonotone and coercive on . Then for any , the penalized problem is solvable, and any sequence with for all converges to the unique solution of the problem (4.3) as . *(iii)*In addition, if is strictly pseudo- -monotone on (in particular, is-monotone), and is any stationary point of the mathematical program with
then converges to the unique solution of the problem (4.3) as.*

#### 5. Conclusion

We have considered a class of bilevel pseudomonotone equilibrium problems. The main difficulty of this problem is that its feasible domain is not given explicitly as in a standard mathematical programming problem. We have proposed a penalty function method to convert the bilevel problem into one-level ones. Then we have applied the regularized gap function method to solve the penalized equilibrium subproblems. We have generalized the pseudo--monotonicity concept from -monotonicity. Under the pseudo--monotonicity property, we have proved that any stationary point of the gap function is a solution to the original bilevel problem. As an application, we have shown how to apply the proposed method to the Tikhonov regularization method for pseudomonotone equilibrium problems.

#### Acknowledgment

This work is supported by the National Foundation for Science Technology Development of Vietnam (NAFOSTED).