Two New Weak Convergence Algorithms for Solving Bilevel Pseudomonotone Equilibrium Problem in Hilbert Space

Li, Gaobo

doi:https://doi.org/10.1155/2022/2208280

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Fixed Points and Computational Optimization

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Research Article | Open Access

Volume 2022 | Article ID 2208280 | https://doi.org/10.1155/2022/2208280

Two New Weak Convergence Algorithms for Solving Bilevel Pseudomonotone Equilibrium Problem in Hilbert Space

Gaobo Li¹

Academic Editor: Sun Young Cho

Received13 Jan 2022

Accepted19 Feb 2022

Published13 Apr 2022

Abstract

In this paper, we introduce two new subgradient extragradient algorithms to find the solution of a bilevel equilibrium problem in which the pseudomonotone and Lipschitz-type continuous bifunctions are involved in a real Hilbert space. The first method needs the prior knowledge of the Lipschitz constants of the bifunctions while the second method uses a self-adaptive process to deal with the unknown knowledge of the Lipschitz constant of the bifunctions. The weak convergence of the proposed algorithms is proved under some simple conditions on the input parameters. Our algorithms are very different from the existing related results in the literature. Finally, some numerical experiments are presented to illustrate the performance of the proposed algorithms and to compare them with other related methods.

1. Introduction

Let be a real Hilbert space and be a nonempty closed convex subset of . Let be a bifunction with for all . The equilibrium problem (EP for short) is associated with and to find such that

The solution set of (1) is denoted by .

If for all , where is a mapping from into itself, then the problem (1) becomes the following variational inequality problem (VIP for short):

The solution set of (2) is denoted by .

The EP (1) has a simple form and is very general in the sense that it includes, as special cases, the variational inequality problem, fixed point problem, complementarity problem, optimization problem as well as the Nash equilibrium problem; see,for example [1,2]. Many methods have been proposed for approximating a solution of the EP (1). Mastroeni [3] used the auxiliary problem principle which was first introduced for solving the optimization problems to solve EP (1) and presented the iteration algorithm in the formwhere the stepsize . For obtaining the convergence of this algorithm, the bifunction is required to be strongly monotone and Lipschitz-type continuous. To avoid the hypothesis of the strong monotonicity, Quoc et al. [4] first proposed the extragradient method (or the proximal-like methods) in which two strongly convex problems are solved at each iteration. The extragradient method is as follows: and

In 2018, Hieu [5] presented a new extragradient method for solving the EP (1.1) as follows: andwhere is a nonincreasing sequence and is a strongly pseudomonotone and Lipschitz-type continuous mapping.

In 2011, Censor et al. [6] proposed a new method, which is called the subgradient extragradient method, for solving the VIP (2). In 2016, Hieu [7] extended this method to the EP (1.1). In 2019, inspired by [5,7], Liu and Kong [8] introduced the following subgradient extragradient method for solving the EP (1): andwhere and , and is a pseudomonotone and Lipschitz-type continuous mapping.

The advantage of equations (5) and (6) is that only one value of at is computed at each iteration. On the recent methods for solving the EP (1), we refer the readers to [9–15].

In this paper, our interest is the bilevel equilibrium problem (BEP for short) which consists of the following:where with for all . The BEPs are the special cases of mathematical programs with equilibrium constraints and also are the generalization of variational inequality over equilibrium constraints, hierarchical minimization problems, and complementarity problems. The methods for solving BEPs have been studied extensively by many authors. Moudafi [16] introduced a proximal method and proved the weak convergence to a solution of the BEP (7). Dinh and Muu [17] proposed a penalty and gap function method for solving the BEP (7). Quy [18] introduced an algorithm by combining the proximal method with the Halpern method for solving bilevel monotone equilibrium and fixed point problem. Yuying et al. [19] presented an extragradient method as follows:where , with , and . Anh and An [20] proposed the following subgradient extragradient method for solving the BEP (7):where with , and are two nonnegative sequences.

Observe that in the works mentioned above, the bifunction is monotone or pseudomonotone while is strongly monotone, and then, the algorithms have a strong convergence. In this paper, inspired by [8,20], we propose two new subgradient extragradient methods for solving the BEP (7) where both the bifunction and are pseudomonotone. The first method needs the prior knowledge of the Lipschitz constants of the bifunctions while the second method uses a self-adaptive process to deal with the unknown knowledge of the Lipschitz constant of the bifunctions. The weak convergence of the proposed algorithms is proved under some sufficient assumptions. Finally, some numerical experiments are presented to illustrate the performance of the proposed algorithms and to compare them with other related methods.

2. Preliminaries

Let be a real Hilbert space, be the set of all real numbers, and be the set of all positive integers. We list some well-known definitions and properties which will be used in our following analysis.

Definition 1. A mapping is said to be(i)monotone if(ii)pseudomonotone if(iii)-Lipschitz continuous if there exists a constant such that

Definition 2. A bifunction is said to be(i)pseudomonotone on if(ii)Lipschitz-type continuous on if there exists the constants and such that

Remark 1. If is -Lipschitz continuous on , then for each , is Lipschitz-type continuous with the constants ; see [21] for details.
Let be a nonempty closed and convex subset of . For each , there exists a unique point in , denoted by , such that is said to be the metric projection from onto . The following lemma characterizes the property of .

Lemma 1. Let be the metric projection. Then,(i) if and only if(ii)for all and ,

Remark 2. For any given and with , let . Then, for all , the projection is defined byThe formula (18) gives us an explicit manner to compute the projection of any point onto a half-space; see [22] for details.

Definition 3. (1)The normal cone of at is defined by(2)The subdifferentiable of a convex function at is defined by

Lemma 2 (see [23]). Let be a convex subdifferentiable and lower semicontinuous function on . Then, is a solution to the following convex problem:if and only if , where denotes the subdifferential of and is the normal cone of at .

For a proper, convex, and lower semicontinuous function: and , the proximal mapping of with is defined by

Lemma 3 (see [24, 25]). For all and , the following inequality holds:

Remark 3. From Lemma 3, we note that if , then

Lemma 4 (see [26]). Let and be two sequences of nonnegative real numbers satisfying the conditionIf , then exists.

3. Main Results

In this section, let denotes the set of all positive integers, , be a real Hilbert space, and be a nonempty closed convex subset of . The notation denotes the weak converge. Let be two bifunctions satisfying the following conditions:(A1) and are pseudomonotone on (A2)for each , and for every sequence (A3) and are convex, lower semicontinuous, and subdifferentiable of for each (A4) and are Lipschitz-type continuous on with the constants and , respectively; that is, for all ,

In this section, the solution set of the BEP (7) is denoted by ; that is, , and assume that .

Now, we introduce the first algorithm for finding a point .

	Initialization: Choose and the parameters and . Put .
	Step 1. For given , solve the strongly convex problems:
	where
	Step 2. Solve the strongly convex problems:
	where
	Step 3. If , then the algorithm stops, ; otherwise, set and return to Step 1.

Remark 4. By using the notation “prog” in Section 2, and may be rewritten asNote that since and are convex and lower semicontinuous on for each , for any given , , and the closed convex subset , from [27], Proposition 12.15, and Definition 12.23, it follows that bothare a singleton. Hence, and in Algorithm 1 are obtained uniquely at each step.

Remark 5. From (A1)–(A4), it follows that (i) and are closed and convex; see [4]; (ii) and for all ; see [28].
The following remark shows that the stop criterion in Step 3 is meaning.

Remark 6. Suppose that for some . By , the definition of , and Lemma 3, we getwhich with Remark 5 implies that . Similarly, by , the definition of , and Lemma 3, we can prove that . By the proof of Lemma 4, we see for all . So for all . It follows that .

Lemma 5. Assume that Then, , for each .

Proof. We first show that for each . By Lemma 1 and the definition of , we haveThus, for , there exists such thatSo,Since , we have for all . Hence, for all , which implies that for all . This shows that for each .
Next, we show that for each . By Lemma 3 and the definition of in Remark 4, we haveNote, we have proved that for each . So substituting any into (33), we obtainSince and is pseudomonotone on , we have . Then, (34) implies thatNow applying (A4) to , we haveCombining (35) and (36), we getOn the other hand, by the definition of , we haveSince , we havewhich with (38) implies thatFrom (37) and (40) andit follows thatHence,In particular, from and , it follows thatwhich implies that . Since is arbitrary, it follows that for each .
Finally, we prove that for each . By the definition of in Remark 4 and Lemma 3, we haveThus, for , there exists such thatIt follows thatSince , we have for all . Hence, for all and , which with the definition of implies that for each . This completes the proof.

Lemma 6. Assume that and Let be the sequence generated by Algorithm 1. For all , the limit of exists, and

Proof. Since , from the definition of , it follows thatthat is,By and the definition of subdifferential, we haveReplacing in (51) with , we getCombining (50) and (52), we haveBy Lemma 3 and the definition of , we haveSubstituting into (54), we obtainNote that (A1) implies that , which with (55) leads toOn the other hand, by the Lipschitz-type continuity of , we haveBy (56) and (57), we obtainwhich with (53) implies thatSinceby (59) we getFix . For all with , by (61) we haveHence,which with leads toIt follows thatBy (65) and the triangle inequality of norm, we obtainandNote that (61) impliesFrom (62) and (69) and Lemma 3, it follows that the limit of exists.
Finally, by (54), (56), and (52), we getNote that Lemma 4 has shown that for each . So by (70), we haveThis completes the proof.

Theorem 1. If the parameters and satisfy the conditions:then the sequence generated by Algorithm 1 converges weakly to the point .

Proof. Since for each , by (66) we haveFurthermore, by (68) and (73), we getFrom Lemma 5, it follows that is bounded. This fact with (74) implies that is also bounded.
Take and put . By (43) and (74), we havewhich with and impliesSince is bounded, there exists a subsequence of weakly converging to . By (73), we can conclude that also weakly converges to . Since is closed and for all , it follows that . We show that . In fact, by (33), (36), and (40), we getLetting in (77), by (74), (76), and (A2), we getwhich with implies that .
Next, we prove that . To end this, we need to show thatIn fact, by (73), we haveLetting in (80), by (65)–(67) and (A2), we havewhich with implies that for all . So, .
Now, we prove that the whole sequence converges weakly to the point . Indeed, assume that there exists a different subsequence of converging weakly to with . By arguing similarly as above, it follows that . Note that in the proof of Lemma 5, we have shown that the limits of and exist. So by Opial’s theorem [29], we haveIt is a contradiction. Hence, . Therefore, the whole sequence converges weakly to the point .
Finally, we prove . Let for all . It is easy to see that is bounded from the boundedness of . We show that is a Cauchy sequence. By Lemma 1 and the definition of , we haveSince , replacing in (69) with , we getFrom (83) and (84), it follows thatFrom (64), (85), and Lemma 3, it follows that the limit of exists. For all with , since , by (69), we deduceFrom and , by Lemma 1 and (86), we haveSince exists, letting in (87), by (64), we get . Consequently, is a Cauchy sequence. Since is closed, converges strongly to some . Now, we prove that . In fact, it follows from Lemma 1, and that . Since and , we have . This shows that . This completes the proof. □
In Algorithm 1, and need to be known as the input parameters. The following algorithm is a modification in which and do not need to be known.

	Initialization: Choose , the parameters and . Put .
	Step 1. For given , solve the strongly convex problems:
	where
	Step 2. Solve the strongly convex problems:
	where
	Step 3. Modify by the following formula:
	Step 4. If , then the algorithm stops, ; otherwise, set and return to Step 1.

Remark 7. By (A4), we havewhere . If , thenNote that from the definition of , it follows that still holds even if . Since is nonincreasing and bounded from below by , there exists such thatSimilarly, we can conclude that there exists such that

Theorem 2. The sequence generated by Algorithm 2 converges weakly to the point .

Proof. Repeating the proof of (37) and (40), we can get, for all ,andBy (92) and (93), we haveBy the definition of , in the case when , we haveIt is emphasized here that (95) still holds even ifSo, combining (94) with (95), we obtainSince , it follows that . Thus, for a fixed number , there exists such thatBy (97) and (98), we havewhich is a similar result with (61).
On the other hand, for all , repeating the proof of (35) and (38), we haveBy the definition of , if , thenNote that the definition of implies that (101) still holds even if . So by (100) and (101), we obtainNow, by (102) and we haveSince , it follows that . Thus, for a fixed number , there exists such thatBy (103) and (104), we haveFinally, by arguing similarly to the proof of Lemma 5 and Theorem 1, we can obtain the desired conclusion. This completes the proof.
As an application of the results above, we consider the following bilevel variational inequality problem (BVIP for short):where and be the mappings from into itself. We denote the solution set of (106) by , that is,

Corollary 1. Let be a real Hilbert space and be a nonempty closed and convex subset of . Let be the pseudomonotone and Lipschitz continuous mappings with the Lipschitz constants and satisfy the following conditions:Assume that . Take the parameters , the initial points and generate the sequence in the following manner:where , , and is defined as in Algorithm 1. Then, the sequence generated by (109) converges weakly to the point .

Proof. Let and for all . Since is pseudomonotone on , it follows that . So is pseudomonotone on . It is obvious that satisfies the condition (A3). In addition, if , by (B), we haveSo satisfies the condition (A2). Finally, since is -Lipschitz continuous, is Lipschitz-type continuous with the constant ; see Remark 1. Thus, satisfies the conditions (A1)–(A4). Similarly, also satisfies the conditions (A1)–(A4). In particular, satisfies (A4) with . So, the conditions on and in Lemma 5 become the ones in Corollary 1. On the other hand, by Algorithm 1,is equivalent to . Similarly, , and in Algorithm 1 are equivalent to , and in Corollary 1, respectively. By Theorem 1, the desired conclusion can be obtained. This completes the proof.
Since the proof process of the following corollary is similar to the one of Corollary 3.1, we give the following corollary and omit the proof process.

Corollary 2. Let be a real Hilbert space and be a nonempty closed and convex subset of . Let be the pseudomonotone and Lipschitz continuous mappings with the Lipschitz constants and satisfying the condition in Corollary 3.1. Assume that . The parameters , , the initial points are taken, and the sequence is generated by the following manner:where , and are defined as in Corollary 1,and is modified by

Then, the sequence generated by (112) converges weakly to the point .

Remark 8. Since , , and are half-spaces, from Remark 2, it follows that , , and in Corollary 1 and 2 can be computed explicitly.

4. Numerical Examples

In this section, we give two examples to illustrate the convergence of Algorithm 1 and 2. The programs are written in Matlab 2016b, and the examples are computed on a PC Intel(R) Core (TM) i5-4260U CPU, 2.00 GHz, Ram 4.00 GB.

We first give the following example to illustrate the effectiveness of Algorithm 1 and 2.

Example 1. Let and . Let be defined byIt is known that satisfies the conditions (A1)–(A4). In particular, is Lipschitz-type continuous with the constants ; see [30] for details. Let be defined bywhere . It is easy to see that satisfies the conditions (A1)–(A4). In particular, is Lipschitz-type continuous with the constants . The solution set of the bilevel equilibrium problem (7) in this example is found as .
We choose the initial points randomly from the interval (0,5) for Algorithm 1 and 2, the input parameters for Algorithm 1, and for Algorithm 2. The maximum iteration of 100 as the stop criterion is used for Algorithm 1 and 2. The numerical results with the different dimensions are shown in Figure 1. In this figures, the -axis represents the number of iterations while the -axis is for the value of generated by Algorithm 1 and 2, whereFrom the computed results, we see the effectiveness of Algorithm 1 and 2.
The next example was ever used in [20]. Here, we use this example to illustrate the convergence of Algorithm 1 and 2 and compare the computed results with Algorithm 2.1 in [20].

(a)

(b)

(c)

(d)

Example 2. Let and be defined bywhere withand is a vector in .
Let be defined bywhere is a vector in and withIt is known that and satisfy all the conditions required in [20] and Section 3 of this paper. In particular, is Lipschitz-type continuous with the constants , where and is Lipschitz-type continuous with the constants ; see [20].
We choose the initial point for Algorithm 1 and 2 and for Algorithm 2.1 in [20]. The stop criterion is , wherefor the three algorithms. The computed results are presented in Tables 1–3 for Algorithm 1, 2, and Algorithm 2.1 in [20], respectively. In Table 3, .
From the computed results, we see that Algorithm 2 needs more CPU times and iterations over Algorithm 1 and Algorithm 2.1 in [20]. The course may be that Algorithm 2 involves a self-adaptive process of computing the values of and .

5. Conclusion

We have proposed two iterative algorithms for finding the solution of a bilevel equilibrium problem in a real Hilbert space. The sequence generated by our algorithms converges weakly to the solution. Furthermore, we reported some numerical results to support our algorithms. How to obtain the strong convergence of Algorithm 1 and 2 without the additional assumptions is our future investigation.

Data Availability

The data used to support the findings of this study are available from the author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Gaobo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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