Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems

Abstract

We consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point such that , for all , where , are two monotone operators and is the solution of the equilibrium problem of finding such that , for all . We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (HEVP); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (HEVP).

1. Introduction

Let be a real Hilbert space with inner product and norm , respectively, and let be a nonempty closed convex subset of . Recall that a mapping of into is called monotone if

(1.1)

for all and is called-inverse strongly monotone mapping if there exists a positive real number such that

(1.2)

for all . It is obvious that any-inverse strongly monotone mapping is monotone and -Lipschitz continuous.

Recently, the following problem has attracted much attention: find hierarchically a fixed point of a nonexpansive mapping with respect to a nonexpansive mapping , namely,

(1.3)

Some algorithms for solving the hierarchical fixed point problem (1.3) have been introduced by many authors. For related works, please see, for instance, [1–9] and the references therein.

Remark 1.1.

It is not hard to check that solving (1.3) is equivalent to the fixed point problem

(1.4)

where stands for the metric projection on the closed convex set . By using the definition of the normal cone to , that is,

(1.5)

we easily prove that (1.3) is equivalent to the variational inequality

(1.6)

At this point, we wish to point out the link with some monotone variational inequalities and convex programming problems as follows.

Example 1.2.

Setting , where is -Lipschitzian and -strongly monotone with , then (1.3) reduces to

(1.7)

a variational inequality studied by Yamada and Ogura [10].

Example 1.3.

Let be a maximal monotone operator. Taking and , where is a convex function such that is -Lipschitzian (which is equivalent to the fact that is cocoercive) with , and . Then (1.3) reduces to the following mathematical program with generalized equation constraint:

(1.8)

a problem considered by Luo et al. [11].

Example 1.4.

Taking , where is the subdifferential of a lower semicontinuous convex function, then (1.8) reduces to the following hierarchical minimization problem considered in Cabot [12] and Solodov [13]:

(1.9)

Let be a nonlinear mapping, and let be a bifunction of into . Consider the following equilibrium problem of finding such that

(1.10)

If , then (1.10) reduces to

(1.11)

The solution set of equilibrium problems (1.10) and (1.11) are denoted by and , respectively. The equilibrium problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, fixed point problems, minimax problems, Nash equilibrium problem in noncooperative games, and others. We remind the readers to refer to [14–30] and the references therein.

Motivated and inspired by the above works, in this paper, we consider the following hierarchical equilibrium problem and variational inequality problem: find a point such that

(1.12)

where are two monotone operators. The solution set of (1.12) is denoted by .

Remark 1.5.

It is clear that the hierarchical variational inequality problem and equilibrium problem (1.12) includes the variational inequality problem studied by Yamada and Ogura [10], mathematical program studied by Luo et al. [11], hierarchical minimization problem considered by Cabot [12] and Solodov [13], as special cases.

For solving (1.12), we propose a double-net algorithm which generates a net . We prove that the net hierarchically converges to the solution of (1.12); that is, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem, and as , the net converges in norm to the unique solution of (1.12).

2. Preliminaries

Let be a real Hilbert space. Throughout this paper, let us assume that a bifunction   satisfies the following conditions:

(F1) for all ;

(F2) is monotone, that is, for all ;

(F3) for each , ;

(F4) for each , is convex and lower semicontinuous.

On the equilibrium problems, we have the following important lemma. You can find it in [31].

Lemma 2.1.

Let be a real Hilbert space, and let be a bifunction of into satisfying conditions (F1)–(F4). Let , and . Then, there exists such that

(2.1)

Further, if , for all , then the following hold:

(1) is single-valued;

(2) is firmly nonexpansive; that is, for any ,

(2.2)

(3);

(4) is closed and convex.

Below we gather some basic facts that are needed in the argument of the subsequent sections.

Lemma 2.2 (see [32]).

Let be a real Hilbert space. Let the mapping be α-inverse strongly monotone, and let be a constant. Then, one has

(2.3)

In particular, if , then is nonexpansive.

Lemma 2.3 (demiclosedness principle for nonexpansive mappings, see [33]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to , and if converges strongly to , then ; in particular, if , then .

Lemma 2.4.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let the mapping be α-inverse strongly monotone. Let , and . Then the variational inequality

(2.4)

is equivalent to the dual variational inequality

(2.5)

Proof.

Assume that solves (2.4). For all , set

(2.6)

We note that

(2.7)

Hence, we have

(2.8)

which implies that

(2.9)

Letting , we have

(2.10)

which is exactly (2.5).

Assume that solves (2.5). Hence,

(2.11)

Noting that and are monotone, we have

(2.12)

It follows that

(2.13)

which implies that

(2.14)

This implies that solves (2.4). The proof is completed.

3. Main Results

In this section, we first introduce our double-net algorithm.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let the mappings be-inverse strongly monotone and-inverse strongly monotone, respectively. Let be a bifunction from   , and let and be two constants. For , we define the following mapping:

(3.1)

where is defined by Lemma 2.1. We note that the mapping is a contraction. As a matter of fact, we have

(3.2)

which implies that the mapping is contractive. Hence, by Banach's contraction principle, has a unique fixed point which is denoted ; that is, is the unique solution in of the fixed point equation

(3.3)

Below is our main result of this paper which displays the behavior of the net as and successively.

Theorem 3.1.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let the mappings be α-inverse strongly monotone and β-inverse strongly monotone, respectively. Let and be two constants. Let be a bifunction from satisfying (F1)–(F4). Suppose the solution set of (1.12) is nonempty. Let, for each , be defined implicitly by (3.3). Then, the net hierarchically converges to the unique solution of the hierarchical equilibrium problem and variational inequality problem (1.12). That is to say, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem (1.10). Moreover, as , the net converges in norm to the unique solution . Furthermore, also solves the following variational inequality:

(3.4)

We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions of Theorem 3.1 are satisfied.

Conclusion.

For each fixed , the net is bounded.

Proof.

Take any . It is clear that . Set for all . Since , and are nonexpansive (by Lemmas 2.1 and 2.2), we have from (3.3) that

(3.5)

This implies that

(3.6)

It follows that for each fixed , is bounded, so are the nets , and . Note that we use as a positive constant which bounds all bounded terms appearing in the following.

Conclusion.

as .

Proof.

From Lemma 2.2, we have

(3.7)

By (3.3), we have

(3.8)

This together with (3.7) implies that

(3.9)

It follows that

(3.10)

Therefore

(3.11)

Using Lemma 2.1, we obtain

(3.12)

which implies that

(3.13)

From (3.3), we have

(3.14)

Hence,

(3.15)

It follows that

(3.16)

Next, we show that, for each fixed , the net is relatively norm-compact as . It follows from (3.8) that

(3.17)

It turns out that

(3.18)

Assume that is such that as . By (3.18), we conclude immediately that

(3.19)

Since is bounded, without loss of generality, we may assume that as , converges weakly to a point . Note that also converges weakly to a point .

Now we show that . Since , for any , we have

(3.20)

From the monotonicity of , we have

(3.21)

Hence,

(3.22)

Put for all and . From (3.22), we have

(3.23)

Note that . Further, from monotonicity of , we have . Letting in (3.23), we have

(3.24)

From (F1), (F4), and (3.24), we also have

(3.25)

and hence

(3.26)

Letting in (3.26), we have, for each ,

(3.27)

This implies that .

We can then substitute for in (3.19) to get

(3.28)

Consequently, the weak convergence of to actually implies that strongly. This has proved the relative norm-compactness of the net as .

Now we return to (3.19) and take the limit, as , to get

(3.29)

In particular, solves the following variational inequality:

(3.30)

or the equivalent dual variational inequality (see Lemma 2.4)

(3.31)

Notice that (3.31) is equivalent to the fact that . That is, is the unique element in of the contraction . Clearly, this is sufficient to conclude that the entire net converges in norm to as .

Conclusion.

The net is bounded.

Proof.

In (3.31), we take any to deduce

(3.32)

By virtue of the monotonicity of and the fact that , we have

(3.33)

It follows from (3.32) and (3.33) that

(3.34)

Hence,

(3.35)

Therefore,

(3.36)

In particular,

(3.37)

which implies that is bounded.

Conclusion.

The net which solves the variational inequality VI (3.4).

Proof.

First, we note that the solution of the variational inequality VI (3.4) is unique.

We next prove that ; namely, if is a null sequence in such that weakly as , then . To see this, we use (3.31) to get

(3.38)

However, since is monotone,

(3.39)

Combining the last two relations yields

(3.40)

Letting as in (3.40), we get

(3.41)

which is equivalent to its dual variational inequality

(3.42)

Namely, is a solution of VI (1.12); hence, .

We further prove that , the unique solution of VI (3.4). As a matter of fact, we have by (3.36)

(3.43)

Therefore, the weak convergence to of implies that in norm. Now we can let in (3.36) to get

(3.44)

It turns out that solves VI (3.4). By uniqueness, we have . This is sufficient to guarantee that in norm, as . The proof is complete.

Proof.

By Conclusions 1–4, the proof of Theorem 3.1 is completed.

Take . Then (1.12) reduces to the following: find a point such that

(3.45)

The solution of (3.45) is denoted by .

Corollary 3.2.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let the mapping be α-inverse strongly monotone. Let be a constant. Let be a bifunction from satisfying (F1)–(F4). Suppose the solution set is nonempty. Let, for each , be defined implicitly by

(3.46)

Then, the net hierarchically converges to the unique solution of the hierarchical equilibrium problem and variational inequality problem (3.45). That is to say, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem (1.11). Moreover, as , the net converges in norm to the unique solution . Furthermore, solves the following variational inequality:

(3.47)

Taking in Theorem 3.1, we have the following corollary.

Corollary 3.3.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let the mapping be β-inverse strongly monotone. Let be a constant. Let be a bifunction from satisfying (F1)–(F4). Suppose that the solution set of (1.10) is nonempty. Let, for each , be defined implicitly by

(3.48)

Then, the net hierarchically converges to the unique solution of the equilibrium problem (1.10). That is to say, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem (1.10). Moreover, as , the net converges in norm to the unique solution . Furthermore, solves the following variational inequality:

(3.49)

Taking in Theorem 3.1, we have the following corollary.

Corollary 3.4.

Let be a real Hilbert space. Let be a -contraction with coefficient . Let be a bifunction from satisfying (F1)–(F4). Suppose the solution set of (1.11) is nonempty. Let, for each , be defined implicitly by

(3.50)

Then, the net hierarchically converges to the unique solution of the equilibrium problem (1.11). That is to say, for each fixed , the net converges in norm, as , to a solution of the equilibrium problem (1.11). Moreover, as , the net converges in norm to the unique solution . Furthermore, solves the following variational inequality:

(3.51)