Abstract

Consider the variational inequality of finding a point satisfying the property for all , where is a level set of a convex function defined on a real Hilbert space and is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of ) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projection (calculating by computing a sequence of projections onto half-spaces containing the original domain ) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of (i.e., Lipschitz constants on some bounded subsets of )), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

1. Introduction

Let be a real Hilbert space with inner product and norm , let be a nonempty closed convex subset of , and let be a nonlinear operator. We consider the problem of finding a point with the property This is known as the variational inequality problem , initially introduced and studied by Stampacchia [1] in 1964. In recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see [121] and the references therein. Using the projection technique, one can easily show that is equivalent to a fixed-point problem (see, e.g., [15]).

Lemma 1. is a solution of if and only if satisfies the fixed-point relation: where is an arbitrary constant, is the orthogonal projection onto , and is the identity operator on .

Recall that an operator is called monotone, if Moreover, a monotone operator is called strictly monotone if the equality “” holds only when in the last relation. It is easy to see that (1) has at most one solution if is strictly monotone.

For variational inequality (1), is generally assumed to be Lipschitzian and strongly monotone on ; that is, for some constants ,   satisfies the conditions In this case, is also called a -Lipschitzian and -strongly monotone operator. It is not difficult to show the following result.

Lemma 2. Assume that satisfies conditions (4) and (5) and and are constants such that and , respectively. Let (or ) and (or ). Then and are all contractions with coefficients and , respectively, where .

By using the well-known Banach contraction mapping principle, this fact together with Lemma 1 leads to the following classical result.

Theorem 3. Assume that satisfies conditions (4) and (5). Then has a unique solution. Moreover, for any , the sequence with initial guess and defined recursively by converges strongly to the unique solution of .

Attempts are worth making to weaken the Lipschitz condition (4) or the strong monotonicity condition (5) so that existence of solutions of variational inequality (1) is still guaranteed. In 2009, He and Xu [14] weakened the Lipschitz condition (4) successfully to the bounded Lipschitz condition. A mapping is boundedly Lipschitzian on if it is Lipschitzian on each bounded subset of ; namely, for each nonempty bounded subset of , there exists a positive constant depending only on the set such that

He and Xu [14] not only proved existence and uniqueness of solutions of under conditions (5) and (7) but also estimated the range of this unique solution.

Theorem 4 (see [14]). Assume that is boundedly Lipschitzian on (i.e., for each bounded subset of , is Lipschitzian on ). Assume also that is -strongly monotone on . Then variational inequality (1) has a unique solution such that where is an arbitrary fixed element.

Similarly, we can also introduce bounded strong monotonicity of an operator. An operator is called boundedly strong monotone on , if, for arbitrary bounded subset of , there exists a positive constant depending only on the set such that So a natural question gives rise to this: is it possible also to replace the strong monotonicity of by bounded strong monotonicity so that the result of Theorem 4 is still guaranteed? Unfortunately, a simple example [14] gives us a negative answer.

He and Xu [14] also consider the iterative algorithms for solving , where is boundedly Lipschitzian and -strongly monotone on . Denote by an arbitrary fixed element in and denote by a positive fixed constant such that Set ( is a closed ball of , i.e., ) and denote by the Lipschitz constant of on the bounded closed convex subset .

Using Theorem 4, it is easy to see that and have the same solution. Thus one can devise iterative methods for and get the unique solution of .

Theorem 5 (see [14]). Define a sequence recursively by the iterative algorithm where . Then converges strongly to the unique solution of .

However, algorithms (6) and (11) all have two evident weaknesses. On the one hand, they involve calculating the projections and , respectively, while the computation of a projection onto a closed convex subset is generally difficult. Particularly, the computation of is maybe more difficult since the structure of is more complicated. On the other hand, the determination of the stepsize depends on the constants (or ) and . This means that, in order to implement algorithm (6) (or algorithm (11)), one has first to compute (or estimate) the constants (or ) and , which is sometimes not an easy work in practice.

He and Yang [22] proposed relaxed and self-adaptive algorithms in order to overcome the above weaknesses of algorithm (6) and proved strong convergence theorems.

In order to overcome the above weaknesses of algorithm (11), new relaxed and self-adaptive algorithms are proposed in this paper to solve , where is a level set of a convex function defined on and is a boundedly Lipschitzian and -strongly monotone operator. Our methods calculate by computing (the computation of is very easy) and a sequence of projections onto half-spaces containing the original level set and select the stepsizes through a self-adaptive way. The implementations of our algorithms are very easy since they avoid computing directly and have no need to know any information about (but is assumed to be known, so our methods partly overcome the second weakness above). The algorithms in this paper improve and extend the above corresponding result of He and Xu.

The rest of this paper is organized as follows. Some useful lemmas are listed in the next section. In the last section, a relaxed algorithm for the case where and are all known and a relaxed self-adaptive algorithm for the case where is known but is unknown are proposed, respectively. The strong convergence theorems of our algorithms are proved.

2. Preliminaries

Throughout the rest of this paper, we denote by a real Hilbert space and by the identity operator on . If is a differentiable functional, then we denote by the gradient of . We will also use the following notations.(i) denotes strong convergence.(ii) denotes weak convergence.(iii) denotes a closed ball in with center and radius .(iv) such that denotes the weak - set of .

Recall a trivial inequality, which is well known and in common use.

Lemma 6. For all , there holds the relation

Recall that a mapping is said to be nonexpansive if is said to be firmly nonexpansive if, for ,

The following are characterizations of firmly nonexpansive mappings (see [7] or [23]).

Lemma 7. Let be an operator. The following statements are equivalent.(i) is firmly nonexpansive.(ii) is firmly nonexpansive.(iii), .

We know that the orthogonal projection from onto a nonempty closed convex subset is a typical example of a firmly nonexpansive mapping, which is defined by It is well known that is characterized by the inequality (for )

The following recent result [22] is likely to become a new fundamental tool for proving strong convergence of some algorithms. Its key effect on the proofs of our main results will be illustrated in the next section.

Lemma 8 (see [22]). Assume is a sequence of nonnegative real numbers such that where is a sequence in , is a sequence of nonnegative real numbers, and and are two sequences in such that(i),(ii),(iii) implies for any subsequence . Then .

Recall that a function is called convex if A differentiable function is convex if and only if there holds the relation Recall that an element is said to be a subgradient of at if

A function is said to be subdifferentiable at , if it has at least one subgradient at . The set of subgradients of at the point is called the subdifferential of at and is denoted by . The last relation above is called the subdifferential inequality of at . A function is called subdifferentiable, if it is subdifferentiable at all . If a function is differentiable and convex, then its gradient and subgradient coincide.

Recall that a function is said to be weakly lower semicontinuous at if implies

3. Iterative Algorithms

In this section, we consider the iterative algorithms for solving a particular kind of variational inequality (1) in which the closed convex subset is of the particular structure, that is, the level set of a convex function given as follows: where is a convex function. We always assume that is subdifferentiable on and is bounded operator (i.e., bounded on bounded sets). We also assume that is a boundedly Lipschitzian and -strongly monotone operator. Using Theorem 4, we assert that in this case has a unique solution, henceforth, which is denoted by .

The computation of a projection onto a closed convex subset is generally difficult. To overcome this difficulty, Fukushima [21] suggested a way to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. This idea is followed by Yang [24] and Lopez et al. and so forth [25], respectively, who introduced the relaxed algorithms for solving the split feasibility problem in a finite-dimensional and infinite-dimensional Hilbert space, respectively. He and Yang [22] also used this idea to devise iterative algorithms for solving variational inequalities governed by Lipschitzian and strongly monotone operators.

In the sequel, we always assume that is known and denote by a selected arbitrarily fixed element. Using Theorem 4, the unique solution of belongs to a closed ball , where is a fixed positive constant such that . We also always denote by the Lipschitz constant of on .

Based on Theorem 4, we are now in a position to introduce a relaxed algorithm for computing the unique solution of , where is given as in (23). This scheme applies to the case where is easy to be determined.

Algorithm 9. Choose an arbitrary initial guess and the sequence is constructed via the formula where where , the sequence in , and is a constant such that .

We now analyze strong convergence of Algorithm 9, which also illustrates the application of Lemma 8.

Theorem 10. Assume that and . Then the sequence generated by Algorithm 9 converges strongly to the unique solution of .

Proof. Obviously, it follows from (24) that is bounded (indeed ) and so is noting the bounded Lipschitz condition of . It is easy to see from the subdifferential inequality and the definition of that holds for all , and hence it follows that . Observing that , a projection is nonexpansive, and is a contraction with coefficient for all (using Lemma 2), where , we obtain using (24) and Lemma 6 that
On the other hand, we also have where is a positive constant such that . Observing that a projection is firmly nonexpansive, we have and the combination of (27) and (28) leads to Setting then (26) and (29) can be rewritten as the following forms, respectively: Since and hold, in order to complete the proof using Lemma 8, it suffices to verify that implies for any subsequence . In fact, if as , then and hold. Since is bounded on bounded sets, we have a positive constant such that for all . From (25) and the trivial fact that , it follows that Take arbitrarily and assume that without loss of the generality; then the of and (35) imply that This means that holds. On the other hand, we assert from that . Moreover, we obtain that and hence .
Noting the fact that and is the unique solution of (i.e., the unique solution of ), it turns out that .

It is worth mentioning that if is easy to be calculated, then in Algorithm 9 can be replaced with and it is easy to see that the whole proof of Theorem 10 is valid for this case. Therefore, if , reduces to the operator equation problem: finding such that , and the following result holds.

Corollary 11. Assume that , , and . Then the sequence generated by algorithm converges strongly to the unique solution of the operator equation .

Sometimes, the constant is difficult to be obtained or estimated in practice (but we assume that has been obtained). In this case, Algorithm 9 is indeed not fit for solving (i.e., ). Then we now turn to introducing a relaxed and self-adaptive algorithm for the case where constant is unknown.

Algorithm 12. Choose an arbitrary initial guess and an arbitrary element such that . Assume that the th iterate () has been constructed. Continue and calculate the th iterate via the formula where is given as in (25), the sequence is in , is a constant such that , and the sequence is determined via the relation

Firstly, we show that the sequence is well defined. Noting strong monotonicity of , implies that and is well defined via the first formula of (40). Consequently, is well defined inductively according to (40) and thus the sequence is also well defined using (39).

Next, we estimate roughly. If (i.e., ), set Observing the fact that from (39), it turns out that Consequently By the definition of , we can assert that (43) holds for all .

Now we analyze the strong convergence of Algorithm 12.

Theorem 13. Assume that and . Then the sequence generated by Algorithm 12 converges strongly to the unique solution of .

Proof. Obviously, is bounded and so is . Setting and , observing and (43), it is concluded that there exists some positive integer such that and consequently Using Lemma 2, we have from (44) that, for all is a contraction with coefficient . This concludes that, for all ,
By an argument similar to getting (27)–(29), we have where is a positive constant, which has nothing to do with . Setting then (46) and (47) can be rewritten as the following forms, respectively: Clearly, and , together with (43)–(45), imply that and .
By an argument very similar to the proof of Theorem 10, it is not difficult to verify that implies for any subsequence . Thus we can complete the proof by using Lemma 8.

Similar to Algorithm 9, if is easy to be calculated, then in Algorithm 12 can also be replaced with and it is easy to see that the whole proof of Theorem 13 is valid for this case. The following result similar to Corollary 11 also holds.

Corollary 14. Assume that , , and . Then the sequence generated by algorithm where is given as in (40), converges strongly to the unique solution of the operator equation .

Finally, we give an iterative algorithm for solving a class , in which the closed convex subset is the intersection of finite level sets of convex functions given as follows: where is a positive integer and is a convex function. We always assume that is subdifferentiable on and is bounded operator (i.e., bounded on bounded sets).

Without loss of generality, we will consider only the case ; that is, , where

Algorithm 15. Choose an arbitrary initial guess and an arbitrary element such that . Assume that the th iterate () has been constructed. Continue and calculate the th iterate via the formula where the sequence is in , is a constant such that , the sequence is given as in (40), and is determined via the relation

By an argument similar to the proof of Theorem 13 (together with the proof of Theorem 3.4 of [22]), we have the following result.

Theorem 16. Assume that and . Then the sequence generated by Algorithm 15 converges strongly to the unique solution of , where and is given as in (55).

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally to the writing of this paper.

Acknowledgments

The authors would like to thank the three referees for their comments and suggestions on improving an earlier version of this paper. This work was supported by the Fundamental Research Funds for the Central Universities (3122014K010) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.