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A new extragradient method for generalized variational inequality in Euclidean space
Fixed Point Theory and Applications volume 2013, Article number: 139 (2013)
In this paper, we extend the extragradient projection method proposed in (Wang et al. in J. Optim. Theory Appl. 119:167-183, 2003) for the classical variational inequalities to the generalized variational inequalities. For this algorithm, we first prove its expansion property of the generated sequence with respect to the starting point and then show that the existence of the solution to the problem can be verified through the behavior of the generated sequence. The global convergence of the method is also established under mild conditions.
Let be a multi-valued mapping from into with nonempty values, and let X be a nonempty closed convex set in . The problem of generalized variational inequalities (GVI) [1, 2] is to find such that there exists satisfying
where stands for the Euclidean inner product of vectors in . The solution set of problem (1.1) is denoted by . Certainly, the GVI reduces to the classical variational inequalities (VI) when F is a single-valued mapping, which has been well studied in the past decades [3, 4].
For the GVI, theories and solution methods have been extensively considered in the literature [2, 5–12], and it is well known that the existence of solutions is an important topic for the GVI . Generally, there are mainly two approaches to attack the solution existence problem of the GVI. One is an analytic approach which first reformulates the GVI as a well-studied mathematical problem and then invokes an existence theorem for the latter problem . The second is a constructive approach in which the existence can be verified by the behavior of the proposed method. The algorithm that is considered in this paper belongs to the second approach.
First, we give a short summary to the constructive approach on the existence theory for the VI. For this approach, the equivalence between the existence of solutions to the VI problem and the boundedness of the sequence generated by some modified extragradient methods was first established by Sun in . Later, Wang et al.  established the same theory by a new type of extragradient-type method. Furthermore, the generated sequence possesses an expansion property with respect to the starting point and converges to a solution point if the solution set of the VI is nonempty. Now a question is posed naturally: as the GVI problem is an extension of the VI, can this theory be extended to the GVI? This constitutes the main motivation of the paper.
In this paper, inspired by the work in , we propose a new type of extragradient projection method for the problem GVI. We first establish the existence results for the GVI under pseudomonotonicity and continuity assumption of the underlying mapping F, and then show the global convergence of the proposed method.
The rest of this paper is organized as follows. In Section 2, we give some related concepts and conclusions. In Section 3, we present the description of the algorithm and establish some properties of the algorithm. The global convergence of the sequence is also established.
For a nonempty closed convex set and a vector , the orthogonal projection of x onto K, i.e.,
is denoted by . In what follows, we state some well-known properties of the projection operator which will be used in the sequel.
Lemma 2.1 
Let K be a nonempty, closed and convex subset in . Then, for any and , the following statements hold:
Remark 2.1 In fact, (i) in Lemma 2.1 provides also a sufficient and necessary condition for a vector to be the projection of the vector x; i.e., if and only if
Definition 2.1 Let K be a nonempty subset of . A multi-valued mapping is said to be
monotone if and only if
pseudomonotone if and only if, for any , , ,
Now let us recall the definition of a continuous multi-valued mapping F.
Definition 2.2 Assume that is a multi-valued mapping, then
F is said to be upper semicontinuous at if for every open set V containing , there is an open set U containing x such that for all ;
F is said to be lower semicontinuous at if given any sequence converging to x and any , there exists a sequence that converges to y.
F is said to be continuous at if it is both upper semicontinuous and lower semicontinuous at x.
For the simplicity of our description, we list the assumptions needed in the sequel.
Assumption 2.1 Suppose that X is a nonempty closed convex set in . The multi-valued mapping is pseudomonotone and continuous on X with nonempty compact convex values.
3 Main results
For , , we first define the projection residue
It is well known that the projection residue is related intimately to the solution set .
Proposition 3.1 For and , they solve problem (1.1) if and only if
The basic idea of the designed algorithm is as follows. At each step of the algorithm, compute the projection residue at iterate . If it is a zero vector, then stop with being a solution of the GVI; otherwise, find a trial point by a back-tracking search at along the residue , and the new iterate is obtained by projecting onto the intersection of X with two halfspaces respectively associated with and . Repeat this process until the projection residue is a zero vector.
Algorithm 3.1 Choose , , .
Step 1: Given the current iterate , if for some , stop; else take any and compute
where , with being the smallest nonnegative integer m satisfying: such that
Step 2: Let , where
Select and go to Step 1.
Now, we first discuss the feasibility of the stepsize rule (3.1).
Lemma 3.1 If is not a solution of problem (1.1), then there exists the smallest nonnegative integer m satisfying (3.1).
Proof By the definition of and Lemma 2.1, we know that
Since , we get
By this and the fact that F is lower semicontinuous, there exists such that
which implies the conclusion. □
The following lemma shows that the halfspace in Algorithm 3.1 strictly separates and the solution set if is nonempty.
Lemma 3.2 If , the halfspace in Algorithm 3.1 separates the point from the set . Moreover,
Proof By the definition of and Algorithm 3.1, we know
which can be written
Then, by this and (3.1), we get
where is a vector in . So, by the definition of and (3.3), we get .
On the other hand, for any and , we have
Since F is pseudomonotone on X, we get
Let in (3.4), for any , we have
which implies . Moreover, it is easy to see that , . □
The following lemma says that if the solution set is nonempty, then and thus is a nonempty set.
Lemma 3.3 If the solution set , then for all under Assumption 2.1.
Proof From the analysis above, it is sufficient to prove that for all . The proof will be given by induction. Obviously, if ,
Now, suppose that
holds for . Then
For any , by Lemma 2.1 and the fact that
it holds that
Thus . This shows that for all , and the desired result follows. □
For the case that the solution set is empty, we have that is also nonempty from the following lemma, which implies the feasibility of Algorithm 3.1.
Lemma 3.4 Suppose that , then for all under Assumption 2.1.
Before proving Lemma 3.4, we present a fundamental existence result for the GVI problem defined over a compact convex region . For the sake of completeness, we give the proof process.
Lemma 3.5 Let be a nonempty bounded closed convex set and let the multi-valued mapping be lower semicontinuous with nonempty closed convex values. Then the solution set is nonempty.
Proof Since the multi-valued mapping F is lower semicontinuous and has nonempty closed convex values, by Michael’s selection theorem (see, for instance, Theorem 24.1 in ), it admits a continuous selection; that is, there exists a continuous mapping such that for every . Since X is a nonempty bounded closed convex set, the , which consists of finding an such that
has a solution (see Lemma 3.1 in ), i.e., the solution set of the problem is nonempty. It follows from that is nonempty. □
Proof of Lemma 3.4 On the contrary, suppose that there exists such that . Then there exists a positive number M such that
Let and consider the . From Lemma 3.5, we know that the solution set of is nonempty. In order to avoid confusion with the sequence , , and , we denote the three corresponding sequences by , and , respectively, when Algorithm 3.1 is applied to with a starting point . We claim that
the set has at least elements: ;
, , for ;
is not a solution of .
Since , using Lemma 3.3, we know that , so , which contradicts the supposition that . □
From Lemma 3.4, if the solution set of problem (1.1) is empty, then Algorithm 3.1 generates an infinite sequence. More generally, we have the following conclusion.
Theorem 3.1 Suppose that Assumption 2.1 holds. Assume that Algorithm 3.1 generates an infinite sequence . If the solution set is nonempty, then the sequence is bounded and all its cluster points belong to the solution set. Otherwise,
if the solution set is empty.
Proof First, we suppose that the solution set is nonempty. Since
by Lemma 3.3 and the definition of the projection, it holds that
for any . So, is a bounded sequence.
Since , it is obvious that
from the definition of the projection operator. For , since
it holds that from Remark 2.1. Thus, using Lemma 2.1, one has
which can be written as
Thus, the sequence is nondecreasing and bounded, and hence convergent, which implies that
On the other hand, by , we get
and by (3.6) we have
Using the Cauchy-Schwarz inequality and (3.1), we obtain
Since F is continuous with compact values, Proposition 3.11 in  implies that is a bounded set, and so the sequence is bounded. By (3.5) and (3.7), it follows that
For any convergent sequence of , its limit is denoted by , i.e.,
Without loss of generality, suppose that has a limit. Then
For the first case, by the choice of in Algorithm 3.1, we know that
for all .
and F is lower semicontinuous on X, such that
where ξ is a vector in . So, by (3.8) we obtain
Using the similar arguments of (3.2), we have
Combining this and (3.9), we know that and thus is a solution of problem (1.1).
For the second case
it is easy to see that the limit point of is a solution of problem (1.1).
Now, we consider the case that the solution set is empty. Since the inequality
also holds in this case, the sequence is still nondecreasing. Next, we claim that
Otherwise, is a bounded sequence. A similar discussion as above would lead to the conclusion that every cluster point of is a solution of problem (1.1), which contradicts the emptiness of the solution set. □
Theorem 3.2 Under the assumption of Theorem 3.1, if the solution set is nonempty, then the sequence globally converges to a solution such that ; otherwise, . That is, the solution set of problem (1.1) is empty if and only if the generated sequence diverges to infinity.
Proof First, we suppose that the solution set is nonempty. From Theorem 3.1, we know that the sequence is bounded and that every cluster point of is a solution of problem (1.1). Suppose that the subsequence converges to , i.e.,
Let . Since , by Lemma 3.3 we have
for all j. So, by the iterative sequence of Algorithm 3.1, we have
Letting , we have
where the last inequality is due to Lemma 2.1 and the fact that and . So,
Thus the sequence has a unique cluster point , which shows the global convergence of .
For the case that the solution set is empty, the conclusion can be obtained directly from Theorem 3.1. □
Certainly, the proposed extragradient method for the GVI in this paper has a good theoretical property in theory, as the generated sequence not only has an expansion property w.r.t. the starting point, but also the existence of the solution to the problem can be verified through the behavior of the generated sequence. However, the proposed algorithm is not easy to be realized in practice as the termination criterion is not easy to execute. This is an interesting topic for further research.
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The author would like to thank two anonymous reviewers for their insightful comments and constructive suggestions, and Professor Wang Yiju for his careful reading, which helped improve the presentation of the paper. This work was supported by the Natural Science Foundation of China (Grant Nos. 11171180, 11101303).
The author declares that they have no competing interests.
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Chen, H. A new extragradient method for generalized variational inequality in Euclidean space. Fixed Point Theory Appl 2013, 139 (2013). https://doi.org/10.1186/1687-1812-2013-139
- generalized variational inequalities
- extragradient method
- solution existence
- multi-valued mapping