A new extragradient method for generalized variational inequality in Euclidean space
© Chen; licensee Springer. 2013
Received: 2 March 2013
Accepted: 13 May 2013
Published: 29 May 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.
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.
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 
- (i)monotone if and only if
- (ii)pseudomonotone if and only if, for any , , ,
Now let us recall the definition of a continuous multi-valued mapping F.
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
It is well known that the projection residue is related intimately to the solution set .
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 , , .
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).
which implies the conclusion. □
The following lemma shows that the halfspace in Algorithm 3.1 strictly separates and the solution set if is nonempty.
where is a vector in . So, by the definition of and (3.3), we get .
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.
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.
has a solution (see Lemma 3.1 in ), i.e., the solution set of the problem is nonempty. It follows from that is nonempty. □
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.
if the solution set is empty.
for any . So, is a bounded sequence.
for all .
Combining this and (3.9), we know that and thus is a solution of problem (1.1).
it is easy to see that the limit point of is a solution of problem (1.1).
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.
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.
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).
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